Syntaxis Mathematica: OR, A CONSTRVCTION Of the harder Problemes of GEOMETRY: With so much of the CONICKS as is therefore requisite, And other more ordinary and usefull PROPOSITIONS intermixed: And TABLES to several purposes.

The Contents follow in the ensuing leaf.

By THO. GIBSON.

Virgil.

Tempora dispensant usus & tempora cultus.

LONDON, Pridted by R. & W. Leybourn, for Andrew Crook, and are to be sold at his Shop at the Green Dragon in St. Pauls Church-yard. 1655.

CONTENTS.

  • Chap. 1. Explanation of the Symbols. Page 18
    • Addition, Subtraction, Page 22
    • Multiplication, Division. Page 23
  • Chap. 2. Of Aequations. Page 25
    • Rules to finde the Roots in the three sorts of Quadratique Aequations. Page 26
    • Demonstrations thereof. Page 27
    • Reduction of Aequations. Page 29
  • Chap. 3. Resolution of Aequations according to the generall method. Page 32
    • Resolution of Mixed Cubiques. Page 40
  • Chap. 4. Of the Genesis of Aequations, number of Roōts, changeing them by Addition, Subtraction, Multiplication and Division. Page 55
  • Chap. 5. Reduction of Solids to one sum. Page 69
  • Chap. 6. Of Surd Numbers. Page 75
  • Chap. 7. Solution of diverse Problemes Page 82
  • Chap. 8. Of mixture, of Metals and Liquors, di­verse Propositions and Rules. Page 94
  • Chap. 9. Of Mensuration, the superficies and so­lidity of diverse Bodies, and fragments of a Sphere. Page 112
  • Chap. 10. To finde an Ellipsis in any proportiō. Page 124
  • [Page]Chap. 11. Definition generall of a Cone, Definiti­ons and descriptions of the Sections, to draw Tangents, Asymptots, and to finde out the Burning points, Centers, and other things. Page 130
  • Chap. 14. To finde two means, trisect an arch, extract the Root of any Cubicall or Biquadra­ticall Aequation, not proposed in numbers, by a portion of a Parabola. Page 154
  • Chap. 15. Two Geometricall Problemes. Page 168
  • Chap. 16. Of Dialling upon any Plain, with de­monstrations of the hardest Page 180
  • Chap. 17. Of Azimuths, Almicanters, Jewish Houres, &c. Page 197
  • Chap. 18. What Section distinguisheth the light and shadow, at any time on any plane. Page 212
  • A Table of semidiurnall arches, for 32 Lati­tudes. Page 220
  • A Table of the Amplitudes, for 27 Elevations. Page 221
  • A Table of Houre-arches, for 21 Elevations. Page 222
  • The making these Tables, and for right Ascen­sion and Oblique, Ascensionall difference, &c. Page 223
  • Two Tables of compound Interest and Rents from 5 to 10 per Cent. for 31 years. Page 233

TO THE READER

ALthough no Book can be so copious, as where­in every Reader may be furnished with every thing which he looks for, yet there is seldome any thing like a Book which may not afford the Reader some­thing which he lookt not for. The hope that I may do so, may be taken for the rea­son why I writ this. But why I writ thus, that is, Analytically, Valesius, Lib. de Philosoph. Sacr. pag. 8. shall answer for me. Si quis velit resolutivum tenere ordi­nem, qui discentium naturae magis se accom­modat, [Page]& petitionibus minus indiget, quia incipit à postremis de quibus primùm om­nium continget dubitare. That is, If any one would hold the Resolutive order, which more accommodates its selfe to the nature of Learners, and lesse needs Peti­tions, because it begins from the last things, of which principally men happen to doubt.

The method here used is the same as in Master Harriot in some places, that is, in such Aequations as are proposed in num­bers. And as in Des Cartes in some other places, that is, in such Aequations as are Solid, and not in numbers. Not that the Book is taken out of them, much lesse that it proceeds continually with them, but disjunctly, as I thought fit to intermix them among other things which are not in them.

I shall use no arguments to commend the Mathematiques, or prefer them before the Dogmatiques, for this is but to write in praise of Hercules.

Yet this may be said of them, that al­though [Page]some bodily exercises conduce more to health, and some mentall labours more to wealth, yet nothing affords the minde more pleasure, or more profit, with lesse repentance.

And therefore in a dull solitude, or va­cancie of businesse (both which may hap­pen to Gentlemen) these are amiable com­pany, which yield a delightfull and inno­cent expence of leasure.

As for that Question (which is fre­quent) What profit is in these hard Studies? it needs no answer, because it imports the ignorance or idlenesse of the Asker, or ra­ther both.

For first (and which may be reprehen­sive to many Writers, that must not be called Authors, which of late have brought up a fashion to write in a Quaerying way) he declares his ignorance, otherwise he needs not aske that which he knows.

Secondly, If some Meats were recom­mended to a man, with which he is yet unacquainted (otherwise they needed not recommendation) if he should first aske [Page]whether they be good or not, that is, whether they would please his palat and stomach? the Question is absurd, for he cannot know that untill he have tasted and digested. So the idlenesse of the asker may from hence be discovered.

Nor is there any profit to be gained by any Science, except the Science be first gained by industry.

Besides, to think others, who being once entered herein, should delight so much in them, as to make them a study all their life, if there were no profit in them; or if it were so, neverthelesse to recommend them to others, signifies ano­ther bad quality or two, which I forbear to name.

As for the difficulty of these Sciences, I must confesse that the first Aspect of them may seeme uncouth and horrid (Radices doctrinae amarae sunt, fructus tamen dulces) yet there is no reason why he should be deterred hereby, and not rather animated with desire to go as far as another, or else with shame to thinke [Page]there should be so many Books in the world, easie to others and usefull, but to him not understood, and therefore use­lesse.

In this following Treatise, my chiefe care hath been to render it all intelligible to every Reader, and I doubt not but it will prove so to every diligent one.

The Symbols and Characters herein used, are such as have been long accepted in the world, without any innovation or fancie of my own, for although every Writer hath equall liberty herein, to adde or alter, as he sees (or rather thinks) fit, yet in my opinion, we ought not to do this without considerable cause, or a kinde of convenience equipollent to necessity: for without doubt, he that increaseth these, increaseth his Readers burthen, especially if such increase be need­lesse.

I leave the rest to the Reader to cen­sure as he findes cause, and it is in vain to do otherwise, for in these demonstrable [Page]things, the Readers detection of any error (of judgement) will be acceptable even to him that writ it, if he be civilly acquainted with it, but the said Readers detraction cannot here hurt any one but the Reader.

T. G.

ERRATA.

PAge 68. line 2. read eee=ccc. p. 70. l. 20. r. cccc3=729. p. 76. l. 21. r. 3 √ 5=√ 45. p. 96. l. 26. r. Proposition. p. 113 at Section 4. dele Eucl. 12.7. and at Section 5. write Eucl. 12.7. p. 124 & p. 129. in the Diagram the letters x and y are to be supplied upon the Diameter ac, see p. 128. l. 19. p. 134. l. 16. r. Ʋmbilicus. p. 138. l. 8. r. fk, go, tl, &c. p. 151. l. 1. r. z=132. p. 178. l. 28. r. for practise. p. 183. l. 1. r. right angles. p. 185. the same diagram as in p. 183. should be used. p. 187. l. 13. r. Circle wne. & l. 14. r. ns. p. 199. l. 7. dele of. p. 203. l. 1. dele the. p. 205. l. 17. r. the arch fb.

Preface.

ALthough in generall the Mathematiques, and e­specially the Analitiques, are easier in the beginning then proceeding (for the hardest is reserved for last) the Principles, Petitions, and Definitions also, sel­dome meeting any opposi­tion, being (for the most) first sight lessons to all: yet I have thought fit, for some mens sake (who expect it in all Books) to premise some initiary things so easie, and so well known already, as must be received by every one. Neverthelesse, that it may not seeme trifling to the already knowing party, I will not be ample.

Common Sentences.

I. Greater.

  • 1 The whole then the part.
  • 2 Equall to that which is greater.
  • 3 Greater then that which is greater.
  • [Page 2]4 Neither lesse nor equall.
  • 5 Double of the whole.
  • 5 Multiplex of the whole.
  • 6 Greater then that which is equall, Euclid 1.16.
  • 7 Any thing commune or equall added to great­er or to lesse, is greater then it. Euclid 4.17.
  • 8 Where the parts are greater then the parts, the whole is greater then the whole.
  • 9 Of two things, that which hath greater pro­portion to a third.

II. Lesse.

  • 1 The part then the whole
  • 2 Equall to the lesse.
  • 3 Lesse then the lesse.
  • 4 Neither greater nor equall.
  • And so by a way contrary to the former may be formed all that's Lesse.

III. Equall things.

  • 1 That which is commune to two others, is equal to it selfe, as in Euclid 1.5. an angle is commune, and in Euclid 1.7.8.9.10.11.12. a side is commune.
  • 2 Those which are equall to the same thing.
  • 3 Which are equall to equall things.
  • 4 Which are equall to nothing.
  • 5 The whole of equall things, added to equall. Ax. 2.
  • 6 The remain of equall things taken from equall things.
  • [Page 3]7 The whole of equall things added to a com­mune thing. Or contra. Eucl. 1.6.9.10.11.12.
  • 8 The remain of equall things when a commune thing is deducted.
  • 9 Verticall angles.
  • 10 The Rectangles of the Means and Extreams.
  • 11 Things which agree among themselves. Ax. 8. this last is proper to Geometry.
  • 12 That which is not unequall, that is neither greater nor lesse: this is proper to homogene­als, for heterogeneals admit no comparison.
  • 13 The whole to all the parts together.
  • 14 The halves of the whole.
  • 15 Whose halves are equull.
  • 16 Whose parts are equall in Number and Mag­nitude.
  • 17 Whose Doubles are equall.
  • 17 Whose Equimultiplices are equall.
  • 18 If the parts be equall to the parts, the whole is equall to the whole.
  • 19 If nothing else be equall besides the thing supposed, that thing is equall.
  • 20 Which have the same proportion to the same thing.
  • 21 Those to which the same thing hath the same proportion.
  • 22 Of four proportionals, if the first be equall to the third, the second is equal to the fourth.
  • 23 If there be twice three Magnitudes, which ta­ken by two and two are in the same proportion, [Page 4]if (of aequality) the first be equall to the third, the fourth is equall to the sixt. Euclid 5.20.
  • 24 If there be twice three Magnitudes, which taken by two and two, are in the same propor­tion, and the proportion be perturbate, if the first be equall to the third, the fourth is equall to the sixt. Euclid 5.21.

IV. Agréeing things.

1 Are such as are equall, and of the same kind.

V, Vnequall things.

  • 1 Greater or lesse,
  • 2 The whole and the part.
  • 3 The vvhole, when a commune thing is added to unequall things.
  • 4 The whole, when an equall thing is added to unequals. Euclid 1. Ax. 4.
  • 5 The Remain, when a commune or equall thing is taken from unequall things.

VI. Double.

  • 1 The double of the halfe.
  • 2 Two equall things taken together are double to one of them.
  • 3 The double of that which is equall.
  • 4 That which is equall to the double,
  • 5 If the parts be double to the parts, the whole is double to the whole.
  • [Page 5]The 6 proportion of like figures to their sides of like proportion, Euclid 6.19.

VII. Halfe.

  • 1 Is the halfe of that which is double.
  • 2 That which is equall to one of two equals, is the halfe of them together.
  • 3 The halfe of an equall thing.
  • 4 That which is equall to the halfe.
  • 5 The proportion of like sides to the proportion of like figures.

VIII. A thing is.

  • 1 If nothing else which can be proposed is the thing, then this which was proposed is. Or,
  • 2 If any thing else besides that supposed be put, and an impossibility follows, then this which was supposed is that which was sought.
  • 3 If that which is supposed be nothing else, then it is what was required. Or,
  • 4 If this which is supposed being put for any thing else, an impossibility follow, then it is what was required.
  • 5 That which necessarily follows from that which is.
  • 6 Which put for not in being, there follows an impossibility.

IX. Something.

  • 1 Is that which if any thing be added to it, it is [Page 6]more, or if any thing be taken from it, it is less, or to which if nothing be added, it is the same.
  • 2 To which if lesse then nothing be added it is lesse.
  • 3 If lesse then nothing be substracted, it is more.
  • 4 Which multiplyed by something is more.
  • 5 Which multiplyed by nothing is nothing.
  • 6 Which multiplyed by lesse then nothing, is lesse then nothing.
  • 6 Or that which divided by
    • Something, is lesse.
    • Nothing, is nothing.
    • Lesse then nothing, is lesse then nothing.

X. Nothing.

1 Is that which added to, or taken from some­thing, or lesse then nothing, leaves it the same it was: and multiplying or dividing some­thing produceth nothing, but takes the thing quite away.

XI. Lesse then nothing.

  • 1 Is that which added to something makes it less.
  • 2 Which substracted from something makes it more.
  • 3 Which added to lesse then nothing makes it still lesse.
  • 4 Which taken from lesse then nothing makes it more.
  • [Page 7]5 Which multiplyed by something gives lesse then nothing.
  • 6 Which multiplyed by lesse then nothing, pro­duceth something.
  • 7 Which divided by something, makes lesse then nothing.
  • 8 Which divided by lesse then nothing, makes something.

XII. Vnity.

  • 1 Is that to which if unity be added it is doubled.
  • 2 From which if unity be taken, it is nothing.
  • 3 If more then unity be taken, it is lesse then no­thing.
  • 4 If lesse then unity be taken, it is lesse then unity.
  • 5 If lesse then nothing be taken, it is more then unity.
  • 6 Is that which cannot be multiplyed or divided by unity without remaining the same.
  • 7 Is the difference of the two greater sides of a rectiline rectangle triangle: or may be so by reduction of the sides to lesser numbers. vide Corol: ad Cap. 7 Prob. 3.

Propositions of EUCLIDE, fit to be known to the ANALIST.

  • In the first Book Prop. 6, 13, 14, 15, 18, 19, 28, 32, 43, 47, 48. In all 11.
  • In the second book all but the eleventh, and last, in all 12.
  • In the third, Prop. 14, 20, 22, 31, 32, 35, 36, in all 7.
  • In the fift Book, Prop. 15, 16, 17, 19, 24, 25, in all 6.
  • In the sixt Book, Prop. 2, 3, 4, 6, 7, 8, 13, 14, 16, 19, 24, 31, in all 12.

Many more propositions out of these and the remaining Books might be usefull: But these 48 last reckoned are such as (in my judgement) ought chiefly to be read, and remembred, for as­sisting to attain and resolve Aequations.

Now whereas it is said in the ensuing Chapter, that vowels are put for things unknown, or sought; and consonants ever for known things, it is to be noted that in a Scheme which imployeth almost all the Alphabet these are promiscuous.

But in abbreviation or demonstration, where­soever one single letter is put (or supposed to be) equal to any line or number, although the same letters which before designed the Diagram be, [Page 9]againe used herein, yet in a different acception; For whereas in the Diagram they signified points. now they stand for lines or things; And ever­more the consonants signifie things given or known before; and the vowels (although all pre­sent) are supposed equal to things which are not yet known, but about to be found.

Onely the vowel o is seldome used in this sense, because it is usurped in another, that is to signifie nothing. As a − b=o. signifies that a want b is equal to nothing: or that a is equal to b where the vowel o stands for a cipher, that is nothing. On the other side the Greek vowal y is usually put for any unknown quantity.

Definitions.

Definition I.

The unknown Quantity of any aequation is called generally Potestas; or a Power, Quantity, or Terme.

Definition II.

A Rectangle is in numbers the Product of two numbers multiplying one another.

In Geometry it is the Area, space, or content of a right angled quadrangular figure, made also by multiplication of two lines, which are called the sides; of which one is the measure of the breadth the other of the length.

Definition III.

A Rectangled Parallelepipedon is the product of a Rectangle multiplied by a right line or number: And if that line or number and the length and breadth of the Rectangle be severally equal it is a Cube, or Die.

Definition IV.

A Prisme is a Solid contained within five su­perficies of which three are Quadrangular, and the other two being opposite, are triangular: Or it is like the top of an ordinary English house cut off by a Plane passing through or parallel to the Eaves.

The rest of this kinde I shall not desine here but referre the Reader to Euclid.

The names of the Potestates or Powers.

1 The first Power is called a Side, or Root: The later word Root is most used here; and it is signified thus, a.

2 The second Power is called a Square, and is thus written, aa.

3 The third is called a Cube, and is thus writ­ten, aaa. or sometimes for brevity, a3

4 The fourth, a Biquadrat or squared square, anciently a Zenzi zenzick, figured thus zz now thus, aaaa, or for brevity, a4.

5 The fift Power is called a Sursolid, and is written thus, aaaaa; or briefly thus a5.

6 The six, a squared Cube, or zenzicube, written thus, aaaaaa or a6.

7 The seventh, a second Sursolid, and is written aaaaaaa, or more short a7.

8 The eight is called a squared square squared, or zenzi zenzi zenzi zenzick, and is written aaaaaaaa; or thus a8, &c.

Consectary I.

Hence it is manifest that these powers uninter­rupted, are in continuall proportion, the propor­tion of them being as a, to unity: or the converse.

Consectary II.

It is also here plaine, that every Power hath so many dimensions, as the letters, with which it is written. For a4 being written with foure let­ters, if one letter stand for one dimension, that is length or breadth, the other three arise by three several Multiplications, and every Multiplication addes a dimension, in this sense.

A Table of the Powers of the Digits.
123456789
111111111
248163264128256512
3927812437292187656119683
41664256102440961638465536262144
525125625312515625781253906251953125
6362161296777646656279936167961610077696
749343240116807117649823543576480140353607
864512409632768262144209715216777216134217728
981729656159049531441478296943046721387420489

In the former Table, the Digits at the top 2, 3, 4, &c. Shew the Columnes of the second, third, fourth, &c. Powers.

The Digits, at the left side, shew the severall Roots or first Powers, and their proportion to Unity. All the rest is evident.

And now because towards the end of this litle Treatise, I shall happen to speak once ortwice of Arithmetical Calculation; The Reader may here­by understand, that such Calculations are usually (and most easily) performed by numbers assumed in Arithmetical proportion, called Logarithmes; of which I intend to say nothing, supposing any Reader conversant about such things wherein I use them, cannot be ignorant of them and their use.

Such as be, should read that first, of which they need no better (nor other) instructions then such as they may have in Mr. Norwoods doctrine of Triangles; which is a Book not very deare.

But to such as have not that, these follow­ing directions may be of some use.

1 In every Spherical triangle which hath one right angle, or one side a quadrant, all the other five parts (for every triangle hath six in all, that is three sides, and three angles) are called Circular Parts.

2 Of these Circular parts, if any two be given, the rest, that is, any one of the rest, may be found ta one operation.

3 For those two are either adjacent, or remote, or opposite: if adjacent, & the part required be also adjacent (or touching) to one or either of them, then that one so touched on the one side by a part given, and on the other side by a part required, may fitly be called

The Middle Part

And it is a demonstrated truth, that,

As the tangent of the known part adjacent, is to the right sine of the middle part;

So is the Radius or Semidiameter, to the tangent of the unknown, or required part; being also adjacent to the middle part, as before.

And therefore. if instead of the naturall sines and tangents, the Logarithmes be used, they being in Arithmeticall proportion, the summe of the two middle termes is equal to the summe of the two extreames; And so here, the fine of the mid­dle part plus Radius is equal to the tangent of the of the adjacent part known plus the tangent of the part required.

I hope the word plus needs no interpretation.

Note 1. It is notwithstanding to be ever re­membred that every of the five circular parts must be considered two wayes; that is whether it be contiguous to the right angle, or quadrant; if so then all before is right and unalterable.

But if not so, that is, if some other part lie betwixt them, then all that hath been said of their sines and [Page 15]tangents, which were then supposed contiguous, must be performed by the sines complements and tangents of the complements of such of the parts respectively as are remote from the right angle or quadrant.

4 If now the two parts given be remote, and the part required lie betwixt them, then make the part required the middle part, and it may be found as easily as in the former case.

5 If the two known parts be contiguous, and the part required adjacent to neither of them, but opposite to one of them, then (working stil by Logarithmes) make the part required the middle part, and then the sine of the middle part plus Radius, is equal to the sines complements of the opposite parts given, if therefore from those two sines complements added, be taken Radius, the rest is the sine of the thing required.

Note 2. It is further to be noted that the sines complements of those parts which by the former note are accompted complements, are the sines of the things themselves.

Example.

In the triangle zps let p be the Pole, s the Sun and z the Zenith zps, the hour from noon in Winter, or the hour from Midnight in Summer; pzs, the Azimuth from the North; and zsp, the angle of position: and sz be 90 Deg. as at Sun-rise.

[mathematical diagram]

Then first, sine z plus Radius, is equal to tan­gent complement zp plus tang. s.

Secondly, sine compl. p plus Radius is equal to tang. compl. pz plus tang. compl. ps. per Not. 1.

Lastly, (by the 5 direction) sine compl. sp plus Radius is equal to the sine of pz plus the sine complement of z; because pz is accompted a compl. and z not so. Note 1.

And if instead of a quadrant (as zs) there were a right angle, at p, or at z, or at s, the afore­going directions serve.

If there be neither right angle nor quadrant, there must be two operations to do this, having first supposed a circle to passe from one of the angles to cut the opposite side (produced when need is) at right angles.

This perpendicular circle shall fall sometimes within the triangle, sometimes without.

Within, when the other angles are both obtuse: Or both acute, as at the hours between six and noon are the angles at p and s.

In letting fall the perpendicular, Mr. Norwoods advice is to do it.

  • 1 From the end of a side given, being adjacent to an angle given, let it fall opposite to that angle.
  • 2 And touching in some part the side required.
  • 3 And opposite (if it may be) to the angle requi­red.

One of the most difficult cases in oblique an­gled sphericall triangles is this.

[mathematical diagram]

In the triangle pzs. Let there be given,

comp. Elevation pz=38.28′

comp. Declinati. ps=66.29′

comp. Azimuth pze=70.0′

To find the complement of the Suns Altitude zs=75.45′ having produced sz and from p, let fall pe, perpendicular to it: then,

First, making pze (the middle part) s.c. pze. plus Radius 195340517

From which taking the tangent of the Elevation, t.c. zp. 100999135

Remains tang. of ze=15.12′t. ze 94343382

Secondly, to sine compl. ze 99845347

Adde sine of Declination, s.c. ps 96009901

The summe of them is 195855248

From which taking sine Eleva. s.c. pz 98937412

Remains (the sine of 29.27′) s.c. 60.33′ 96917696

To whose comple. 60.33′ adding ze=15.12′, the sum is 75.45′ the thing required.

Whereas, if the Azimuth it selfe pzs were 70d, then ze being taken from 60.33′, the rest 45.21′ is the Suns Altitude.

CHAP. I. An explanation of the Characters and Symbols used in this Work.

FIrst, One single letter of the Alphabet is usu­ally put for any quantity whatsoever, as well Line as Number, whether known or un­known.

But for the most part, where any quantity is [Page 19]sought, there a or some other Vowel is put for it; and the other Quantities known, are signified by Consonants.

These letters are multiplyed one into another by joyning them together without any pricke or comma between, nor doth it import at all which is first or last written: for bcd, bdc, and cbd; are all one.

So a multiplyed by a produceth aa.

And a multiplyed by b, produceth ab.

And ab multiplyed by c, produceth abc.

The like of all others whatsoever, except Fra­ctionall quantities; as, ab+fg/d and bc−fb+rc/b+c

If the first of these were to be multiplyed by d, it is done by taking away the d under the line, and the product is ab + fg.

If the second were to be multiplyed by b + c, it is done by taking away the Denominator b + c, and the product will be bc − fb + rc.

For all Fractions aswell in Plaine as in Figu­rate Arithmetick, are nothing else but Quotients of one number divided by another; and are mul­tiplied again by taking away their Divisor, and line of Separation.

Division is done in Figurative Arithmetique, most commonly by applying some line of sepa­ration between the Dividend and the Divisor. [Page 20]So a / b is a divided by b, and abc / f signifies that abc is divided by f.

But yet if the letter f had been found in the Dividend, the application of this line had not been necessary, for it might have been better done by taking away that letter out of the Dividend.

So afc divided by f quotient is ac and ffcc divided

  • by fc quotient is fc
  • by ff quotient is cc
  • by cc quotient is ff
  • by ffc quotient is c
  • by fcc quotient is f
  • by f quotient is fcc
  • by c quotient is ffc

And the like may easily be understood of all the rest.

Symbols of

  • Majority >
  • Minority <
  • Aequality =
  • Addition +
  • Substraction −
  • Root of a quantity √

  • Proportionality continued ′ ″ ‴ ''''
  • Proportionality disjunct ′ ″ ′ ″

[Page 21]

  • So b > c signifies b greater then c
  • b < c....... b lesse then c
  • b=c .......b equall to c
  • b + c .......c added to b
  • b − c.......c taken from b

√ 72 signifies the square root of 72, &c.

And b′ c″ d‴ f'''' signifieth that as b is to c, so is c to d, and so d to f.

Likewise b′ c″ f′ g″ signifies that as b is to c, so is f to g.

These things before expressed are almost gene­rally received: and used not only for brevity in writing, but perspicuity in proving, as will be seen hereafter.

Note that wheresoever − is not expressed, there + is understood, though it be not expressed.

Also in Trigonometrie. I use, s. pzs, for the sine of an angle pzs, and s.c. zp for the sine of the complement of a side pz to 90. Also, t. zp and t.c. spz for tangent of zp and tangent of the com­plement of spz, &c. Also for Radius I use r.

If the signe of Addition, namely + stand be­fore any quantity, it shewes that quantity, to be more than nothing; that is something.

But if the signe of Substraction, to wit − stand before any quantity; it shews that quantity to be lesse than nothing: or a want of the said quantity.

Soe + 4, signifies four of any thing: but − 4, signifies a want of four, or four lesse than nothing.

In ADDITION.

The addition of a want of any thing, is all one with the subtraction of the same thing.

  • So if to + 12 you adde − 5 it makes + 7
  • And if to + 12 you adde − 16 it makes − 4
  • But if to + 12 you adde + 16 it makes + 28

In SƲBTRACTION.

The subtraction of − is all one with ad­ding +

So if from + 12 you subtract − 5 remain. is + 17.

And if from + 12 you subtract − 16 remain. is + 28.

Addition of + to + and Subtraction of − from − is all one with Common Addition and Subtraction. And generally for both.

In Addition, add the quantities together with the same signe.

In Subtraction, adde them also, but all the signes of that which is to be subtracted from the other, must be changed.

Example.

If to + 6 − 2 + 3, be added + 5 + 1 − 3 the sum is + 6 − 2 + 3 + 5 + 1 − 3=10.

But if from + 6 − 2 + 3, be subtracted + 5 + 1 − 3, the remain is + 6 − 2 + 3 − 5 − 1 + 3=4. This Rule is generall, and generally known.

In MƲLTIPLICATION.

  • + multiplied by + ever produceth +
  • + multiplied by − ever produceth −
  • − multiplied by − ever produceth +

More Varieties there are not.

The quantities that are accompanied with these signes of + & − (in both Multiplyers be­ing placed one under another, as in cōmon mul­tiplication) must be multiplyed every one below into very one above, and then this work is done.

So if, + bb + b − c, be multiplied by + f − g place them thus.

+bb+b−c

+F−g

Saying, + f multiplyed into + bb gives +bbf

And + f into + b gives +fb

And + f into − c gives −fc

And − g into + bb gives −bbg

And − g into + b gives −bg

Lastly, − g into − c gives +cg

Which added together is, bbf−bbg+fb−fc−bg+cg

Which is the true product.

In DIVISION.

If the line of separation doe not serve the turne that is, if any desire, (and it may be done) other­wise, [Page 24]it must then be by seeking what quantity may be multiplied by the Divisor to produce the Dividend.

So if bb + bc − bf − bg − cg + fg, were to be divided by, b + c − f, triall must be made what mixt quantity multiplying b + c − f will produce bb + bc − bf − bg − cg + fg,

In which there is this of Compendium, that seeing the Dividend consists of six members, and the Divisor of three, the quotient must be of two; that is a Binomial only.

And because the quantity g is found in the Di­vidend, and not in the Divisor, it must be in the quotient.

The said quotient therefore must be one of these, b + g, b − g, or g − b.

It cannot be the first, for + g, into − f would have produced − fg: but in the Dividend it is + fg, therefore it must be − g.

By the same reason it cannot be the last, as also because − b, into + b produceth − bb, but it is + bb, in the Dividend.

The quotient sought, must therefore be b − g.

Some further Rule for saving labour herein might be given: but every one likes that best which he finds out himself. Nor is it my purpose to write a Booke of Algebra; but to premise so much of the Rudiments thereof, as the Reader may stand in need of in perusing the following Treatise.

Wherein because Division is seldome needed; If I have a litle exceeded already, and shall a litle more in treating (but very briefly) of resolving some few Rooted Aequations, I shall aske the Readers pardon for both together.

CHAP. II. Of Aequations.

AN Aequation is when one or more special quantities, are equal to one or more other speciall quantities, and written with the signe of equality betwixt them; As aa=bb.

This is called a simple Square Aequation. And bb, being a known square, the square root thereof being extracted, is equal to a. And that is the thing required.

But, aa + ba=cc, and aa − ba=dd, and lastly − aa + ba=ff; are all of them of that kind, which are called mixed aequations, because a (the thing required) is multiplied not only into it self, but into another known quantity, namely into b.

And note that this known quantity in all mix­ed aequations is called the Coefficient.

Note also that the three sorts of mixed aequati­ons above expressed are all that can happen in quadratiques: And by some one of these, all Pro­blemes [Page 26]whatsoever transcending plaine Division, and not reaching Solids, are to be resolved by finding the Root a, according to these Old Rules.

In the First, aa + ba=cc.

Unto the quantity given namely cc, adde the Square of half the Coefficient, it makes +cc+bb/4 Which if it be in lines, may be reduced into one Square, and from the side of that Square, take half the Coefficient, and the remainer shall be a. Which was the thing desired.

In the second, aa − ba=dd.

Unto dd adde bb / 4 as in the former, and the sum thereof being always in numbers a Square, or in lines to be reduced to a Square as aforesaid: Ʋnto the Root or side of that Square, adde halfe the Coefficient, the Summe thereof shall be a, or the Root of the Aequation sought for.

In the last, − aa + ba=ff.

From the square of half the Coefficient, which is bb / 4 take the quantity given, that is ff there will remain bb / 4 − ff, which being put into one Square, and the side thereof known: If that [Page 27]side be either added to halfe the Coefficient, or subtracted therefrom, either the summe of that addition, or the remain of the subtraction, is equall to a.

For all Quadratique Aequations of this kinde (where aa the greatest unknown power is want­ing) have two Roots, which being both together ever equall to the Coefficient, if upon the Coef­ficient, as a Diameter, a Semicircle be described, and the side of ff (the quantity given) be appli­ed therein, perpendicular to the Diameter b, the two segments of b are the two Roots sought.

For in the Aequati­on − aa + ba=ff, it is by the 14th. of the 6th. of Euclid, as fol­loweth.

[mathematical diagram]

b − a′ f″ f′ a″.

Wherefore either Segment may be a, and the o­ther will be b − a, and f a mean betwixt them.

Likewise in the two former Aequa­tions, the worke may be effected Geometrically, & proved also by this present Scheme.

[mathematical diagram]

In which, as the figure intimates, the Perpen­dicular [Page 28]represents the side of cc in the first Ae­quation, and the side of dd in the second.

[mathematical diagram]

Draw a line bg from the center b to the top of the perpendicular, the center b being first ta­ken in the middle of the line b, to wit, of the Coefficient, for so it is usually called.

And first, let the pricked line be put for a, Therefore, by the before recited Proposition, ‘It is, a + b′ c″ c′ a″. Eucl. 6.14.’ And if (as the Rule prescribeth) to the square of halfe b you adde the square of c, the totall shall be the square of the line bg: By the 47th. of the first of Euclide,

If therefore from the line bg, or (which is all one) bf, you take the line br, which is half the Coefficient b (for the whole Coefficient b, is the same with sr) the rest, namely the prick­ed line rf, shall be equall to a. For, rf=ns=a.

In like sort concerning the second Aequation, [Page 29] aa − ba=dd: If according to the Rule, you adde the squares dd, and bb/4 together, it gives the square of the line bg, to the Root of which, to wit, bg, if you adde halfe the Coefficient, to wit, br, or bs, the sum shall be fs or nr, either equall to a. And then, as nr, that is a, is to d, so is d to rf, or a − b, as it ought to be.

I intend anon to write something of Extra­ction of Roots, according to the generall Method of resolving all manner of Aequations of Powers, how high or composed soever. I do not mean to exemplifie them any further then the Cubique order. There are Authors enough, whom they that desire the full of that Artifice, may at their own leasure in Books confult.

And now because I shall herein make some use of Aequations, though not higher then Cu­biques, or at the most the Biquadratique order: I think fit to admonish the Reader, that in put­ting a alwayes for the thing sought, and work­ing therewith, as if it were known, quite through as the question requires, he shall at last come to an Aequation, but it may be such a one as wants reducing: of which a little.

REDƲCTION

Of Aequations is done by adding all that's ne­cessary, or subtracting all that's not necessary on [Page 30]both sides the signe of aequality: Or by subtract­ing contradictories if they happen on one and the same side, untill the Aequation, purged of all unnecessary members, remain with all that's ab­solutely known on one side, equall to (as little as may be) all that's unknown on the other side.

One example of this shall serve as followeth:

In the Aequation aa − ba + dc + ba=gg + ba − dc.

To reduce this, you must remember what hath been said before; that the taking away a Want of any thing, is all one with the addition of that thing.

Therefore seeing there is on the first side a Want of ba expressed by − ba, if you take away that − ba, you thereby adde ba on that side.

Wherefore that it may still be an Aequation, you must adde ba on the other side also.

Then it will be, aa + dc + ba=gg + 2ba − dc

Again, subtract ba on each side, then it is, aa + dc=gg + ba − dc

Once more, subtract ba on each side, that you may bring it to that side where aa stands.

Then it is, aa − ba + dc=gg − dc.

Lastly (that the Consonants, or known things may come all on one side) subtract dc on each.

Then it will be, aa − ba=gg − 2dc.

Take the Rectangle 2dc out of the Square gg, and let the rest be a Square, namely ff.

Then it is reduced, aa − ba=ff.

Having gone a little about, only for exercise of them that are quite unskilfull herein, now they shall see this Reduction might have been quickly done another way, that is, seeing in the Aequation aa − ba + dc + ba=gg + ba − dc There are in the first part Contradictories, to wit, − ba and + ba, they (destroying one ano­ther) might be taken away both at once,

So it will be, aa + dc=gg + ba − dc. Then if you subtract dc and ba on both sides, it will be reduced to aa − ba=gg − 2dc, as it was before. And gg − 2dc being put in­to one Square ff, the Aequation aa − ba=ff, may be resolved as the aequation aa − ba=dd [Page 32]was, by the second Rule for plain Aequations, a little before expressed.

And as here the Reduction was made by Ad­dition and Subtraction only, so sometimes it is made by Multiplication, sometimes by Division; in both or either of which, this is generall: that Whatsoever is done to any one Member, must be done to every Member quite through the Aequa­tion.

CHAP. III. Of the resolution of Aequations, according to the generall Method composed by Mr. Tho. Harriot.

ALthough (having before shewed Rules for all sorts of mixed squares) it may seeme preposterously done hereafter to speake of Simple Squares; yet forasmuch as I pretend not much to Method or Order, and because the gene­rall Method of Mr. Harriot begins with Squares, I will do so, but only with one Example. That is, Let there be an Aequation of aa=ff.

Or let it be exhibited in numbers, aa=69169

First, take notice that all Squares whether simple or mixed in Numbers, are to be marked with points, the first alwayes over the place of Unity or unities, and so successively every Bina­ri or second figure.

Cubes with every ternary figure.

Biquadratiques with every quaternary.

Sursolids, every quinqnenary, and so forwards.

This square number so pointed is 6̇91̇69̇

In which because there are three points, there are three figures in the Root.

So that a being a single letter cannot fitly re­present that Root, but some trinomiall, as is b + c + d should be put equall to a, and the square thereof should be equall to aa, or 69169.

But because it may be done aswell by adding the Gnomons, that is repetition of the second working, (as they are commonly called) so often as the points are more then two; a binomiall will serve (with lesse trouble) to do the same.

Let that Binomiall be b + c.

And put b + c=a.

Their Squares shall be therefore equall.

That is, bb + 2bc + cc=aa.

That is, bb + 2bc + cc=69169.

The Resolution.

The homogeneall number given 6̇91̇69̇

First single Root b=2 and bb=4.0000

Which 4.0000 being subtracted from the number given 69169, then there Remains of the Number given 291̇69̇

Remains of the Number given, 291̇69̇

Root decuplate b=20

Divisor 2. b 40.00

The second single root c=6

2.bc240.00
cc36.00
 276.00

Subtract 276.00

Remains of the Number given 15669̇

The Root increased b=26

Root increased and decuplate b=260

Divisor is 2 b=520

The third single Root c=3

2bc1560
cc0009
Totall1569

Subtract 1569

Remains of the Number given 0000

The Root increased 263, is therefore the true Root, as may be proved by recomposition, or multiplying 263 by 263, for the Product will be 69169, which was the number given.

The Ciphers which are put after in the Divi­sors and Subtracts, are only to fill up the num­ber of places, by which the number given, or ra­ther the remaining Points would else exceed.

For the like purpose is used the decuplation of the Roots, as only to supply a place untill another fi­gure succeed in place of the Cipher.

And in nothing else doth this work differ from the ordinary Extraction of the Square Root, commonly taught and known.

The reason of it depends upon the 4th. Prop. of the second Book of Euclide, where it is de­monstrated, that If a right line be divided by chance into two parts, the Square made of the whole, is equall to the Squares of the parts, and to the Rectangle made of the parts twice.

So it is here as followeth.

The Square of the greater part, that is, of 260 bb=67600
The Square of the lesser part, that is, of 3. cc=00009
The Rectangle of the parts, that is, 260 into 3 twice. 2bc=01560
Equall to the whole Square. 69169

Nor do these letters represent so naturally the things themselves in a divided Superficies only, but as properly and clearly the parts of Solid Bo­dies, [Page 36]of which, two or three Examples for satis­faction.

In which I admonish the Reader, to be intent to the severall pointings of the quantities accord­ing to their due order, as is before expressed, and also to the placing of the Divisors and Substracts by Ciphers, as before also is intimated: for this to the Ingenious is enough, & a long verbosity to others will scarce be so.

Of Cubicall Aequations.

  • Let there be a Cube aaa=fff
  • Or proposed in Numbers aaa=41781923
  • Put (as before) b + c=a
  • Then their Cubes also shall be equall.
  • That is bbb + 3bbc + 3bcc + ccc=41781923

The Resolution.

The Homogeneall Number given 41̇781̇923̇

The first single Cubique Root b=3

And bbb=27.000000

Subtract 27.00000

Remains of the Number given 14781̇923̇

Remains of the Number given 14781̇923̇

The first Root decuplated b=30

[...]

Divisor 2790.000

Second single Root c=4

[...]

Substract 12304.000

Remains of the Number given 02477923̇

The Root increased b=34

Root increased decuplate b=340

[...]

The third single Root c=7

[...]

Subtract 2477923

Remains lastly of the number given 000

The Root increased b + c=347 [Page 38]Which is the true Root of the Cube 41781923, as may be proved by recomposition, that is, by Multiplying 347 by 347, and the Product again by 347, the last Product shall be equall to the Cube which was given to be resolved.

And as above in the Square the Canon of the Resolutions was the letters bb + 2 bc + cc, being the true Square of b + c. And those let­ters did answer exactly to the parts of the Square divided alike in both the Dimensions: So here also the Canon of Resolution, or the letters bbb + 3bbc + 3bcc + ccc, do exactly answer to the Parts or Members of a Cube, divi­ded into two parts, alike in all the three Dimen­sions, as any one may prove upon a Cube made of some slender matter, and cut through all three wayes, for he shall finde the whole Cube (sup­posed equal to 41781923 as before) justly made up of the two Cubes of the two segments, that is, bbb and ccc, and three Parallelepipedons, whose length and breadth are equall to b, and their thicknesse to c, those three are the 3bbc. And lastly, three other Parallelepipedons, whose length and breadth are equal to c, and their thick­nesse to b, such are the 3bcc.

See the following Schematisme.

The Cuhe of the greater Segment which is 340, bbb39304000
The three greater Parallelepipe­dons, 3bbc·2427600
The three lesser Parallelepipedons, or 3bcc···46980
The Cube of the lesser Segment, which is 7, ccc·····343
The whole Cube given41781923

Note, That the greater Segment is the aggre­gate of all the single Roots except the last, being duly valued by a Cipher, as here it is 340, but the lesser segment is the last single Root only, as here 7,

I have done this to let the Reader see, that he may be sure let the quantity to be resolved be great or little whatsoever, if he be carefull to make his Canon right, the letters themselves will di­rect him how to frame his Divisors and Subtracts in order to the finall resolution, especially in these unmixed Quantities, where the points li­mit how far the subtract shall advance at every operation, beginning first at the point next the left hand, not further, and to the second point only at the second work, and not otherwise in all that follow.

And in Mixed Aequations, if they be made up of Cube with addition of certain Squares, or certain Roots, or both Squares and Roots, or by Subtraction of the same, the Canon of the Reso­lution must ever be made by multiplying the as­sumed Root b + c in the place of the quesititi­ous Root a, quite through the Aequation in all the degrees thereof, for so shall arise all the seve­rall parcels of which the severall Subtracts are or­derly to be made.

In a Cubique Aequation, if all the quantities be present, there is no need to point any but the Cubiques and Roots: yet I have here distinguish­ed the places of the Squares also with litle Crosses obliquely; which labour, when the Workman is intent upon his businesse, may well enough be spared.

Of the resolution of Mixed Cubiques.

  • Let the Aequation aaa + daa − ffa=ggg be proposed in Numbers.
  • As let it be aaa + 32 aa − 75 a=29282970
  • Therefore d=32 and ff=75
  • And ggg=29282970
  • Put b + c=a

And make the Canon of Resolution by substitu­ting b + c in the place of a quite through the severall quantities aaa + daa − ffa. The Canon rightly made will be + bbb

  • + 3bbc + dbb − ffb
  • + 3bcc + 2dbc − ffc
  • + .ccc + dcc

These severall parcels of the Canon, being rightly subtracted from the homogeneall Number 29282970, the Number shall be thereby resol­ved, and the Root a found.

☞ Note first, That all the parcels in the Ca­non, which have not the secondary Root c in them, as + bbb + dbb and − ffb, are to be subtracted at the first Operation, the other remaining parcels to be all subtracted as often as there shall be points left above.

The Resolution.

The homogeneal Number given [...]

The first single Root b=2

[...]

Subtract 92650.00

Remains of the Number given [...]

Remains of the Number given [...]

The first Root decuplate b=20

[...]

Divisor 1390450

The second single Root c=9

[...]

Subtract 17793450

Remains of the Number given [...]

The Root increased b=29

Root increased decuplate b=290

Remains of the Number given [...]

[...]

Divisor 271655

Third single Root c=8

[...]

Subtract lastly 2224520

Remains of the Number given 000

Whereby it appears that the whole Root 298 is the true Root whereby this Aequation is ex­plicable, as may be proved also by recomposition.

  • For bbb=24389000
  • 3bbc=.2018400
  • 3bcc=...55680
  • ccc=.....512
  • dbb=.2691200
  • 2dbc=..148480
  • dcc=....2048
  • In all=29305320

Frō which subtract ffb + ffc= ...22350

Remains 29282970 [Page 44]Which was the whole Homogeneall Number given.

NOTE.

Whereas in composing the Divisor all the gradual quantities are used, as in the former ex­ample, 3b and d, aswell as 3bb and 2db, it is to be noted that in practice, those smaller par­ticles 3b, &c. May be omitted; the other with­out them ministring light enough for choosing the Secondary Roots.

Having now instanced in an Example where all the powers were present, in these one or two that follow, to make the work shorter, I shall leave out one or other of them.

In the Aequation aaa + ffa=ggg. Propounded in Numbers aaa + 320406a=8348132, It sometimes happens that the Co­efficient abounds with more binarie figures then the homogeneal doth with ternaries, in such a case that there may be rooom made to begin the Extraction. The Coefficient must be devolved to the next point further to the right hand, or to the second, third, fourth, or further, if need re­quire, and there the work is to begin. The Co­efficient is alwayes the known quantity which multiplies any of the unknown inferior quanti­ties.

Example of Devolution.

aaa + 320406. a=8348132

Put b + c=a

The Canon will be

  • bbb + 3bbc
  • + 3bcc + ccc
  • + ffb + ffc

= 8348132

Resolution.

The Homogeneall number given [...]

The first single Root b=2

[...]

Subtract 641612.0

Remains of the Number given [...]

The first Root decuplate b=20

[...]

Divisor 321606

The second single Root c=6

Remains of the number given [...]

[...]

Subtract 1932012

Remains of the number given 000

Wherefore the whole Root is equall to the Root increased, 26, as may be proved in manner as beforesaid.

It sometimes happens also in the Aequation

  • aaa − ffa=ggg Put into Numbers.
  • As aaa − 105000. a=203125.

That the Coefficient abounds with more binarie figures then the homogeneall with ternaries: Wherefore that there may be place for the Reso­lution, put before the homogeneall, toward the left hand, so many Ciphers as will afford that to receive as many Cubicall points, as the Coeffici­ent doth Quadraticall: And at the first empty point, as it were by anticipation, begin the Reso­lution. In which there is this of Compendium, that the first Square Root extracted out of the Coefficient, is either equall to the first single [Page]Root of the homogeneall sought, or lesse then it by Unity.

But if the Aequation had but two Dimensions, As aa − 254a=65024, then the first figure of the Coefficient, namely 2, is the first Root.

Example of Anticipation.

The homogeneall Number given [...]

The Canon is

  • b + c=a
  • bbb + 3bbc + 3bcc + ccc − ffb − ffc

The Resolution.

The first single Root b=3

[...]

Subtract the differēce which is −45000.00

Remains of the Number given [...]

The first Root decuplate b=30

[...]

Divisor 165000.0

[...]

Subtract [...]

Remains of the Number given [...]

Remains of the Number given +1035125

The Root increased b=32

Root increased decuplate b=320

Divisor [...]

The third single Root c=5

Subtract +366800.0

Remains of the number given [...]

Which sheweth that the Root increased, b + c=325, is the true Root of the Aequation, And it may be proved by recomposition as for­merly.

In the Aequation − aaa + ffa=ggg, Which is explicable by two Roots, as shall be shewed in the next Chapter, Section 5, to finde them both. Put the Aequation into Numbers.

As − aaa + 52416a=1244160

Therefore ff=52416 & 1244160=ggg

Putb + c=a

Therefore [...] = 1244160

Extraction of the greater Root.

The Homogeneall Number given 1244160

The first single Root b=2

Subtract [...]

Remains of the Number given [...]

The first Root decuplate b=20

Divisor [...]

The second single Root c=1

Remains of the number given −1239040

Subtract −736840

Remains of the number given [...]

The Root increased and decupled b=210

Divisor −79884

The third single Root c=6

Subtract −502200

Remains of the number given 000

Root increased b + c=216, which is the true Root sought.

2. Eduction of the lesser Root by Devolution.

The homogeneall Number given [...]

[...]

Subtract +1040320

Remains of the Number given [...]

The Root increased and decupled b=20

Divisor .51216

The second single Root c=4

[...]

Subtract +203840

Remains of the Number given 000

The Root increased b + c=24

Wherefore 24 is the true Root sought, as may be proved by recomposition, as hath been shew­ed before.

So this Aequation is explicable by two Roots, that is, 216, and 24.

VIETA, Lib. de Recognitione aeqnatio­num, Cap. 18. Prop. 2. saith, That in the Ae­quation − aaa + ffa=ggg, the Coeffici­ent ff is composed of three proportional Squares, and the Homogeneall ggg is made by Multipli­cation of the aggregate of the two first, or the two last, (for all is one) into the side of the o­ther, and the Root a may be the side either of the first or third. This (or the same in substance) saith that Noble Author, And it is evident, for make

cc′ + dd″ + hh′″=ff

And pnt c=a

Therfore ccc + ddc + hhc − ccc=ddc + hhc

Or put h=a

It is hhh + ddh + cch − hhh=ddh + cch

Both which are manifest

COMPENDIUM 1.

Hence it may be shewed, that either of the quaesititious Roots, as a, being found and called c, the other Root e may be found by a Quadra­tique [Page 53]Aequation only. For supposing

ee + ce=ff − cc, Then

It is ee + ce + cc=ff.

And cc′ ce″ ee′″ Euclide 6.23. But by construction cc′ dd″ hh′″. And cc + dd + hh=ff. So then hh=ee and h=e.

But it was shewed before that h might be a Root of this Aequation − aaa + ffa=ggg And therefore e also is a Root of the same, and the Compendium is proved.

Example also in Numbers.

In the last Aequation aa=cc=46656

And ff=52416

From which take cc=46656

Remains ff − cc=5760

05760

But [...]

And [...]

The Summe is ee + ce=5760

Therfore ee + ce=ff − cc, Which was, &c.

In the Aequation − aaa + faa=ggg, [Page 54]the Coefficient f is composed of three proporti­onall lines, and ggg is equall to a Solid made by a Square (whose side is equall to the two first, or the two last) multiplyed into the remaining line: And the aggregate of the first and second may be a, and the aggregate of the second and third shall be e. Put 1′ 2″ 4′″

And suppose − aaa + 7aa=36

Then a may be 3, and e is 6. Vieta, de Recognit. Cap. 18. Prop. 6.

COMPENDIUM 2.

And therefore the Root a found, and called c, the Root e may be found by a plain Aequation; for suppose the middle proportionall y, it is f − y − c′ y″ c′″.

And fc − cy − cc=yy Or, yy + cy=fc − cc. And making fc − cc=xx, it is yy + cy=xx. And the Root y being found by the first Rule of Chap. 2, It is lastly (making c′ y″ d′″) y + d=e.

I will here adde a few Rules (grounded upon Mr. Harriots 6 Sections) by which the Reader may easily perceive the Fabrique of Aequations, their Roots, increment and decrement, Multipli­cation and Division of them, and their number in any Aequation as followeth.

CHAP. IV.

Rule 1.

EVery Aequation being composed of some known and some unknown quantities hath its Originall by roots eomposed of a quantity known and of one other quantity unknown, and these roots multiplied together produce certain particular Members with + and − respectively signed (for in every aequation both these signes are present) which orderly pla­ced make up the aequation. As the aequation aa − ba − ca + bc=0. is made by multi­plying a − b=0 by a − c=0. And because it was at first a − b=0 therefore a=b and the like of c. And from hence it follows that where the first terme (or highest power) in a quadra­tique Aequation is signed − there the Aequation hath two roots, as here by substracting on both parts + aa − ba − ca, the Aequation will be bc=− aa + ba + ca, and must have 2 roots.

1 These compound quantities so multiplying I shall call Binomialls, whether a + b or a − b. not having any need in this treatise to distinguish betwixt Binomials and Residuals.

2 The aequation aa − ba + ca=bc. If it be, b < c. put c − b=d. then [Page 56]the aequation will be, + aa + da=bc, and is of the first kind mentioned in Chap. 2. but if it be b > c, put b − c=f and the aequation will be + aa − fa=bc, and is like the second sort in the same Chapter.

The Originall of the aequation aa − ba + ca − bc=0 here proposed, is + a − b=0 multiplied by + a + c=0, that is a=b by a=− c. This aequation hath but one true root, which is b, and one false, which is c.

3 By this which hath been said it is plain that some aequations have as many roots as dimensi­ons, some not so many, but none can have more; for the number of dimensions being the same with the number of multipliers (if all diverse) can be but all roots. Nor can the aequation be di­vided by any other thing then one of those Binomials by whose multiplication it was made.

But if the multipliers how many soever be still the same, there can be but one root. For let + a − b=0 be muliplied Biquadratically, the product is + aaaa − 4baaa + 6bbaa − 4bbba + bbbb. where it is plain there can be no other root but b. I mean none greater or lesse then it: because in truth here are 4 roots, but e­very one singularly equall to b.

For if there may, let it be d, and let d be great­er or lefle then b, it imports not which. And see­ing d=a, Substitute d in the place of a, quite through the Aequation, it will be [Page 57] dddd − 4bddd + 6bbdd − 4bbbd + bbbb=0. Which if d > b, or else d < b, is at first sight impossible: For the dif­ference between the + and − is alwayes equall to the power of the difference between b and d, which power is here a Biquadrat, therefore d=b. And again seeing this Aequation may be derived by putting b equall to d, for substituting b in the place of d quite through, It will be + 2bbbb + 6bbbb=4bbbb + 4bbbb Which is manifest, therefore again b=d, which is contrary to the supposition, therefore b is the only Root of this aequation, for indeed, the ae­quation proposed being made only of multipli­cations of a − b=0 cannot be divided, that is resolved, by any other Binomiall then a − b, of which it was made,

4 Hence it is that the last term in every aequa­tion may be called the Homogeneall, because it is naturally made by multiplication of the Roots of the aequation, though the coefficients in some ordinary aequations are disguised with other cha­racters, which happens by Addition or Subtracti­on of them, to reduce the canonicall aequation to fewer members, whereby the redundancie of the signes + and − is to be taken away, this is to be seen above in this Rule, where the aequation + aa − ba + ca − bc=0 is reduced to [Page 58] + aa + da − bc=0, by making d=c − b and + aa + ba + ca − bc=0, reduced to + aa − fa − bc=0, by making b − c=f Where the Coefficient d or f, is not a part of the Homogeneall bc, but a difference by which b is greater or lesse then c: by help of which difference, the aequation which consisted canoni­cally of four Members, hath now but three.

5 And this Reduction is usefull, for as Mr. Des Cartes saith, and which may be seen true by the way of Multiplication above shewed, every aequation hath so many true Roots as the Signes + and − therein are changed, which in the ca­nonicall aequation + aa − ba + ca − bc=0, are changed three times, whereas the aequation hath not three true Roots, but one true and one false, that is b & c, and the common aequation reduced changeth the signes but once, that is, from + da to − bc in the former; or from + aa to − fa in the later: and from thence it may be known that the aequation hath but one true Root. The like con­sideration ought to be in others.

And whereas the said Des Cartes doth often mention false Roots, it is to be noted that such are lesse then nothing, as + a + b=0: Or + a=− b, & if any true root, as + a − c=0 be multiplyed by this + a + b=0, there will arise an aequatiō + aa + ba − ca − bc=0 [Page 59]where the signe + follows twice, the signe − twice, and they are once changed, which should intimate (according to Des Cartes) two false Roots, and one true: for he saith, So many times as + or − come twice together, so many false roots there are, this aequation therefore must be reduced, by making b − c=d if b > c, or else if b < c then make c − b=f, so it will be either + aa + da − bc=0, Or + aa − fa − bc=0 which confirms that which Des Cartes saith of twice + or −: Namely, that there are as many false roots in the aequation, as + or − come twice together, and so many true roots as + and − are changed.

And where the Roots are all false, the aequati­on is impossible, as a + b=0 multiplyed by a + c=0, produceth aa + ba + ca + bc=0 which cannot be. And therefore when there is an aequation pretended like aa + ba + ca=− bc, present judgement may be made.

6 The same Des Cartes saith also that all the false Roots in any aequation, may be turned to true ones, and the true ones to false, by change­ing the signes of the second, fourth, and every even term. And this is evident, for of the aequa­tion a4 − 2a3 + 10aa − 30a − 87=0 by such change is made + a4 + 2a3 + 10aa + 30a − 87=0 where the first had three true Roots, and but one false, the later hath three false and but one true. This Aequation was ta­ken [Page 60]at all adventures, to serve for an Example only, whereas any other whatsoever will doe the like.

Rule 2.

The unknown roots of an Aequation may be increased or decreased, by supposing another un­known quantity + or − the decrement or in­crement, and of that Binomiall composing the aequation as it was before of the first unknown quantity: and if this increment be put equal to such a part of the coefficient of the second terme, as unity is of the dimensions of the first terme (if the signes of the first and second be both + or both −) or if the Decrement be made equal to such a part of the said coefficient as unity is of the dimensions as aforesaid, (if the signes of the first and second terme be one + the other −) then by such increase or decrease of the root the second terme of the aequation shall be taken away, and auulled.

Example.

In the aequatiō + aaa + baa − bbc=0, the Roote a may be increased by making e − q=a, and substituting e − q in the place of a quite through the aequation, and there­by shall arise a new aequation:

[...]

Which is equall to the former as you see agree­ing in the particulars, and the root e being found, a may be had by casting away q from e.

And because the number of the dimensions of the first terme aaa is 3, if according to the later part of the rule the quantity q be proportioned, by making 3′ 1″ b′ q″ then b=3q and + bee will destroy − 3qee, and so the se­cond term ee wil be quite taken out of the aequa­tion, as is manifest, for the aequation so purged will be + eee − 3qqe + 2qqq − bbc=0 And by subtracting on each patt + 2qqq − bbc having first made bbc − 2qqq=ddd, it will be then + eee − 3qqe=ddd. The manner of such reduction of Solids, shall follow in the next Chapter.

In like sort the Root a, might have been de­creased by any quantity, as x, which if it be pro­portioned to b as aforesaid, would take away the second term of an aequation, where the signes of the first and second terms are not like; as in the aequation + aaa − baa − bbe=0, by putting 3x=b, and e + x=a: the Pro­bleme will be fully performed by making e + x the Root of the new aequation, as before was e − q, observing the same order in composing the particulars, due respect had to the signes + and −, where they ought to be altered.

The former reduced aequation + e3 − 3qqe=ddd might be further reduced (if need re­quire) to + e3 − bqe=ddd.

NOTE.

This augmentation and diminution of the Roots in such manner as to take away the second term of any aequation, is of excellent use in such aequations as have three or four dimensions, and cannot by any division with any binomiall made of a + or − some other known quantity, as b, c, or the like, be reduced to fewer dimensions, whereby it is certain that such an aequation is Solid, and cannot by any artifice already, or like­ly to be invented, be resolved by Ruler and Compass, but by any of the Conique Sections it may; in this case it is either necessary or extream­ly facilitating, to take away the second terme (if there be any) from the aequation, as shall be seen hereafter in its place.

Rule 3.

The unknown Root of any Aequation may be multiplyed (or divided) by any known quan­tity multiplying (or dividing) the second term of the Aequation by the said quantity, the third by the Square, the fourth by the Cube thereof, and so forward continually in this order, as often as there are terms in it, having first assumed ano­ther unknown quantity, so multiplex to the said unknown Root, as is required.

Example.

In the Cubicall Aequation a3 + baa + cca − bcd=0: Let it be required to multiply the Root a by 4.

Assume e=4a and write eee + 4bee + 16cce − 64bcd=0 Which is an aequation, and the root e is quadru­ple to a, as may be proved thus.

Put a=4 b=3 c=2 and d=21⅓

Then [...]

Therefore a3 + baa + cca − bcd=0

Again, Put e=16 All else the same still.

Then [...]

Therfore e3 + 4bee + 16cce − 64bcd=0

And e=16=4a Which was to be proved.

The utility of this Rule wil appear in reducing [Page 64]aequations affected with fractions, to whole num­bers, by multiplying the Roots by the denomi­nator or denominators of the fraction, for by such means the coefficient of the second term is multiplyed by the same as before, multiplying a by 4, multiplyed also b by the same number 4. And many times by this Rule aequations may be freed from Surd Numbers also: especially if such be found in the second term, as is easie to be seen by triall, for if there be an aequation so affected,

As aaa + √8aa + 29/24a − 4√2=0

Put e − √8=a

And write + eee + 8ee + 9⅔e − 128=0

So the Surds are vanished.

But if yet it be required to avoid the Fraction 9⅔e, then make y=3e. And multiplying 8 by 3, 9⅔ by 9, and 128 by 27, there wil be a new third aequation.

+ yyy + 24yy + 87y − 3456=0 Which consists of entire Numbers, having one true Root which is 9, and the Root of the mid­dle aequation was 3, which is the third thereof, and the Root of the first aequation was 3 − √8 And now I hope this Rule and the use of it is plain enough.

NOTE 1.

It may be noted, that if the Surds in the second [Page 65]and last termes of the first aequation, to wit, aaa + √8aa + 29/24a=4√2 had been utterly incommensurable, the reduction had not been so fesible. For although 4√2 multiplied by the cube of √8 that is by 8√8 produceth 32√16. which is equall to the intire number 128, yet if it had been 2√3 or 2√5, or any such primes to be multiplied by 8√8 the pro­duct would have been 16√24 or 16√40. though this last may (by the note after the Con­sectarie in Chap. 6.) be reduced by multiplying it again by √40 unto the intire number 640. Neverthelesse this second multiplication by a Surd, renders the aequation inexplicable, at least by the precedent Rule.

NOTE 2.

It may be further noted, that if instead of e=4a one would put e=fa lines not being so liquid as numbers, the aequation would then be eee − fbee + ffcce − fffbcd=0 in­creasing the dimensions of the lesser termes, for remedy whereof three lines are to be found in proportion one to another as are the magnitudes fb. ffc. fffb. of which let the first line be sup­posed to containe Ʋnity as often as the superfi­cics fb doth (for which purpose Ʋnity must be a line set, and agreed on before.) The names of [Page 66]these lines when found may be called g, h, k. and the aequation may be written

+ eee + gee + hce − kbc=0.

NOTE. 3

But it is againe to be noted, that where the lines f, b, and c, are commensurable in length the three lines k, h, g, may be very easily found, for then they may be signified by numbers and if f be put for Ʋnity then e=a and the work frustrate, but where the said lines are incommen­surable in length this Reduction is alwayes hard if not impossible: For those incommensurable lines doe most commonly represent such surd numbers as cannot by any Reduction be com­pared.

Rule 4.

The Aequation aaa − 3bba=2ccc, or any other like it, by putting ee + bb / e=a may if c > b be brought to eee=ccc + ddd or if c=b to eee=ccc, or lastly, if c < b then to eee=ccc + √ − dddddd: Which last may be called an impossible Aequa­tion.

Put e′ b″ bb‴ / e And because a is equall [Page 67]to the sum of the Extreams, which are e + bb / e therefore, From thence it will be

[...] Therefore rejecting the contradictories, and multiplying all by eee, it is, + e6 + b6=2c3 e3.

Therefore + e6 − 2ccce3=− bbbbbb.

And + e6 − 2c3 e3 + c6=+c6 − b6.

Therefore, (for e3 − c3=√e6 − 2e3 c3 + c6) + e3=ccc + [...]

If now in the first case c be greater then b, then put c6 − b6=d6.

Then it will be eee=ccc + √dddddd

That is, eee=ccc + ddd. Which is the Aequation promised in the first case.

Secondly, If b be equal to c, then c6 − b6=0 And it will easily follow, seeing (as is shewed a­bove) [Page 68]that e6 − 2 c3 e3 + c6=o, therfore the Root of it e3 − c3=o, that is eee − ccc the second aequation prescribed.

Lastly, by the third case, seeing c is lesse then b, Put cccccc − bbbbbb=− dddddd: Then it will be eee=ccc + √ − dddddd the equation prescribed in the third case, and (be­cause of the inexplicability of √ − dddddd) impossible.

COMPENDIUN

Whereas Mr. Harriot saith Propter √ − d6 inexplicabilitem, &c. The said quantity √ − d6 is not explicable because − d6 ariseth by multiplying + d3 by − d3 betwixt which two there is no meane; for no one thing can pro­duce d6 but d3 onely, and − d6 is not produ­ced by + d3 or − d3 because by both, this there­fore may serve for a Compendium to save labour which might else be lost, in seeking that which is impossible to be found.

NOTE.

I use b6 for bbbbbb, and b4 for bbbb, and b3 c3 for bbbccc, and the like, (as Des Cartes hath done) only for abridgement, as in the Definitions of the Powers is already shewed. [Page 69]And [...] with that line over to distinguish betwixt √c6 − b6 as one quantity, and √c6 taken by itselfe and − b6 taken apart also, for by such mistakes may great errors succeed.

I will adde no more rules, these 4 may be mul­tiplied by any one that doth not find these suffi­cient for his purpose, at his own pleasure.

CHAP. V. Of Reduction of Solids.

HAving spoken in Chap. 4. Rule 2. of ma­king bbc − 2 qqq=ddd, and in Rule 4. of c6 − b6=dddddd, I think it not amiss here to shew how such Addi­tion and Subtraction of Solids may be performed.

And it may be noted that ddd, is for brevity sake there usurped for ggc. or some other solino­mial rectangle Parallelepipedon. equall to the Binomial rectangle Solid bbc − 2 qqq, for if this Binomial could (by plaine Geometrie be) given in a Cube, as is ddd, something else might be done which here I will not speake of.

Now therefore seeing b=3q as there it is, the aequation may be written 9qqc − 2qqq=ddd, or rather 9 qqc − 2qqq=ggc.

Make qq / c=f, therefore 2qqq=2qfc, Secondly, make 9qq − 2qf=gg, from thence it is plain that bbc − 2 qqq=ggc, which was first to be done.

Secondly, to reduce c6 − b6 into one intire Solid, though not into a Squared Cube as d6, as is usurped by Mr. Harriot for brevity in writing, or facility in reasoning, Pag. 100, supposing that done which cannot be done by streight lines and circles hitherto.

Now therefore seeing c6 − b6 is produced by multiplication of ccc + bbb into ccc − bbb. Make cc / b=f, and bb / c=g, & f + g=q and f − g=p, therefore bcf=ccc, and bcg=bbb, and bcq=ccc + bbb. Se­condly also bcp=ccc − bbb: And there­fore bbccpq=cccccc − bbbbbb, which was secondly to be done.

Example in Numbers.

Put b=2 and c=3: Then ccc3=729, and bbbbbb=64, and then cccccc − bbbbbb, that is 729 − 64=665, which is produced by multiplying 27 + 8 by 27 − 8, that is, 35 by 19. Now make f=9/2, and [Page 71] g=4/3;, then f + g=5⅚=q, and f − g==3⅙=p. And bcq=35, and bcp=19. And lastly bbccpq=665.

Moreover, if you make pq=xx, the Solid is further reduced to bbccxx, which although it be not a Squared Cube, yet it hath a square root, namely bcx, which may be of good use in ma­ny cases to resolve Aequations into Analogismes, of which kinde of Demonstration, by help of Euclide 6.14. some notice is taken before in Chap. 2.

NOTE.

The three cases of the aequatiō a3 − 3bba=2 c3, mentioned in the beginning of the fourth Rule of the last Chap. are called by Mr. Harriot, the first Hyperbolicall, the second Parabolicall, the third Ellipticall, because of some similitude between them and those sections, of which three Cases, the first is resoluble by a Conique Section, the se­cond by a Circle, and the third not at all.

Multiplication and Division of Solids is al­together as easie as Addition or Subtraction, for if one would divide ccc by bb, make cc / b=x, and again make cx / b=z, then z is the Quo­tient required.

Example in Numbers.

Put b=2 and c=3, then ccc / bb=6¾, to finde which, make cc / b=x=9/2, then cx=27/2, & cx / b=27/4=z=ccc / bb=6¾, as it should be.

Again, if c5 should be divided by b, it is now ccc / bb=z, and multiplying by b it is ccc / b=bz: Again, multiplying by cc it is c5/b=bccz, and bccz is the Quotient required.

But if it be required to bring the quotient to a Biquadrat, make bz=dd, then ccdd=bccz And make cd=ff, then the quotient will be ffff.

Multiplication is naturally so easie that there needs no more be said of it, then what hath been said already in Chap. 1.

Now, of aequations consisting of 3 termes in continuall proportion as a4 + bbaa=c4 or secondly a6 − bbba3=c6, or lastly let it be − a8 + bbbba4=c8, let them first be pro­posed in numbers as a4 + 2aa=24, if by [Page 73]Rule 1 of Chap. 2. it be wrought, it will be found √25 − 1=aa, and aa=4 or a=2.

Otherwise if the square of halfe the coefficient be added on both parts, then a4 + bbaa + 1=25.

And their square roots also are equall; that is aa + 1=5 and aa=4 or a=2 as before. and the latter may prove the former.

2 In the second, let it be a6 − 10aaa=459 Adde 25 to each part, then it it is aaaaaa − 10aaa + 25=484.

Now each part of the aequation is a Square & their Roots also are equal; that is aaa − 5==22, that is aaa=27, and a=3.

3 Lastly, If − a8 + 700a4=46875 from the Square of 700/2, that is, from 122500: take the homogeneall 46875, there remains 75625, whose square root is 275. And either 350 + 275. Or 350 − 275, that is either 625 or 75 is equall to aaaa, and a=5. Or √qq. 75=a, which Charrcter √qq. signifies the Biquadraticall Root.

NOTE

The first and last of these three aequations, may be done aswell in Lines as Numbers (by the said three Rules of Chap. 2. and so any aequation of 4, 8, 16, or. 32 dimensions, but aequations of 6, 12, or 24 dimensions, cannot be effected so, [Page 74]because there is ever one or more Cubique roots to be extracted, which without two meanes can­not be done.

For if it may, then I say, that two meanes be­tween any two lines may thereby be found, for in the second aequation a6 − bbbaaa=c6 by Rule 2 Chap. 2c6 + ¼b6 is a square, make cc / b=d, then bd=cc, and bbdd=c4, and bbccdd=cccccc, then make ¼bb / c=f therefore fc=¼bb, and fcb4=¼b6. Now because bb / c=4f, make h=4 f, then fccchh=¼b6.

Make fc=ll, then cchhll=¼b6. Again, make bb + hh=mm, and dd + ll=nn. And then it will be . . ccmmnn=bbccdd + cchhll, that is, c6 + ¼b6, to the square root hereof cmn, add ½bbb: thus, make ½bb / m=p; then mp=½bb, and bmp=½bbb. Make bp / n=q, then mnq=½ bbb. Lastly, make c + q=x, then it is cmn + ½bbb=mnx=aaa, by Chap. 2. Rule. 2. Now if m, n, and [Page 75] x be proportional, then the middlemost is equal to a, but that is uncertaine, and cannot be made otherwise: But by making rr=mn it will be rrx=a3 and a will then be the lesser of two meanes between r and x if r < x or the greater meane, if r > x. And so if r and x had been given, and required to find 2 meanes between them by retrogradation orderly, one might come to the said aequation a6 − bbbaaa=c6 of which if the root a be found, two meanes are also found between r and x which was to be proved.

CHAP. VI. Of Surd Numbers

Rule 1

THe square root of any number being multi­plied by that number, produceth the square root of the Cube of the number.

For √a multiplied by a produceth a√a, but a√a=√aaa for taking Equimultipli­ces they will be equal, as if the first, namely a√a be muliplyed still by √a, the product is a√aa, that is aa. And if √aaa be multipli­ed by √a it produceth √aaaa that is aa al­so, wherefore a√a=√aaa. And therefore 3√3=√27 either of which is the cube of √3, and the like of all others.

Rule 2.

Surd numbers are multiplyed and divided like whole numbers, the Product retaining still the Character of the Root..

That is, √2 multiplied by √3, produceth √6, and so of all others.

NOTE.

Where I shall have occasion (if any be) to speak of a Cubique Root, I shall signe it thus, √c. and the Biquadratique Root thus √qq.

Rule 3.

To multiply, divide, adde or Substract the roots of Surd numbers. And first of

MULTIPLICATION.

Besides that which hath been said in the last Rule above, these roots of Suids may be multi­plied and divided, and known by other names, so as sometimes the products, or quotient shall be rationall. First, therefore any square root doubled is the square root of the quadruple, as

  • 2√5=√20 and 2√20=√80.
  • 3√5=45. 4√5=√80. 5√5=√125.
  • 2√10=√40. 3√10=√90.
  • 4√10=√160, and 5√10=√250,

&c, infinitely still multiplying the Numerator, 2, 3. 4, 5, &c. into it selfe, and the product into the snrd number, as if 3√10=√90, it ariseth from 3 times 3 into the surd number √10: and the like of all others whatsoever.

For put √a=√10, to be multiplied by [Page 77]another number, as by a=10, the product is a√a=10√10, which by the first rule is √aaa=√1000, that is, the numerator 10 into it selfe making 100, which multiplied again by the surd √10, gives √1000.

And if it had been at first √a=√10, mul­tiplied by any other number, as e=3, the pro­duct must by the same method be e√a=e√10 that is (by the same reason as the former) √eea=√ee10=√90.

And it is plain, that if any Root be multiplyed by

  • 2 The product shall be the Root of the Quadruple.
  • 3 The product shall be the Root of the Noncuple.
  • 4 The product shall be the Root of the Sedecuple.
  • 5 The product shall be the Root of the Vigintiquintuple.
  • 6 The product shall be the Root of the Ttigintisextuple.

And so forward infinitely, according to the proportion of the Squares of the Multipliers

Also by Decuplation, as if 5√5=√125, then 5√50=√1250. Or if 4√4=√64 then 4√40=√640. And (as above) if 4√10=√160, then 4√100=√1600.

Also by Subdecuplatiō, if 2√10=√40, then 2√1=√4. Or if 5√20=√500, then 5√2=√50. And (according to that afore­said) 3√37=√333, and 3√36=√324, that is, the square root of 3 times 3 times 36.

And this may often be of use, not only in num­bers but Species, and is therefore to be had in memory by him that would be ready in Multipli­cation of Surd numbers, or Surd quantities.

Furthermore it may be useful to remember that in Reciprocall Surds as 4√5 and 5√4 these two have that proportion one to another as 4 hath to a meane betwixt 4 and 5.

As for example 4√9 hath that proportion to 9√4 as hath 4 to 6, which is a mean betwixt 4 and 9, for 4√9=12, and 9√4=18, but 4′ 6″ 12′ 18″ or more generally a√e′e√a″ a′√ae″ for multiply the Meanes, it is ae√a and multiply the Ex­treames it is a√aee, & divide each of them by a the first is e√a the other is √aee, but by the former part of this rule e√a=√aee where­fore this is proved,

CONSECTARY.

Hence it is evident that roots of themselves inexplicable may be so multiplied as the product may be rational: for if √20, be multiplied by 4√5 the product will be 4√100=40.

For 2√5=√20 and 2√20=√80, there­fore 4√5=√80, but √80 multiplied by √20 gives √1600=40,

I need say nothing of Division, for that is no more but by the same steps to go back again, as √1600 divided by √80 quotient is √20. And so of the rest which hath been said in multi­plication.

NOTE.

These things being so, it will not be hard to find some number to compare with any Surd number so as to make that worke rationall and exprimible which seemed not so: for there is not any surd number can be given which may not by some multiplication be made a rationall number: for let it be √5, √7, √8, or any of these as √7 multiplie it first by √7 that produceth 7, but multiply √7 by any square number what­soever, as by 4 omitting the signe √, it gives 28, then again multiply √7 by √28 it produ­ceth √196=14.

For this is all one as to multiplie one square number by another, which must needs produce a square number.

So heer the square number 4 was multiplied by 7 and after by 7, that is by 49, which multi­pliers cannot produce any other then a square number, to wit 196 Euclid. 9.1.

And whatsoever hath hitherto been said of Quadratiques, may serve for cubiques also; due respect alwayes had to the degree of the quantity and root, for any √c. multiplied by 2 gives 8√c. by 3 it gives 27√c. by 4 it gives 64√c. that is 2√c. 8=√c. 64 and

3√c. 8=√c. 216, and

3√c. 27=√c. 729 the proportion still [Page 80]increasing as the Cubes of their nultipliers.

And the like consideration had, this may be applicable to Biquadratiques, or any higher or­der.

And still whatsoever hath been said of multi­plication, serves in a retrograde way for division also.

Rule. 4 For ADDITION.

Surd roots are usually added and substaacted by the signs + & − as the square root of 2 added to the square root of 8, sum is √2 + √8 or substracted rest is √8 − √2.

But these may be added into one summe for seeing 8 is quadruple to 2 therefore 2√2=√8. And the summe is 3√2 and the remaine is √2. Likewise the reciprocall Surds 8√2=2√8, are capable of addition, substraction, multiplication or division; for they are being added 3√32 that is √288; substra­cted, √32; multiplied √4096; divided √4: but such as are neither commensurable nor reci­procall cannot be amassed into one summe.

And the summe of the former addition of √8 + √2 being already reduced to 3√2 may be yet further reduced to √18 for 3√2 is equall to the square root of three times three [Page 81]times two as hath been more then once shewed.

And generally when the Surds given are de­nominated by numbers in quadruple proportion, as √2 to √8, and √3 to √12, &c. the lesser and the greater twice being added together, as 2 to 16, or 3 to 24, the square Root of the sum is equall to the sum of the two square Roots gi­ven to be added, that is, √2 + √8=√18, and √3 + √12=√27.

The reason is, √1 + √4=√9, which 9 is composed of the lesser once and the greater twice, that is, as often as the √1 is contained in the √4.

But if the numbers be prime one to another, they must be added or subtracted by the signes + and −, for these Rules reach not to primes.

And having said this little to acquaint such as are wont to be afraid of operations where Surds are present, with this which will render some things easie which perhaps seemed hard, and o­thers which were hard, lesse difficult. I will now leave this ragged Subject, and recreate a little with a few easie Propositions; the performing of which may serve to recall into use and practice that which hath been spoken of Solids in the for­mer Chapter.

CHAP. VII.

Prob. 1.

ANy right line being given, to divide it into two parts, so as the Rectangle of the whole and one of the parts; may be to the square of the other part, in such proportion as is betwixt any two right lines given.

Let the right line given be b.

The segment to be squared a.

Then the other Segment is b − a.

And let the two lines given be r and s.

Then bb − ba′ aa″ r′ s″.

And raa=sbb − sba. per 16.6. Euclide.

That is, raa + sba=sbb.

Make sb / r=d, and divide all by r.

Then it is, aa + da=db. Make db=ff.

Then lastly it is aa + da=ff. And a is easily found by Rule 1. of Chaep. 2.

And if it had been required to have had the Rectangle + or − some other plain to have had any limited proportion to the square aa, the work had been almost the same, with some small addition.

Prob. 2.

To make a Scalenon Triangle, of which the Base, Perpendicular, and proportion of the other Sides shall be given. (I account that the Base which subtends the divided angle.)

Let the base given be b, the perpendicular c, and the proportion of the other sides, as r to s.

Of which let r be the lesser.

And for the lesser segment of the base put a:

Therefore by supposition,

[...]

So that the squares of them are also proportional, That is.

rr′ ss″, cc + aa′ cc + bb − 2ba + aa″.

And by multiplying the means & extreams, It is,

ssaa + sscc=rrcc + rrbb − 2 rrba + rraa. That is,

ssaa − rraa + 2rrba=rrcc + rrbb − sscc.

Make ss / r=x.

And divide all the aequation by r, Then it is,

xaa − raa + 2rba=rcc + rbb − xcc

Secondly, make x − r=f, & gg=bb + cc.

Then it is, faa + 2rba=rgg − xcc.

Again make rg / f=h, and xc / f=k, and di­vide faa + 2rba=rgg − xcc by f.

Then it will be aa + 2rb / f a=hg − ck.

Lastly, make 2rb / f=q, and hg − kc=mm

The Aequation finally reduced will be then aa + qa=mm, and a may be found by the first rule for square Aequations. Chap, 2.

Prob. 3.

Any number being given, to finde two other numbers, so as all the three may constitute a Rectangle Triangle.

Unto the Square of the number given adde unity, the halfe of the summe shall be the hypo­thenuse, or from the said square take unity, the halfe of the remain shall be the middle side.

For let the number given be a, the square is aa, to which adding unity, the sum is aa + 1, the halfe whereof is ½aa + ½, for the Hypote­nuse.

Secondly, from aa take unity, the rest is aa − 1, the halfe whereof is, ½aa − ½ for the middle side.

But the lesser side (by supposition) is a.

The square of the lesser side is aa.

The square of the middlemost is ¼aaaa − ½aa + ¼

Both those squares are ¼ aaaa + ½aa + ¼:

But the square of the hypotenuse, viz, of ½aa + ½ is equall to these, that is, ¼ a4 + ½aa + ¼.

Therefore by the 48. of the first of Euclide the Proposition is proved.

COROLLARY.

Hence it is plain, that the two greater sides of any rectangle Triangle differ by unity, for if two Squares differ by 2, their halves differ by 1.

NOTE.

If it be required to have all the three sides in whole numbers, then the lesser side must be an odd number.

Probl. 4.

The difference of the sides of a Rectangle, with the Area and Diagonall in one sum, being given in numbers, to finde out the sides.

Let the difference of the sides be 7

And the Area and Diagonall together 73.

And put the lesser side equall to a.

Then the greater is a + 7.

These two multiplied produce aa + 7a, which is equall to the Area.

And therefore 73 − aa − 7 a is the Diagonal

The square of which is

  • + 5329 − 146aa
  • + aaaa + 14aaa +
  • + 49aa − 1022a

Which reduced and rightly ordered, Is

+ aaaa + 14aaa − 97aa − 1022a + 5329

Which by the 47. of the first of Euclide, is equall to the two squares of the other sides a, and a + 7, whose squares are aa, and aa + 14a + 49.

That is, + aaaa + 14aaa − 97aa − 1022a + 5329=2aa + 14a + 49.

That is + aaaa + 14aaa − 99aa − 1036a + 5280=0.

That is, − aaaa=14aaa + 99aa + 1036a=5280.

In which aequation, because aaaa hath four dimensions, and the Homogeneall 5280, but four places, the root a cannot consist of more then one place, or figure, which must be found out by trying every one of the nine Digits, if need be, and will be found at last to be 5, therefore the other side is 5 + 7=12, the Area 60, and the Diagonall 13.

But if a had been more or lesse then 5 yet (ex­cept something else lead a readier way) it is good to try 5 at first, if it be too little then 7, if that [Page 87]also too little, then 9, so there will be no need to try the even numbers, 6, 8, &c. for if 5 be too little and 7 too great, it must be 6, the like reason will serve for 8, 4, 2, so that he which guesseth most unfortunately, needs not try above four or five digits, which is no great matter, the like happening sometimes in seeking the quotient in plain division, for no man is sure to guesse right at first.

But that we may exemplifie this in bigger numbers, where a may consist of two or more places.

Let the difference of sides be 71

The area and diagonall together 1177.

Working as in the former example, there will arise an aequation, which being reduced and or­dered as before, will be

− aaaa − 142aaa − 2685 aa + 167276a==1380288.

And putting b + c=a:

Then the Canon of the resolution will be

− bbbb − 4bbbc − 6 bbcc − 4bccc − cccc − 142bbb − 426bbc − 426 bcc − 142ccc − 2685bb − 5370bc − 2685 cc + 167276b + 167276c.

To be orderly subtracted from the homogeneall number given 1380288, as followeth.

The number given [...]

The first single root b=1.

[...]

Subtract (the difference of + & −) +1252260

Remains of the number given [...]

The first Root decuplate b=10.

[...]

Then [...] is all the −

And [...] is all the +

Divisor [...] is their difference.

The second single Root c=3

Remains of the number given [...]

[...]

All the − being 373800

The differēce is +128028

Which being taken from the remaines of the number given + 128028, there remains final­ly nothing, so that the given aequation is justly resolved by the Root b + c=13.

The lesser side a is therefore 13, to which if the difference given, namely, 71, be added, the middle side 84 is thereby composed.

Again, if to that middle side 84 be added unity, the hypotenusa of a right angled triangle is com­posed, whose three sides are 13, 84, 85.

The Superficies of this Triangle is halfe the parallelogram or rectangle required.

For 84 multiplied by 13, gives 1092 for the area of the rectangle, to which adding 85 the Diagonall, composeth the number 1177, as was required in the Proposition.

COMPENDIUM.

Seeing the two greater sides of any rectangle triangle, exceed one another by unity (as by the former Corollary) the difference betwixt the two lesser sides being given, the difference betwixt every two sides is also given.

So that putting a for the lesser side of the rect­angle, the greater side is a + 71, and the diago­nall a + 72, whose square is + aa + 144a + 5184, to which the two squares of the sides, being aa + aa + 142a + 5041, are equall:

That is, 2aa + 142a + 5041=aa + 144a + 5184

And subtracting from each part aa + 144a + 5041

There will remain + aa − 2a=143.

And a will be found 13, by the second Rule of Chap. 2.

RESUMPT.

In the second Probleme of this Chapter it hath been shewed how upon a Base and Perpendicular [Page 91]and proportion of the remaining sides given, to describe a Triangle.

It is there to be understood of an acute angled triangle, in which the perpendicular falls within the triangle.

Now therefore let it be otherwise.

[mathematical diagram]

As in this figure, of the triangle adb, let there be given, the base ab=b, the perpendi­cular de=c, and the proportion of the other sides, namely bd=r, and da=s, whereas in the second Probleme, the lesser segment of the base was called a, and the greater b − a. Now here the line be may be called a, and ae shall be b + a, and the rest of the work will be like that before in Probl. 2.

NOTE.

And it may be noted, that if from a and b [Page 92](the ends of the line ab) be drawn other binarie lines, how many soever, so as they hold the same proportion as s to r and concur in other points, as c, f, g, &c. Those points are all in the circum­ference of a circle whose center is in the line ab, produced towards g.

For upon ab describe the rectangle triangle abc whose two sides ac, cb, may be as s to r, and divide the angle acb into two equal parts by the right line cx, and draw cq perpendicular to ac in c, then the angle xcq=90 − xca likewise the angle cxq=90 − xcb, but xcb=xca therefore xcq=cxq; and cq=xq.

And because of the similitude of the triangles acq and cbq it is aq′ cq″ bq′″ that is aq′ xq″ bq′″.

Now by supposition ag′ bg″ ac′ bc″

And it hath been proved aq′ xq″ bq′″.

Therefore by composition also, it will be aq + xq′ xq + bq″ xq′ bq″

But ac′ bc″ aq′ xq″.

And xq′ bq″ aq′ xq″.

Therefore ag′ bg″ aq + xq′ xq + bq″.

And therefore ag=aq + xq. And xq=gq.

But it hath been proved that xq=cq.

Therefore xq=cq=gq.

Againe, making the center q, and the [Page 93]

[mathematical diagram]

space xq, cq, or gq, describe the circle xcg, and prolong db to the circumference in z, and draw qz, and because it hath been proved above, that ag′ bg″ ac′ bc″.

That is ag′ bg″ xq′ bq″ therefore also By division ag − bg′ bg″ xq − bq′ bq″.

That is ab′ bg″ xb′ bq″.

Therfore the rectangles abq=xbg Eu:. 6.16.

And because by supposition ac′ bc″ ad′ bd″.

And ac′ bc″ qc′ qb″.

Therefore ad′ bd″ qc′ qb″ but qc=qz, therefore ad′ bd″ qz′ qb″ and the angles abd, zbq being equal, the triangles abd, zbq are equiangular, Euclid. 6.6.

And therfore ab′ db″ bz′ bq″ Eucl. 6.4. and the rectangles dbz=abq: Eucl. 6.16. But it was now proved that abq=xbg. there­fore xbg=dbz.

So that the points x, z, g, being in the circum­ference of the Circle zxg, the point d must be in the same circumference. Euclide 3.35.

The like proofe may serve to shew that the point f is in the same circumference; which is all that was to be proved.

This Circumference, however desired by the Ancients; and effected by modern Mathematiti­ans, seems yet to have little use, more then to help the construction of the triangle, which (but now I shewed) may be done without it.

CHAP. VIII. Of Mixtion.

DEFINITION. 1.

STandard finenesse, Is that finenesse which the current Gold and Silver Moneys are made of. In England the Gold is 22 Carects of fine Gold, and two Carects of Allay. The silver Monies are made of silver so as the pound weight contains eleven Ounces, two peny weight of fine silver, and 18 Peny weight of Allay.

DEFINITION 2.

If any Ingot be finer then Standard, it is cal­led better, if courser worse, and this betternesse and worsenesse is reckoned by Carects and Grains [Page 95]in Gold, and by Peny weights in Silver, and is summed by multiplication of the betternesse or worsenesse in the pound weight, or pound weights of the Ingot.

DEFINITION 3.

The Temper is that which when two or more quantities of Liquors, or Herbs, or Minerals, or any thing used in Medicine, of differing degrees of Heat, Cold, Drouth, or Moisture, are mixed together, so as the whole Masse so made by mix­ing have none of these four Qualities.

NOTE.

The Standard and the Temper differ in this, the first respecteth but two qualities, to wit, bet­ter and worse: the lattet respects four qualities, namely, hot, cold, dry, and moist: yet the Mix­tor dealing but with two of these at once, that is, such two as are opposite, as are the two first or the two last mentioned before, or any two which are alike, as both better, both worse, both hot, or both cold, may use the same way in both.

Prop. 1.

If there be two Ingots of equall weight, the one better then Standard by a certain finenesse, the other as much worse, those two Ingots molten together shall produce a Fusion or Masse which shall be of Standard finenesse

Let the first Ingot b be better by c.

The Second Ingot d worse by f.

And let it be b=d and c=f.

Multiply b by c it gives bc equal to all the betternesse of the Ingot b. By Def. 2.

Likewise cf is equal to all the worstnesse, but bc=df.

Therefore the whole Fusion b + c is asmuch better then standard as worse.

Wherefore it is neither better nor worse, but just Standard finenesse.

Prop. 2

If two Ingots to be molten differ in weight, quality, and degree of quality reciprocally, that is, if as the weight of the first, to the weight of the second, so the degree of worstnesse of the se­cond to the degree of betternesse of the first, the whole fusion shall be Standard finenesse.

Let there be quantity b better by c.

And quantity d worse by g.

And let it be b′ d″ g′ c″.

Therefore bc=dg Eucl. 6.16.

Namely all the betternesse equal to all the worstnesse, and therefore the mixture of the masse neither better nor worse. The same arguments will serve if the Propisition had been in Liquors, to prove the mixture to be temperate.

Prop. 3.

If there be two quantities of Silver or Liquor, of divers qualities, or divers degrees of the same qualitie, if all the betternesse or all the worst­nesse, all the heat, or all the cold be found out by multiplying each quantitie by its qualitie, and taking the difference of them if they be opposite; or the summe of them if they be alike; that diffe­rence or summe divided by the summe of the quantitie, gives (as some call it) the forme re­sulting or the degree of betternesse, worstnesse, heat or cold, of the whole fusion or mixture.

Let there be quantity b hot in g.

And quantity d cold in h.

Then bg is equal to all the heat of b.

And dh equal to all the cold of d.

If bg > dh then bg−dh/b+d is the degree of the form resulting, Hot, Or if bg < dh, then dh−bg/d+b is the same in coldnesse.

Now although this is plaine from Def. 2. be­cause all the heat bg − dh, or the coldnesse dh − bg of the whole mixture, ariseth by mul­tiplication of the severall qualities by their re­spective quantities, and therefore that whole heat, or whole cold divided by all the weight of the several quantities, gives the quotient equal to the degree of heat or cold of any part of the weights, which in respect of the whole weight may be called one, which degree of heat here [Page 98] bg−dh/b+d being multiplied by the whole weight namely by b + d gives bg − dh, that is all the heat of all the weight, and therefore bg−dh/b+d is that which we call the forme resulting, and dh−bg/b+d if bg < dh. Yet this may be further confirmed by that Rule given by Mr. John Dee, in his Mathematicall Preface before Euclid.

The Rule which there he sheweth is this.

What proportion is of the lesser quantity to the greater, the same is of the difference be­tween the degree of the forme resulting and the degree of the greater quantitie to the difference between the degree of the said forme and the degree of the lesser quantitie.

Here therefore let be b < d for that is free. Also let it be bg > dh.

It is to be proved by the said Rule, that b′ d″ h′ + [...] Multiplie the two later by the cōmon denomina­tor b + d, the first gives bg + bh, the second dg + dh. [Page 99]And therfore b′ d″ bg + bh′ dg + dh″ Multiply the means it gives dbg + dbh, like­wise the extreams multiplied is dbg + dbh: And therefore the Analogisme which was to be proved is true, by the 16. of the 6. of Euclide.

In like sort if it were bg < dh, and b < d, it might be proved, that [...]

Lastly, if g and h were like qualities, that is, both Hot, or both Cold, and b < d, It is then to be proved that [...] And reduced b′ d′ bg − bh′ dg − dh″, Which is manifest.

Example in Numbers.

First, Let it be put b=5, g=4, d=7, and h=2.

Then bg=20, & dh=14, & bg−dh/b+d = ½, Now because the heat bg is greater then the cold dh, the whole mixture shall be hot, and that heat shall be in the middle of the first degree, and according to Mr. Dees Rule it will be [Page 100]5′ 7″ 2½′ 3 ½″, for the difference be­tween the form resulting which is hot in ½ and the greater quantities degree, which is cold in 2, is 2½, likewise the difference between the les­ser quantities degree, hot in 4, and the forme hot in ½ is 3½: So that this is right.

Secondly let be b=5, g=2, d=7, h=4, in opposite qualities h and g: dh−bg/b+d = 18/12, It will be 5′ 7″ 4 − 1 ½′ 2 + 1½ (for 18/12=1½. That is, 5′ 7″ 2½′ 3½″ as before, and the forme resulting cold in 1½ degree.

3 Lastly, let be b=5, g=4, d=7, h=2, in like qualitie, for example both hot then bg+dh/b+d = 2 10/12 for the forme.

And 5′ 7″ 10/12′ 14/12″ for the Anologisme, exactly agreeing in all cases with Mr. Dee.

And this is so plain that it needs not be exem­plified in metals, it not being my purpose to write much of them nor of the Standarding of Gold and Silver, because it is so neatly and fully done already in a little Treatise put forth in Anno. 1651. by Mr. Jobn Reynolds of the Mint.

Yet the Reader may take notice that he which brings but common Arithmetique with him, may by some one or more of these three foregoing propositions, perform any plain Pro­bleme [Page 101]that can be required concerning mixtures in valuable metals or liquors. For first

Rule. 1.

If the weight of the masse be not limited, if any quantity with any quality (which exceeds not the degree of the greatest finenesse) be given, a like quantity of the just opposite quality, will cause all to be Standard or Temper.

Rule. 2.

If the quantity of the masse be limited, and the two oppositive qualities given, then divide the quantity of the masse into two parts proportionall with the qualities, and taking them reciprocally the mixture shall be Standard, or Temper, by the second Prop.

Rule. 3.

If there be two Ingots of Silver to be molten the first better by a certain difference, and the second also better not by the same difference, if each weight be multiplied by its betternesse, the two products added together make the betternesse of the whole masse; which being divided by the summe of the two weights, gives the forme result­ing of the masse by the third proposition, which [Page 102]masse may be made Standard by Allay as follow­eth.

As the fine silver in the pound Standard, Is to the forme resulting:
So is the weight of the masse, To the weight of the Allay.

But this Rule is not pertinent to the mixture of liquors, because in them there is nothing agreed on for Allay.

NOTE.

If the two Ingots molten produce a masse worse then Standard, out of any Silver which is better, a quantity may be limited by the second proposition, to make it Standard

But if there be given the weight of an Ingot worse by a certaine difference; and the weight of the whole fusion be limited, and the finenesse, wheher Standard, or better, or worse; This Rule doth it. Multiply the weight by the worstness, and divide the product by the betternesse of the Sil­ver to be added, the quotient shall be the weight of that to be added to make it Standard. And if it be required to have the fusion better, or worse then Standard (but not worse then the Ingot given) it is easily done by taking more or lesse weight of the fine silver to be added, or of more or lesse finenesse as the case requires, and which needs no more then hath been shewed.

If the fusion consist of more then two quanti­ties, all that hath been said of two things misci­ble, is appliable to other miscibles how many so ever, by repetition of the working with two at a time.

Prop. 4

If there be three like solids equal in Magni­tude, and differing in weight, the middlemost being composed of some of the matter of the first, and the rest of the matter of the third mixed, if the rectangle made of the weights of the first and third, Minus the rectangle made of the weights of the said first and second, be divided by the weight of the third want the weight of the first, the Quotient shall be equal to all the mat­ter of the first (that is to the weight thereof) which is contained in the mixed solid.

Let the first be b, the second c, the third d.

And the weight of the first q, of the second r, and of the third s. And the common magnitude Unity, and make a equal to the weight of all the matter of the first contained in the second, as a­foresaid.

And make q′ 1″ a′ [...]

And s′ 1″ r − a′ [...] [Page 104]Therefore a / q is equall to all the said matter of the first, and r−a/s to all of the third in the Mixture, I mean to the Magnitude of it. And [...] Multiply all by qs (or first by q, and the pro­duct by s) it gives sa + qr − qa=qs, That is, sa − qa=qs − qr. But qs − qr is the Dividend required. And s − q the Divisor required.

And qs−qr/s−q = a by the last Aequation, And a equal to the weight of the matter in the first contained in the second, wherefore the Pro­position is proved.

Example in Numbers.

Put q=97 r=73 and s=63 Then qs−qr/s−q = 28 9/17, which is all of the matter of b contained in c. [Page 105]And the residue 73 − 28 9/17=44 8/17 is all of d contained in c.

Now 97′ 1″ 28 9/17 ′ 485/1649″ = a / q

And 63′ 1″ 44 8/17 ′ 756/1071″ = s−a/1

But [...].

As it ought to be, the like proofe serves for any Numbers.

Prop. 5.

If there be three like Solids of which the se­cond is composed of divers matters, to wit of parts of the first, and parts of the third, and the three solids equall in weight, but not in magnitude, if the rectangle made of the Magnitudes of the first and third, lesse the Rectangle made of the Mag­nitudes of the first and second, be divided by the Magnitude of the third want the Magnitude of the first, the Quotient will be equall to all the matter composing of the first, I mean to the magnitude thereof, which is contained in the second Solid.

The proofe of this is the same with the for­mer, Mutatis mutandis.

Prop. 6

If there be three such Solids as before in the fifth, and the magnitudes of the parts composing found, if the magnitude of the parts of the first composing the second, be divided by the magni­tude of the first, the quotient is the weight of those parts.

For the common weight being Unity, As the first magnitude is to its weight, which is uni­ty; So is the magnitude of the parts of the first Solid, to the weight of the said parts. (Not to re­tern the same form, but diffused in mixture, and compared in minute parts commensurable with the whole.

Let the magnitudes of the intire Solids be f, g, h, and their common weight Unity, and let the magnitude of the parts of the first com­posing the second, be put equall to e. Then f′ 1″ e′ f″ / e which f / e is the weight of the said parts composing. This is plain.

Prop. 7.

If there be three Solids, and the first and third composing the second as before, differing all in weight and magnitude: if the rectangle Paral­lelepipedon made of the weights of the first and third, and the magnitude of the second (all mul­tiplied together) want the rectangle Parallele­pipedon made of the first and second (I mean the [Page 107]weights of them) and the magnitude of the third (all multiplied together) be divided by the rectangle made of the magnitude of the first, and the weight of the third want the rectangle made of the weight of the first, and the magnitude of the third, the quotient shall be the weight of the parts of the first composing the second: which weight multiplied by the magnitude of the first and the product after divided by the weight of the first, this later quotient shall be equall to the magnitude of the said parts.

Let the Solids be in

  • Weight f, g, h,
  • Magnitude b, c d,

And f the weight of the first, and put the weight of the parts composing of the first equall to a.

Therefore f′ b″ a′ [...] that is, as the weight of the whole first to its magnitude, so the weight of part or parts of the said first, to their magnitude.

Now because the weight of the parts of the first composing the second Magnitude g, are a, the weight of the parts composing of the third are therefore g − a.

Therefore secondly [...]

That is, As the weight of the whole third, is to [Page 108]the weight of the parts thereof, so is the Magni­tude of the said whole, to the magnitude of the parts thereof. So then the magnitnde of the parts of the first, more the magnitude of the parts of the third, are equall to the whole magnitude of the second.

That is [...] Multiply both parts by f h the rectangle of the denominators, it gives + hba + fdg − fad=fhc

That is hba − fda=fhc − fdg

But fhc − fdg is the dividend proposed.

And hb − fd the divisor desired.

And [...] by the last equation.

But by supposition a is equal in weight to the parts of the first composing the second, wherefore the proposition is proved, as to the first part.

And the second part is manifest out of the first Analogisme [...] to wit that ba / f is equal to the magnitude of the parts of the first.

Example in Numbers.

  • Let the weight of the first solid be 85
  • [Page 109]Let the weight of the second 60
  • Let the weight of the third 54
  • The magnitude of the first 49
  • The magnitude of the second 50
  • The magnitude of the third 48

And the weight of the parts of the first a as before.

Then [...]

And [...]

And therefore [...].

That is the magnitude of the parts composing the second taken together, must be equal to the magnitude of the whole second.

Multiply each part by 85 times 54, that is by 4590.

It produceth 2646 a + 244880 − 4080 a=229500

That is (rrduced) − 1434 a=− 15300.

And a=10 960/1434

Now the rectangle parallelepipedon of 85, 54, and 50, is 229500, from which taking 244800 (which is the rectangle parallelepipedon of 85 60 and 48) there remaines − 15300 for the Dividend proposed.

Secondly, If from the Rectangle of 49 and 54, which is 2646, be taken the Rectangle of 85 and 48, which is 4080, remains − 1434 for the divisor proposed. And (by the last Ae­quation) 19300/1434=a, and therefore the Exam­ple in Numbers is cleared.

In the Aequation before hba − fda=fhe − fdg, the quantity a is easily found by this analogism, namely, hb − fd′ hc − dg″ f′ a″, if one make hb − fd=mm and hc − dg=nn. And m′ n″ p′″, for then m′ p″ f′ a″.

Upon the same way of reasoning which hath been used in this Chapter, is grounded the Rule of False Position, and also that of Alligation:

For if the two degrees of the qualities of any two Miscibles, be called the two false positions, and the two respective quantities of the said Miscibles, be called the two Errors, then the de­gree of the form resulting is the true point sought. For if any one would work by the Rule of False, and go the neerest way, he must divide the di­stance betwixt the false positions into two parts proportionall with the Errors, and the work is thereby done sooner then by the common way of Crosse Multiplication.

As if it were required to part 48 in two, and [Page 111]one of the parts againe into 3, and the other into 4, so as the thirds of the one may be (in number) quadruple to the fourths of the other. Suppose first 40 and 8, and dividing 40 by 3, quotient is 13⅓, and 8 by 4 is 2, whose quadruple should be 13⅓, but is but 8, so the first Error is − 5⅓. And putting the second time 30 and 18, the second Error will be found to be + 8. Make therefore 8 + 5⅓′ 40 − 30″ 8′ 6″ if this 6 be added to the second position 30 whose error we here worked with (namely with 8) the summe is 36 for the part required, and the other part is 12.

And as for the Rule of Alligation, which is to adde all the betternesse and worstnesse of each particular component severally taken into one summe (which there is called the summe of the differences;) And then to worke by this Anola­gisme, viz,

As sum of all the betterness and worstnes mixed, Is to the whole Mass, or mixture to be made;
So is any particular betterness or worstness. To all that which is to be taken and mixed of that respective quality.

All this being manifest, shall not need any proof.

☞ As before, Page 106, so here again, I let the Reader know that the word Magnitude in this Chapter, is to be taken for the number of small parts or atomes of a Body, and not for a line or Superficies.

CHAP. IX. Of Mensuration.

IN this Chapter I shall demonstrate little, as not intending to write much new, but (for the most part) such as hath been already exhibited by Archimede and others, yet put here because the Book should not want somethingfor the Reader which hath not read such Authors, and for such as stand in need of the thing rather then the Proofe.

[mathematical diagram]

If there be a Cube whose side is bc, and a Sphere whose Axis ue is equall to bc (which here we put 7,) and an upright Cone adf, whose base is df, equall also to bc, and its altitude ao=bc.

1 Then the superficies of the Cube (being e­quall to the Square bcfd multiplied by 6 is 294.

2 And the superficies of the Sphere (being quadruple to the Circle uaeo) is 154.

3 And the Superficies of the Cone (being made by multiplying the side ad=√61¼ by the semicircumfe­rence auo=11) is 86, and not considerably more, to which adding the superficies of the Base 38½, the whole superficies of the Cone is 124½

4 And if there be a Prisme, whose Base and altitude are severally equall to the base of the Cube, or of any other rectangular Parallelepipe­don, the Prisme is the halfe of the Parallelelipi­pedon in solidity. Eucl. 12.7.

5 And if a Piramis insist on the same Base with the Prism, having equall altitude, the Pira­mis is two third parts of the Prism, or 2/4 of the Parallelepipedon.

6 The solidity of a Cone or Piramis is found by multlplying its altitude by ⅓ of the area of the Base.

The solidity of these other is found thus.

For the Cube, Multiply the side [7] by the square of the side [49] it gives the solidity of the Cube, which is 343.

For the Sphere, Multiply the Cube of the Diameter [343] by 11, and di­vide the product by 21, it makes the solidity of the Sphere, which is — 179⅔

The solidity of a Cylinder, whose Diameter and Altitude are the same with the Diameter of the Sphere, is made by multiplying the superficies of the base [38½] by the altitude [7] whereby the solidity is produced==269½

For the Cone, Euclide having pro­ved it to be the third part of the Cy­linder, Euc. 12.10. the solidity there­of is — 089⅚

A Fragment.

The superficies of the Fragment of a Sphere is found by multiplying the superficies of the whole Sphere by the altitude of the fragment, and divi­ding the Product by the Diameter of the Sphere, and adding to the quotient the superficies of the base of the Fragment.

The solidity of a Fragment (lesse then half a Sphere) is found thus.

From the Semidiameter of the Sphere, subtract the altitude of the Fragment, and by the remain [Page 115]multiply the area of the Base, and subtract the product from that which is made by multiplying the semi-axis of the Sphere into the convex super­ficies of the fragment: Lastly, divide the residue by 3, the quotient shall be the solidity sought for. If the fragment be more then halfe a Sphere, subtracting this from the whole, the greater frag­ment is thereby had.

This last Rule presupposeth the Axis of the Sphere to be known; but if it be not so, it may easily be found by the following analogie.

Let the altitude of the fragment be b the semidiameter thereof c, Make [...] and make f=cc/b Then it is manifest by the 13 of the 6 of Euclide, that b + f is equal to the diameter of the Circle, or to the Axis of the Sphere.

It is manifest by the former work, that the So­lidities of the Cone, Sphere, and Cilinder, being 89⅚ 179 4/6 269 3/6 are in proportion one to another as 1, 2, and 3, for the Cone is ⅓ and the Sphere⅔ of the Cilinder, but the superficies of the Sphere and Cilinder are equal excepting both the bases of the Cilinder

So by that which hath been said afore the Py­ramis, Prisme and Cube of equal base and alti­tude, are in solidity also as 1, 2, and 3.

There may be other parts of a Sphere beside those which here are called Fragments, (not to speak of those which are irregular & Multiform) which are either Cones or Piramids, whose bases lie in the superficies of the Sphere, and their Ver­tices at the center, the solidity of one of these is found by multiplying the third part of the Base by the altitude, (which here is the semiaxis) the product is the solidity: These fragments are those which are usually called Solid angles.

Example.

Let there be a Piramis of three sides, whose Base is 19¼, equall to ⅛ of the superficies of the Sphere, and the vertex thereof in the center, it is plain enough that this Piramis is the eight part of the Sphere in solidity.

Multiply 6 5/12 (the third of the base) by the perpendicular 3 6/12, the product is 22 11/24, which is the solidity of the Piramis, and multiplied by 8 gives 179⅔ equall to the whole Sphere.

The like for Cones in this case.

If the superficies of the Base be not wholly given, if any three things be given (if they be not the three angles) the three angles may yet thereby be found. And then, the Rule (which I had from my learned friend Mr. John Leeke) is, If the excesse of the three angles above 180 deg. [Page 117] be multiplied by halfe the Diameter of the Sphere, the superficies of any Sphericall Trian­gle is thereby produced.

[mathematical diagram]

This may be thus demonstrated, let the su­perficies of the oblique Triangle abc be requi­red, from the greatest angle c, let fall to the base ab, a perpendicular cz, produced towards f, and continue the base ab till it cut zc, produced in n, and ac produced in o, then the triangle nfo is equall to acz, being equiangular, and having the sides cz and nf equall, for either of them is equall to the diameter of the Sphere, or rather a Semicircle want the line cn.

Now if the angle at a be multiplyed by the diameter of the sphere, it makes the superficies, acobz, likewise if the angle fco that is acz, be multiplyed by the said diameter, they produce the superficies fnco. And both these superficies are equal to zbon + acz + fno, that is to a fourth part of the superficies of the sphere plus twice the triangle acz.

But zbon is produced by the diameter mul­tiplied by the right angle nzb, wherefore the di­ameter multipyed by the angles acz + caz is greater then the said diameter multiplied by a right angle by twice the triangle acz, and there­fore the angles acz + caz − azc multipy­ed by the diameter, produce twice the superfici­es acz or by the semidiameter, they produce it justly once.

And by the fame reason the angles bcz + cbz − bzc multiplyed by the Radius produce the Superficies bcz, which added to acz make the whole abc.

But the angles acz + caz − azc + bcz + cbz − bzc.

Are the same as the angles abc + bca + bac − 180d. which is the dif­ference whereby the three angles given exceed two right angles. So that this excesse multiplyed by Radius as aforesaid, produceth the superficies of the whole Triangle at first given, namely abc, which was to be demonstrated.

Many more such things might be taken out of Archimedes, as to measure the Superficies made by revolution of a Spirall line, and o­thers, which seldom occurre to any vulgar use: And for that cause, and also because the recitation of them would not benifit the other sort of Rea­ders which know them already, I medle no further, but will leave this Subject after one short Rule for measuring Hogsheads or Barrels, which is this.

From the area of the Circle of the greater di­ameter, multiplied by the length of the Vessel, subtract the area of the lesser, multiplied also by the length of the Vessell, and take the third part of the difference from the greater area multipli­ed by the length as before; the rest is the content sought for, in such measure as was the length, of the Vessel. That is Inches, if the Scale were so: of which 231 make a Wine Gallon, and 288¾ a Beer Gallon, or rather an Ale Gallon, according to some accounts, but not yet resolved fully.

Otherwise thus.

Let the square of the greater diameter in Inches be bb.

The square of the lesser diameter in Inches cc.

The length in Inches d.

And let the content in Inches sought for be aaa.

Then it will be [...]

Example.

Let it be

  • b=6
  • c=3
  • d=10
Then22 dbb = 7920
And11 dcc = .990
In all8910

Divide 8910 by 42 the quotient is 212 1/7, which is equall to aaa the content in inches, which was required. The very same number will come forth if one work by the former way, put­ting circumference to Diameter, as 22 to 7. But although this should be exactly true in one Ves­sel (which cannot be proved because of the irre­gularity of the Vessel) it would not be so in others, because of the irregularity (or diversity) of this irregularity.

In Mr. Spidals Extractions there are many Propositions of worth, and all undemonstrated, I will therefore in this place bestow a Demon­stration on one of the hardest of them, which is this.

Let it be required to divide any Triangle, as [Page 121] cng from a point without the Triangle as q, into two parts, of which one part shall have any pro­portion to the whole, given between two right lines, as here the lines cd and cg.

[mathematical diagram]

By the point q draw qa parallel to the nee­rer side cn, cutting gc, produced, in a. Make aq′ cn″ ac′ cf″, And ac′ bc″ cf′″.

And part cf into two equall parts in the point h, and draw the Diagonall bh, And make he=bh. Lastly, draw the line qe.

Then I say the Triangle cng is divided by the line qe into two parts cme and mnge so, that as cme is to the whole cng, so is cd to cg.

DEMONSTRATION.

Forasmuch as the right line cf is divided into two equall parts, in the point h, and to it is ad­ded another line fe, therefore the Rectangle cef + hfh is equall to the Square heh (by Euclide 2.6.) but he=hb, and therefore hbh=cef + hfh. But hc=hf, therefore hbh=cef + hch, and hch + bcb=hbh: Take away hch common to both Aequations, Then it is plain, that cef=bcb, because either of these is equall to hbh − hch: so then ce′ bc″ ef′″, Euc. 6.17. But ac′ bc″ cf′ ″ by construction, wherefore acf=cef, and seeing by construction it is aq′ cn″ cd′ cf″, therfore aqfc=den, & because ef′ cf″ ac′ ec″ therefore by composition ce′ cf″ ae′ ce″, but ae′ ce″ aq′ cm″, because the two last are parallels, Eucl. 6.2. And therefore aq′ cm″ ce′ cf″, and aqfc=mce, but also aqfc=ncd, as is proved before, and there­fore the Rectangles ncd=mce, and also the halves of them are equall, namely the triangles ndc and mce, but the Triangles ndc and ngc have that proportion as have their Bases, cd and cg, wherefore mce′ ngc″ cd′ cg″, and the line me is drawn from the point q which was to be proved.

I know this is demonstrated by others already, but I may aswell insert a Demonstration diffe­ring, [Page 123]as Mr. Spidall might write the same Propo­sition without proof.

Some Corollaries might be deduced from this Proposition, by considering the various analogies therein, which, I leave to the invention of the Reader.

Now if it were required to draw a line from a point given without a Circle given, through the Circle, so as to cut off an arch equall to an arch given, that may very easily be done in this man­ner.

The Cirele crd being gi­ven,

[mathematical diagram]

and a point without it at b, let it be required to draw from b to the in­ner Circumference at c, a line bdc, so as that the arch thereby cut off crd, may be equall to an arch given, for example, to 120 degrees.

From b draw the tangent br, and make

  • br=c
  • cd=b
  • db=a

Then it will be b + a′ c″ a′″, Euclid 3.36. Therefore aa + ba=cc, Euclide 6.17. Wherefore a may he found by the first Rule for plain Aequations, Chap. 2.

CHAP. X.

THe superficies of an Ellipsis may be easily found as neer the truth as that of a Circle, because it hath been proved by diverse to be a mean proportionall between the two Circles de­scribed severally upon the diameters of the Ellip­sis, and it is almost axiomatically evident by meer inspection of the figure following.

And therefore it is as easie to give an Ellipsis in any proportion to another Ellipsis, as to de­scribe any Ellipsis at all.

[mathematical diagram]

As for Example, Let the greatest Diameter of the Semi-ellipsis adc, be ac=28. then the semicircle described thereon shall be abc=88/2 and let the lesser diameter of the said Ellipsis be 2 do or fg=14.

Lastly, let it be required to describe an Ellipsis which should be to the Ellipsis adc, as 1 to 4.

Upon the line ac from o both wayes, set off [Page 125] fo and go, each of them equall to do, and divide do into two equall parts in h, then describe the Ellipsis which shall passe by the three points f, h, g, I say that the Ellipsis fhg is to the Ellipsis adc as 1 to 4.

For seeing the Circle abc is to the Circle fdg in diameter double, therefore abc=4 fdg, and of what parts soever abc is 16, of those fdg shall be 4.

And seeing the Ellipsis adc is a mean betwixt them, the said Ellipsis is 8 of the same parts.

Again, by the same reason the Circle fdg is quadruple to the Circle nhk.

Therefore of what parts soever fdg is 4, of those nhk shall be 1.

And seeing the Ellipsis fhg is a mean betwixt them, the said Ellipsis is 2 of the same parts.

But the Ellipsis given adc is 8.

And 2′ 8″ 1′ 4″, which was to be done.

In like sort having duly proportioned the Dia­meters of Circles, may be made Ellipses, in any proportion one to another, or in any proportion to a Circle given.

And the works may be proved by induction, as this also might have been, for seeing the circle abc=616, the Circle fdg=154, the Ellip­sis adc, a mean betwixt them, must be=308.

Again because the Circle fdg=154.

And the Circle nhk=038½

The Ellipsis fhg being a mean betwixt them must be=77.

But 77′ 308″ 1′ 4″, &c.

NOTE. 1

Herein I make use of that proportion which is betwixt 22 and 7 for the Circle to the Dia­meter for easinesse in accompt, small and whole numbers being also better attended and under­stood sooner by the Reader; and for no other cause: the more exact proportion being as 355 to 113, or (which is more used) as 360 to 114 5915492/10000000.

NOTE. 2

Hence it is manifest that the Content of the Lunula adbc comprehended by the Circle abc and the Ellipsis adc, (being according to this account halfe the Circle abc, that is 308.) As also the mixed figures adf and cdg (being here the residue of the Semicircle fdg, to the Semi-Ellipsis adc) may be found out as exactlie as the Superficies of a Circle. with which, until a further discovery, we must be content. And I have here noted it, to shew that investigation is not yet to be contemned, as if the thing sought [Page 127]were (not onely impossible but) uselesse, when so many neat Propositions might thereby be start­ed, as would (although not so absolutely necessa­ry for present use, yet) delight the modest eye with the novelty.

NOTE. 3

Moreover if the said Lunula adbc were com­posed of two Circles, there might be a rectiline figure given equal to the superficies thereof; That is, if the superficies of a Circle adc, were dou­ble to the superficies of the Circle abc, the lines ab, bc, being drawn, the triangle abc, would be equal to the Lunula adbc; as might be pro­ved if it were not easie, and well enough known already.

So that some figures of crooked lines, either differing in kind, or in quantity may be equalled with Rectiline figures, or numbers; And yet where Cirles of equal quantity include any Lu­nula or other figure, this cannot yet be done; So thinne is that Curtain which is drawn be­tween us and our desires.

NOTE 4.

Whereas in the former figure, the making of the Ellipses, adc, fhg; is not shewed; this may be here usefull to some: and it is as follows.

The greater Axis of the Ellipsis being equall to the diameter of a Circle abc, namely, to the right line ac, the other Axis to be taken at plea­sure according to the occasion; having here as­signed the line do for the halfe of the lesser Ax­is, draw from the Circle to the diameter ac per­pendiculars as many as you please. Then lastly, dividing each perpendicular into two parts pro­portionall with bd and do, in certain points, if by those points (of which the more, the better) a line be drawn with an even hand, that line shall passe also by the point d and be the Ellipsis required.

Otherwise, and more for Mcchanick use.

Having chosen the two Axes ac, and 2do and made them cut one another into two equal parts, and at right Angles in the point o; take the halfe of ac, and apply it both ways from the point d, to be Diameter ac in x and y, then in the points x and y which points x and y are cal­led the burning points, fix two pinns, of Iron, or wood (as the greatness of the Plain shall require) And upon the Plane place a string that com­passing both pinns shall reach just to the point d, or c, (for all is one) and there fasten the ends of the string together by a knot or otherwise, at which knot holding a Pencill, and carrying the Pencill round upon the Plane, about the pinns [Page 129]

[mathematical diagram]

with the string allways straight, the Ellipsis (whose halfe is adc) shall be thereby described.

Moreover (although I will not meddle much with this kinde of Geometry) seeing these things are already richly treated in Greeke and Latine, and not much more then named in any English Book that I have seen, I will write a little here of a Cone, and all the Sections thereof, comprehen­ded in one figure, and after take some principall Definitions, and one or two wayes of descri­bing the Sections, and drawing tangents to them, and some few other Problems out of Claudius Midorgius, not word for word, but as it shall seem convenient here.

CHAP. XI. Definition generall of a Cone.

A Cone is a Solid Body made by turning a Triangle round about upon a Plain, one side remaining alwayes in its place, the second side describing a Circle upon the plain, the third side describing the superficies of the Cone in the

[mathematical diagram]

aire; the first side is the Axis, the Circle, the Base, and the Vertex of the Triangle is the Vertex of the Cone. As let the plane Triangle abc repre­sent half a Cone, which is made by the Motion of the Triangle abx, about the Axis ax, the side xb describing a Circle upon the plane bc, of which bc is the Diameter and doth here repre­sent the base of the Cone, which added to the Su­perficies [Page 131]described by ab, moving about in the aire, composeth the whole superficies of the Cone: of which a, is called the Vertex, or top. And if the Angle axb, be (as here) a right An­gle, the Cone is called an upright, otherwise Scalenon, either of them may be cut by Sundry Planes, as first by a Plane passing through the Vertex a, and perpendicular to the plane of the Base bc, and this Section in upright Cones is an Isosceles Triangle, in the other Cones, a Scalenon Triangle, except it be ab=bc, for then in up­right Cones the Section is an Equilateral Trian­gle, in Scalenon Cones an Isosceles.

1. Now let this Triangle abc represent halfe the Cone as aforesaid, and then if a plain, as eboaz touch the Cone all along from b to a, and make right angles with bc the diameter of the base, and again, another plain fd parallel to eboaz cut the semi-cone bac, the section ir in the superficies of the Cone is halfe a Pa­rabola, the other halfe underneath, if the Cone be supposed entire, and is not to be projected in plano.

2 Again, if the Semi-cone bac be cut by another plain gkz, parallel to the Axis ax, the section in the superficies of the semicone, to wit gk shall be halfe an Hyperbola, and the like for the other halfe underneath, if the Cone were supposed intire, and further, whatsoever plain cutting the Semicone as aforesaid being [Page 132]produced shall concur with the plaine ba pro­duced towards z.

Thirdly, If the said Semi-cone be cut by a plain nph, neither of the former wayes, nor parallel, nor subcontrary to the base, the line in the super­ficies, namely nh is a Semiellipsis.

Subcontrary position is that where two like triangles are joyned at an equall (and then ver­ticall) angle, yet have not their bases parallel.

Lastly, if it be cut by a plain lorq parallel to the plain of the base, the section or is a Semi­circle.

Definition 1.

Opposite Sections are two Hyperbola's in op­posite superficies cut by the same plain.

Definition 2.

The Vertex of a Section is a point in the great­est curvature thereof, but more generally the point where any diameter cuts the Section, and where the Axis cuts is called the highest Vertex.

Definition 3.

Any two lines applyed within the Section, and equidistant, are called Ordinately applied, in re­spect of some diameter which divides them into two equall parts.

Definition 4.

Any line drawn so as it cuts the section, and divides the Ordinates into two equall parts, is called the Diameter of the Section, and if it di­vide them as aforesaid, and at right angles, it is the Axis, and so much of the Axis or diameter as lies betwixt the Vertex and any ordinate is called (in respect of that ordinate) the intercepted Axis, or intercepted Diameter, and those two diametets which mutually divide lines applied in the Section and parallel to the Diameters, into two equall parts are called Conjugate diameters, of which, as likewise of the oppsite Sections, I intend to say no more in this Tract.

Definition 5.

The transverse Diameter of an Hyperbola, is a right line in the intercepted diameter conti­nued without the Section, and is equall to the double of that line intercepted betwixt the Vertex and the center, and connects the Vertices of op­posite Sections: In an Ellipsis; or Circle, it is any whole Diameter: in the Hyperbola and El­lipsis, if it be the continuation of the Axis, or the Axis (in the later) it is called the transverse Axis. But the Parabola whose Diameters are all equidistant, hath no transverse Diameter, nor any center.

Definition 6.

The Center is a point where all the Diameters meet.

Definition 7.

The Figures of Hyperbola's, and Ellipses, and Circles are paralleligrams included between the transverse Diameter, and the contiguous Pa­rameter, of whicb those are called transverse sides, and these Coefficients by some.

Definition 8.

The said Parameter is a right line drawn to touch the Section at the end of the intercepted Diameter, to which all the Ordinates are paral­lel, and according to which they are compared, and valued, which is therefore called juxta quam possunt: and if it be contiguous to the Axis, it is called the right Parameter.

Definition 9.

The umbilicius, focus, or burning point in the Parabola, is a point in the Axis distant from the Vertex by a fourth part of the right Parameter.

But in the other two Sections, the burning points are assigned in the Axis of either Section, distant from either end of the transverse axis by the space of a right line that is the square root of the fourth part of the figure produced by the said transverse axis, and the right Parameter, which applied to the transverse axis is in the Hyperbola excedent in the Ellipsis deficient.

The same points in any Ellipsis whose diame­ters or diameter are given, may easily be found by the mechanique way of describing an Ellipsist a little before shewed. Wherein also it is plain that these points are as it were Centers proper to the generation of the Section.

CHAP. XII. Of the description of the Sections.

MAny are the methods Generall and Speci­all, which Midorgius shews to describe these three Sections, I will only mention one or two.

1. To describe a Parabola about any Diameter given with one of the ordinate lines.

Let the Diameter given be ab, and let bc be one of the ordinate lines applied unto it, by which the angle abc being given, joyn a and c by the right line ac, And let ab be divided into as many parts as you please, and through every such division draw right lines parallel to bc, and produce them, and make dk=√bc in dg, likewise el=√bc in eh, and fm=√bc in fi, and so of all the rest, and the points c, k, l, m, a, &c. shall be all in the same Section, so that a line drawn with an even hand by all the said points, shall be by the first [Page 136]Prop. of the second of Midorgius, the Parabola required.

[mathematical diagram]

And bc on the one side, is equall to bc on the other side, because by supposition, that, and all the parallels to it kd, le, mf, &c. ate those lines which are called Ordinates, or Ordinately applyed, and so ab in respect of bc, also ae in respect of le, &c. are the intercepted Diame­ters, or if the angle abc were a right angle, the intercepted Axes. Def. 4.

And if you make ad′ dk″ aq′″, and draw aq parallel to dk, then aq shall be the contigu­ous Parameter in respect of the intercepted dia­meter ad, and so may the Parameter by ab, or [Page 137]any other diameter given, be found, and there­fore the Parameter aq only being given, the Pa­rabola by points may easily be described.

2. About any Diameter, and one Ordinate line, to describe an Hyperbola known in kinde, in a plain by points.

Let ab be a diameter of the Hyperbola, and bc an Ordinate to it, comprehending the angle given, abc, and let the Section be of such a kinde, as that the transverse diameter to the contiguous parameter may be as r to s.

[mathematical diagram]

Make ab′ bc″ bd′″, And s′ r″ bd′ be″ and joyn the points d and e, and in the line ab take points how many soever, and by them points f, g, t, &c. draw lines parallel to bc, as fh, gn, ti, &c. the more the better, and making the triangle ded compleat, produce these paral­lels both wayes to the sides de, in the points h, n, i, &c. Lastly, making fk, gd, tl, &c. the square roots of the rectangles afh, agn, ati, &c. the points k, o, and l, shall be in the Hyperbola required: per 5. of 2. Midorg. And therefore a line drawn with an even hand to passe by the said points, shall be the Hyperbola required.

And the transverse Diameter thereof is the line ae. Wherefore, by the Proposition, if you make it r′ s″ ae′ ax″, then ax shall be the con­tiguous parameter to the intercepted diameter ba, supposing it drawn parallel to bc, fk, go; &c. Def. 8.

And the lines fk, go, tl, are all of those which are called Ordinates to ba, Def. 3.

3. About a diameter given, and an Ordinate, to describe by points in a plain; an Ellipsis known in kinde.

Let ab be any transverse diameter of the Ellipsis required, and cd one of the Ordinates drawn at any angle given, as acd, to describe as aforesaid; &c.

Make the rectangle bce equall to the square of cd, and by a and e draw the line ae, and pro­duce it to f, that is, so far as till it meets with bf, being made parallel to cd, and in ab take other points gh, through which draw lines parallel to cd. Lastly, to every Rectangle bgi, bhl, &c.

[mathematical diagram]

make squares equall, as the square of gk equall to the first, and of hm to the latter Rectangle, and so as many as you please; the points m, k, d, shall be (by the 3. of the second of Midorgius) in the same Ellipsis of which ab is the transverse diameter, and bf the contiguous parameter, [Page 140]wherefore a line drawn with an even hand by those points m, k, e, shall be the Ellipsis requi­red.

COROLLARY.

Hence it is evident, that having the transverse diameter of an Hyperbola, or an Ellipsis, the pa­rameter contiguous is easily found by applying any Ordinate to the Diameter, as kg, and drawing a parallel to it from b, for making bg′ kg″ ig′″ a line drawn from a to i shall meet bf in the point f, so as it shall there­by determine the parameter bf in the Ellipsis, and the work (though not the letters) is the same in the other.

RƲLES.

In a Section given, the Diameter is found by applying two ordinate or equidistant lines divi­ded both into halves, through which divisions the Diameter must passe. Def. 4.

Secondly, drawing two other equidistants different in situation from the former, and divi­ding them as aforesaid, you have another dia­meter.

Thirdly, produce both, and where they con­cur is the center of the Section. Def. 6.

Fourthly, Produce them still (in the Hyper­bola) till the space betwixt the Vertex and the Center be doubled, that doubled space is the [Page 141]transverse diameter: the Vertex is here meant at large, for that point of the Section through which the diameter passeth. Def. 5.

Fifthly, Having the center, an arch of a circle, any where within the Section, bisected, gives a point by which from the center must passe the Axis. Midor. 1.54.

I have shewed already how the burning points may be found in an Ellipsis. Def. 9.

Sixtly, In the Hyperbola let the transverse axis be b, the right parameter c produced till it make x equall to a mean betwixt them, bisect this mean in a, an arch drawn from a to the axis (the center being the center of the Section) shall there give the point desired. Midorgius 1.58.

The burning point of the Parabola is obvious out of the 9. Def.

To finde the Axis of a Parabola.

Seventhly, Because it hath no center, but all the Diameters are parallels, finde any one dia­meter by help of two ordinates, as aforesaid, and to it within the Section draw a perpendicular, which being produced both wayes just to the Se­ction, divide into two equall parts, and through the point of that division, draw a line parallel to the diameter found before, that parallel line is the axis required: The thing is so easie it needs no Example.

CHAP. XIII. To draw a tangent to any point assigned in any Section, or from any point without the Section.

NOt to trouble this little book with two ma­ny Figures, let the first Figure viz. of the Parabola be here resumed, which may serve by supposing the Diameter ab to be the Axis.

[mathematical diagram]

First let it be required to draw a line to touch the Parabola in the point m, and from m draw mf perpendicular to the Axis: produce the Axis ba, to z, making az=af and from z to m, draw the pricked line zm; the [Page 143]said line zm, is by 55 of the first of Midorgius the Tangent required.

If the point m, had been coincident with the point a, a perpendicular to ba, in a had been the Tangent, per. 17 of the first Ejusdem.

Now let there be a point given without the Section, (not in the Axis) at x, from which let it be required to draw a line to touch the Section.

From x draw xp parallel to ab cutting the Section in some point, as here at m.

And draw the Tangent zm, as aforesaid and make mp=mx.

And from p, draw a line parallel to zm, cut­ting the Section in r, and draw xr, then xr shall be the Tangent required by the said 55 of the first.

Secondly, let it be required to draw a Tangent to any point in the Hyperbola dac, which shall be repeated here also, wherein let the Diameter of the Section ab be supposed to be drawn, and the line go any ordinate, and the point o to be touched by a right line to be drawn as follows.

Having found the Center y, as is shewed in the former Rules, make yg′ ya″ yr′″, lastly from r to o draw the line roq for the Tangent required per ditto 53.1.

And so by conversion of the work, if the point r were given without the Section in some Dia­meter or Axis, there might from thence be drawn a right line to touch the Section in some place, as here it doth at [...].

NOTE.

If the point to be touched were in the Section, and in the vertex a, then by finding another diameter, another Vertex comes in place, and a in respect of this other Diameter will be a point in the Section, and a tangent to it as easily drawn as to o: As may be seen in Fig. 2.

Thirdly let the Ellipsis akb, be here represent­ed, and let it be required from any point given in, or without the Section, to draw a Tangent.

First in the Section at m. And from m to the other limb of the Section, draw here also any line as mm, and divide it into two equal parts in the point h, and finding the center g, draw by g, and h, the Diameter ab and produce it, making gh′ ga″ gx′″: lastly, from x draw xm for the tangent required.

Secondly, If the point x had been given without the Section, and required from thence to draw a Tangent to the Section in the point r, or where it falls, by conversion of the work. Make gx′ ga″ go′″ so have you the point o, from which a parallel to bf gives the point r, where a line drawn from x shall touch the Se­ction.

The working of these things in an Ellipsis is the same as in the Hyperbola, only seem unlike to them that consider not fully, because the [Page 145]center and transverse diameter of the Ellipsis lies within, and of the Hyperbola without the Se­ction.

[mathematical diagram]

And if h, or any point within a Section be given, and required through it to draw an Ordi­nate, that may be easily done, because it must be parallel to a tangent at the Vertex a.

Any Section given, to find that diameter thereof, which shall make an angle with the Ordinate to it, equall to an angle given.

If first the Section given be a Parabola, finde any diameter, and from the end or vertex thereof, [Page 146]draw a right line to the Section, making an an­gle with the said diameter equall to the angle given, to which if a parallel through the middle of the other right line be drawn, that parallel is the diameter required.

[mathematical diagram]

Let there be given therefore the Hyperbola bac, and the angle z, to finde the diameter eg, which with the Ordinate af shall make the angle ega=z. Finde the transverse axis ad, and the center e, and upon ad describe (by the 33. of the 3. of Euclide) a portion of a Circle dfa capable of an angle equall to z, then draw df and af, and through the middle of af draw eg the diameter required.

The work is altogether the same in an Ellipsis, only the lesser axis is to be used. Midor. 3.67.

Any Hyperbola being given, to finde the Asymptoti.

Finde any diameter of the Section, as ac, and the center e, and the transverse diameter am, and by the Vertex a draw the parameter ag contiguous to the diameter ac, that is, touching the Section in the Vertex a, and unto the rect­angle or parallelogram mag, make the Square or

[mathematical diagram]

Rhombus of ah equall, and divide ah into two equall parts in the point n, then the right line en drawn and produced shall be one of the A­symtotes, then producing ha untill ao be equall to an, the line eo shall be the other Asymtote, as appears by Prop. 57. Lib. 1. of Midorgius, Which he demonstrotes out of Propositions of his own Book thus.

Because ah, toucheth the Section, it is equi­distant to the Ordinates, per Coroll. 2, ad 17 pri­mi, But to the Rectangle or Parallelogram mag, that is to the figure comprehended of the two sides ma, and ag, is made equal the Square or Rhombus of ah, and an, is half of ah, therefore the square or Rhombus of an, is equal to a fourth part of the Square or Rhombus of ah, that is to the quadrant of the figure mag, and therefore by the 38. of the first and Coroll. to it, by con­version it may be shewed that the right line en, drawn from the Center and produced how far soever shall never meet with the Section bac and by the same reason, and because an=ao, eo drawn from the Center shall doe the like, &c.

From hence it appears, that the Asymptotes are lines drawn from the center of the Section; and produced, so as that inclining toward the section still more, shall never be coincident therewith.

More for the Parabola. Numerically.

Let the base be given in Numbers 20, that is, of what lenght soever, let it be parted into 20 equal parts.

And at any inclination to it, let there also be given a diameter; which divide into 100 parts.

And through all the other 9 divisions of the [Page 149]Semi-base, draw lines equidistant to the Diameter shortening them in this proportion, viz.

Of such parts as the Diameter is 100 let the next be 99, the next 96, the next 91, the fourth 84, the fift 75, the sixt 64, the seventh 51, the eighth 36, the ninth 19. A line drawn with an even hand by the ends of these lines shall be a Semiparabola.

The said Numbers are made thus, 10 in 10. 11 in 9. 12 in 8. 13 in 7. 14 in 6. 15 in 5. 16 in 4. 17 in 3. 18 in 2. and 19, in 1. Prop. 62. lib. 2.

And they differ just as the square Numbers im­mediately succeeding to Unity, viz. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. &c, by the quantity of the odd numbers intercepted, as 1, 3, 5, 7, 9, 11, 13, 15, 17, &c.

Which is the same proportion by which the degrees of Velocity of the falling of any thing to­ward the center of the earth are increased, as Ga­lileo hath sufficiently proved in his Dialogues.

And therefore the course of every Projectile or thing shot from Gun or Bow may easily be proved to be a Parabolical line.

And the making a Rectilone figure equal to a Parabola might be facilitated from hence, if it were not needlesse, the thing being already often done.

Moreover it is to be noted, that the equidistant lines thus drawn, may represent squares, because they differ as the Square numbers doe.

For an Hyperbola, Numerically.

The burning points and transverse axis being given, the Vertex is also given. Let the trans­verse axis be 80, the distance of each burning point 20 of the same parts, the said points a and b, the center a space 23, and the other cen­ter b, and space 103, describe arches, which shall cut where the Section is to passe, and so describing from the center a other arches, 34, 57, 100, and from the center b, with distances 114, 137, 180, other arches, so as the dist­ances from b may exceed 100, as much as the distances from a exceed 20.

Those arches of Circles shall intersect, and thereby give points by which the Hyperbola is to passe, by the 26. of the 2. of Midorgius.

For an Ellipsis, Numerically.

The burning points and Vertices being given (as they were before) the Ellipsis also may be described by numbers as followeth, let the one burning point be at a, the other at b, and let the diameter be z, the distance betwixt a and b let that be x equall to 100, and let it be [...] x″ 16′ 100″, Therefore also [Page 151] z=232, wherefore making the center b at se­verall spaces (more then 16, and lesse then 116, of such parts as z is 132) as 110, 97, 81, &c. describe arches. Again, making the center a with distances 22, 35, 51, and others, still the correspondent complements of the former di­stances to 132, draw other arches, which shall cut the former in points whereby the Ellipsis de­sired must passe, by the said 26 of the second. And it is plain from the generation of an El­lipsis, shewed in the instrumentall way before in this Book: for the string which describes it is alwayes equall to z + x, that is, 232, and so is 100 + 110 + 22 and 100 + 97 + 35, &c. wherefore this is evident.

And thus they that like this last way better, may accomplish the Section by number.

Moreover, put the diameter of a Parabola [...] of an inch ferè.

And let the whole base (inclined to the Dia­meter at angle 84 ferè) be c=150/64

Lastly, Let the perpendicular from the Ver­tex to the base be d=64/64

Multiply 150/64 by 64/64 the Product is 9600/4096 Of which two thirds is equall to the supersicies of the Parabola, and is 6400/4099 Of these parts the middle Parallel which was before 75 (when the diameter was supposed 100) is 50/64 which doubled is 100/64 that is 6400/4096 as be­fore.

So that in this case the residue of the Rectangle or Parallelogram, when the superficiall content of the Parabola is taken from it, and the length of the middle parallel, are both denominated by the same number, but this is left to the Reader to try by a Figure delineated by himselfe.

But what use might be made of this (if it were further urged) either in naturall or artificiall num­bers, I leave at this time also to the Readers in­quiry and study.

Here it may be noted, that a line being drawn to touch a Section in any point, if from that line in that point be raised a perpendicular, that per­pendicular is said to cut the Section in the said point at right angles.

For Example, let the right line ab touch the Ellipsis qpr in the point p, and let po be made perpendicular to ab in p, I say po is usually said [Page 153]to cut the Section qpr, at right angles. For, if any line drawn from the diameter qr to the Se­ction, may divide it at right angles, let op be sup­posed to do so, and ab a tangent in p, as before.

[mathematical diagram]

Now if the angle apo be a right angle, all is proved, if not, draw the line qs by the point p, to cut po at right angles. It is manifest that qs shall cut the Section in p, because it cuts the tan­gent there.

Wherefore the same right line po cuts two lines qpr, and qps at right angles in the point p, namely, where those two lines cut one the other, which is absurd.

And into like absurdities will that opinion lead one which affirms that any crooked line can make any angle with any line whatsoever which toucheth that crooked line.

For although Clavius against Peletarius and others may say that the angle of Con­tact (as they call it) made between a Circle and his Tangent is lesse then any acute [Page 154]angle made of right lines, yet seeing it is not di­visible into parts aliquotas or aliquantas, which can have any other measure then the whole, that is, that each of them is lesse then any acute angle made of right lines, for ths cause, and because it seems improper language to give the name of an angle to any space lying betwixt two lines, which although infinitely produced would never meet, I refuse to call it so.

Which space being rejected as nothing, or at most non-angular, then the angle apo being a right angle, the angle qpo is equall to it, and is a right angle in any Circle, or Section of a Cone, or any other crooked line, how much soever composed.

For herein it is the same with them, as with a Circle, namely, that in the point of any conta­ction, the angles on both sides immediately begun are equall.

CHAP. XIV.

I Shall here adde a little to shew the resolution of such Problemes, which seeming to require two unknown points at once, are without help of a Conique Section (in lines) inexplicable. And other Problemes may happen higher then these infinitely requiring four mean proportio­nals, or five or six. Or to divide an angle into [Page 155]five, seven or eleven parts, and appearing in Ae­quations of five, seven, or eleven dimensions, as need requires.

I will begin with the most simple of these, Namely:

Probl. 1. Betwixt any two right lines given, as r and s, to finde two mean proportionals.

Put a for the lesser mean required. And r < s Then rr′ aa″ a′ s″, Euc. 6.20.

And aaa=rrs Eucl. 6.16.

[mathematical diagram]

Describe the Semi-parabola adg, so as r may be equall to the right parameter of it, which [Page 156]may be done by Chap. 12. hereof and in the axis ah make ac=½r, and from c raise the perpen­dicular ce=½s. And making the center e, and Radius ea describe the circle fadx, cutting the section in d, and from d let fall a perpendicular to the axis ah in b, then the lines bd, ba are the two means required. For make r=1, and bd=a, then because of the Parabola ba=aa, for the ordinate db is a mean betwixt the para­meter which is unity, and the intercepted dia­meter ab, Chap. 12. And it rests only to be proved that s, or twice ce is equall to aaa, for 1′ a″ aa‴ aaa'''' that is r′ a″ aa‴ s'''' Draw dq, parallel to ba. Likewise draw de.

[mathematical diagram]

Then qe=½s − a, for qc=db=a. and ec=½s, likewise dq (that is bc) is aa − ½. [Page 157]the squares of which two, are, ¼ss − sa + aa, and aaaa − aa + ¼. which together are aaaa − sa + ¼ss + ¼. equal to the square of de: Eucl. 1.47. but de=ae, and the square of ae is ¼ss + ¼. And therefore also it is aaaa − sa + ¼ss + ¼=¼ss + ¼.

That is subtracting from each ¼ss + ¼ the re­sidue is aaaa − sa=0. or aaaa=sa that is aaa=s, which was to be proved.

This and all other solid aequations not trans­cending the biquadratique order, are explicable (as Des Cartes saith) by a small portion of any of the three Sections. Yet seeing he holds the Parabola the most convenient I make use of that also, and of his Examples in this former and the next succeeding Probleme, aswell because a Parabola is much easier fitted to the worke re­quired, as also for that the demonstration thereby is not so anxious as by the other.

Prob. 2 Now secondly let it be required, to divide any rectiline angle given into three equall parts, as the angle bag.

Suppose it already done by the lines ae, af, and draw the chord bg, and also ec parallel to fa, lastly draw be.

Now because of the similitude of the Triangles bae, bde, and ced, (for the angles are adg=aeb=bde=ecd) it it ab′ be″ de‴ dc''''.

Put bg=b and bc=a. And let the Radius ba be Unity.

Then de=aa and dc=aaa.

And because bg + dc=3be.

Therefore b + aaa=3a, or 3a − aaa=b

Which Aequation Pitiscus hath also in making of the Sines.

Now suppose the Parabola akf drawn so as aq the right parameter may be equall to ba, that is Unity, and the part of the axis ac may be e­quall to ½, and ac=2, then from e raise em perpendicular to the axis, and equall to ½b, and upon the center m, and the space ma, describe the Circle akftp, which shall cut the Parabo­la on that side remote from m, in two points kf, from which perpendiculars in g and d, shall be true roots of this Aequation, of which kg is the subtense of the third part of the arch required, and is equall to be, that is to a, and fd is the subtense of the third part of the complement thereof to a circle, and if on the same side where m is, as from the intersection at p be let fall pl perpendicular also to the Axis, then pl, is a false root of this Aequation, and equall in Magnitude to both the true ones, that is pl=fd + kg.

[mathematical diagram]

But if in stead of − aaa + 3a=+ b, as in this Example it was, the aequation had beene − aaa + 3a=− b, then the true roots had fallen on the side where the center m was, and the false root on that part remote from the said Center.

NOTE.

This Aequation − aaa + 3a − b=0 is naturally without the second terme aa, which is the cause that it hath the false Root not discern­ed [Page 160]by twice + or twice − succeeding, as hath been spoken of Chap. 4. If therefore one would have it so, he must fill up the second term, by aug­menting the root never so little, putting e − x=a.

The Demonstration of this Problem is as fol­loweth.

It is to be proved that kg, in the Section is e­qual to be the subtense of the third part of the an­gle given.

Put kg=y.

Then because of the Section. ag=yy.

From the center m, draw mk and ma, which are equal because of the circle.

And draw kn, parallel to ae, and produce me to it in n.

Then it is kn=ge=2 − yy.

The square therfore of kn, is 4 − 4yy + yyyy.

And mn, being equall to y plus halfe the sub­tense bg, call bg by the single letter, b, as before, then mn=y + ½b, the square of which being yy + by + ¼bb, add to it the former square of kn, that is 4 − 4yy + yyyy, it makes + 4 − 4yy + yyyy + yy, + by + ¼bb, equall to the square of the Hypothenusal mk.

Againe the square of ae, is 4, and the square of me, is ¼bb, which two squares are equall also to [Page 161]the square of mk, because mk=ma. Therfore 4 − 4yy + yyyy + yy + by + ¼bb==4 + ¼bb.

That is,

− 3yy + yyyy + by=0.

That is, (by adding on each part 3yy, and sub­tracting yyyy) + 3yy − yyyy=by.

Or lastly (dividing all by y) + 3y − yyy=b.

But this aequation is alike graduated and like affected as the first aequation + 3a − aaa=b.

Wherefore y=a.

But a=be and y=kg.

And therefore kg=be. Which was to be proved.

In like sort it might be proved that fd is a true root of the aequation 3a − aaa=b (in the first figure) and the subtense of the third part of the complement of the angle given bag to a Circle.

And by such working one may finde it evident that when a Circle cuts a Parabola in points, how many soever (the Vertex excepted) perpendicu­lars let fall from all those points to the axis, are all the severall roots of one and the same aequa­tion. Nor hath that aequation any more roots then those perpendiculars to the axis.

NOTE. 1.

In the aequation + aaa − bca=bbd, the construction differs somwhat from the former, for b being reputed unity, if c as here be signed with − the axis of the Parabola must be produced from the point c in the axis within the Section distant from a by ½ beyond the vertex, till the continuation be equal to ½c, and at the end there­of raise a perpendicular equal to ½d, at the end of that is the center of the circle desired. And ac­cording to this method may any aequation, not above biquadratical be resolved, after by taking away the second term (if there be any) by the se­cond Rule of Chapter the fourth it is reduced to such a form as this, aaa * bca=bbd, if the quantity unknown hath but three dimensiōs: or if it have four then thus aaaa * bcaa * bbda==bbbf. Or else taking b, for Unity, then thus aaa * caa=d, nd thus aaaa * caa, * da=f the signes + and − are here omit­ted: for they must be supplied as the nature of the Aequation requireth.

NOTE. 2.

Note that in this breviate the line b, is that which was ba in the example of trisection, and that which was r or unity in the example of two Meanes: Also the line c is that which in the for­mer example of trisection was 2ce, or 3. And if this quantity be nothing, then the perpendicular [Page 163]

[mathematical diagram]

equal to half d, is to be erected at the end of half b, or ½ set off from the vertex upon the Axis with­in, but if c have any length, then at the distance of ½c from that end, upon the axis. And this which hath been said is enough for all Cubiques.

Prob. 3.

But where the equation is aaaa − caa + da=f so placed as here, if there be + f and the Probleme be to find the value of the root a, then producing ma towards a, [Page 164]Make as equall to the right parameter of the Section, and make ax=f, and upon the dia­meter xs describe the Circle xhs cut by a per­pendicular to ma, namely ah in h, then making the center m and the space mh, describe the Cir­cle desired.

But if it be − f, as in this Example I put it, then after ah is found as before: upon the diame­ter am describe a circle, and in it from a apply a line ai=ah, and making the center m and the space mi describe the circle fik, which is the circle sought for.

Now this Circle fik may cut or touch the Parabola in 1, 2, 3, or 4, points, from all which perpendiculars let fall to the axis give all the roots of the Aequation, as well the true as false ones. Namely, if the quantity d be marked − then those Perpendiculars which are on that side the Parabola where the center m is, are the true Roots, but if it be + d, as here, the true roots are those of the other side, as gk and no, and those of the center side as fz, pq, are the false.

DEMONSTRATION.

Put ce=c/2 and draw me perpendicular to ag, and gl equall and parallel to it: lastly, [Page 165]Put gk=a, then ag=aa, and taking from it ae, that is ½c + ½, then ge=aa − − ½c − ½, whose square is aaaa − caa − − aa + ¼cc + ½c + ¼.

[mathematical diagram]

And because by construction gl=½d, there­fore kl=a + ½d, and the square of it is aa + da + ¼dd, which added to the former square of ge, it gives the square of km, that is a4 − caa + ¼cc + da + ¼dd + ½c + ¼.

Again, the square of ae is ¼cc + ½c + ¼.

To which adding the square of me, that is ¼dd the whole is the spuare of ma, to wit ¼cc + ¼dd + ½c + ¼.

But the square of ah, that is ai, is equal to f because sa=1 and xa=f between which ah, or ai, is a mean.

Therefore the square of mi is

¼cc + ¼dd + ½c + ¼ − f

But mi=mk.

Therefore their squares are equal, that is. aaaa − caa + ¼cc + ¼dd + da + ½c + ¼=¼cc + ¼dd + ½c + ¼ − f.

That is aaaa − caa + da=− f. or else aaaa=caa − da − f. which is the same aequation which was to be resolved, of which therefore gk, is a true root.

In like sort might no, be proved a true root which was to be demonstrated.

Des Cartes, demonstrates of all this no more but the case where the aequation is aaaa=caa − da + f, and leaves the Reader to please himself in finding proofes for the rest: I have chosen this case, to demonstrate, & have demon­strated the cases of the two Meanes, and trisecti­on not onely because some Readers may be as much pleased to have this done to their hand as left to doe themselves: but also that all might see [Page 167]that the generall way of demonstrating all sorts of cases, depends on these two things; first that the right parameter of the Parabola being always Unity, if any of the roots be put equal to a, the intercepted diameter will be always aa. Se­condly, there may be ever found two squares e­qual to two other squares, and either the first two, or second two equal to the square of Radius. By help of these two things may any case hereof be proved.

I will conclude with a Breviat of such equati­ons as are not resoluble by Ruler and compasse.

1 If there be as many vowels as consonants, and the vowels unequal.

As, ae − da=db, ae + da=db Or, − ae + da=db.

2 Though but of two dimensions and in fewest termes, as ae=bb, though such are solvable yet it may be by infinite ways, and therefore cannot be applied to any limited Proposition.

3 If there be but one Vowell, but cubically multiplied, or higher.

As, aaa=bbc. Or, aaaa=bbbc.

Where the Aequation being already in the least termes, and not to be brought down by any common divisor nor the homogeneal bbc, re­ducible to any solid more regular, as, to fff, it is irresoluble.

CHAP. XV.

HAving said (in the conclusion of the former Chapter) that the Aequation ae + da=db is (as by right lines and Circles only) irresoluble, I will here shew a Probleme, by resolving where­of the said aequation will be happened on, which is this following.

Probl. 1.

In any rectangle (bdca) given, from an angle in it [c] to draw a right line [cf] cutting one opposite side [bd in o] and concurring with the other [bq] produced in [f,] so as the intercepted line [fo] may be equall to [z] any other right line given.

[mathematical diagram]

Put bd=b ed=d do=a and bf=e

And because the Triangles bof, doe, are like, Therefore it is b − a′ a″ e′ d″, And db − da=ae, that is, ae + da=db.

So we are quickly come to the Aequation re­quired, which aequation having as many un­known quantities (as a, e) as known (to wit, b & d) is hitherto uselesse.

That the Probleme therefore may be solved, we must work another way, and bring it to a Solid Aequation, by making (for more conveni­ence) cd=b. fo=c. and bd=d. and bo=a.

Then d − a′ a″ b′ [...] and [...], And the square thereof [...] is equall to cc, by the 47. of the 1. of Euclide. That is, [...] Multiply all by the denominator dd − 2da + aa

It makes ddaa − 2daaa + aaaa + bbaa==ddcc − 2dcca + ccaa, That is, aaaa − 2daaa + bbaa − ccaa + ddaa + 2dcca=ddcc.

Make bb + dd − cc=ff, then ff shall be signed + because hereby supposition it shall be bb + dd > cc. And the aequation will c, aaaa − 2daaa + ffaa + 2dcca=ddcc.

Expunge the second terme which is − 2daaa, by the second Rule of the 4. Chap. And because the Rule is not fully exemplified there in the o­perosity thereof, I will here work it at large. Be­cause aaaa hath foure dimensions.

Therefore make 4′ 1″ 2d′ ½d″

Again, because the first and second term have different signes, therefore put e + ½d=a Chap. 4. Rule 2.

The new Aequation arising thereof will be.

[...]

The homogeneal − ddcc, is here put on the same side with the rest, because (for the present) it seems better to stand so, that it may be the last term, in relation to that which is gone before Chap. 4. Sect. 4. of the second Rule.

In this last aequation, it is manifest that the se­cond term 2deee, is (through contradiction of + and −) abolished, as was required.

And now because the quesititions root e must be found by help of a Parabola, as before in the like case was used, it is necessary to reduce the aequa­tion to some such form as hath been shewed be­fore, in the Note of the former Chap.

First therefore to reduce the third term, because d > f, and + 6/4dd, taken from − 3dd rests − 6/4dd > ff, make 6/4dd − ff=gg, so all of the third term shall be − ggee.

Likewise for the fourth term, if + 4/8ddd − 6/4ddd + dff, be summed up together the aggregate will be − ddd + ffd, make dd − ff + 2cc=hh, then all the fourth term will be + dhhe.

Now for the last term − ¼dddd − 1/16dddd=3/16dddd and therefore making 3/16dd − ¼ff=ll, the aggregate of the last terme is thereby − ddll, for ddcc is through contradiction of the signes annulled.

And now the Aequation is eeee − ggee + dhhe − ddll=o, make gg/d = m and hh/d=n, and ll/d=p, then the aequation will be eeee − dmee + ddne − dddp=o and making d=1, then the aequation fully redu­ded and rightly prepared is + eeee − mee + ne − p=o. (In reducing this or the like consider Chap. 5. Note. 2.)

Or eeee=mee − ne + p, which is alto­gether the same with that in the former Chapter, and the working of it is there shewed. Except on­ly because there the quantity f is signed −, and here the like quantity p is signed +, I shall (although this case only is demonstrated in Des Cartes) here demonstrate it thus.

[mathematical diagram]

Describe the Parabolaf ap, according to the pa­rameter d, (that is as) & let ae be the axis, & make ac=½d, ce=½m, and at right angles at e make me=½x, and draw the line mas, making as=d, and ax=p, and upon xs as a diame­ter describe the semicircle xhs, and from a to h [Page 173]raise the perpendicular ah, cutting the Circle in h, and with radius mh describe the arch hky, cutting the Section in k, and from k let fall a perpendicular to me produced in q, and draw the lines mk and mh.

DEMONSTRATION.

To prove gk=e Suppose it done, and because kg=qe=e, and me=½n, therefore mq=½n + e, and the square of it is ¼nn + ne + ee.

And because ae=½d + ½m, and gk=e, & because of the Parabola ga=ee, therefore also eg or qk=½m + ½d − ee, & the square thereof is + ¼mm + ½md + ¼dd − mee − dee + eeee. That is (because d is equall to Unity) + ¼mm + ½m + ¼ − mee − ee + e4 to which adde the former square of ½n + e,

And then the whole is + ¼mm + ½m + ¼ − mee − ee + eeee + ¼nn + ne + ee, That is, ¼mm + ½m + ¼ − mee + e4 + ¼nn + ne equall to the square of Radius mk.

Again, because me=½n, the square of it is ¼nn, And because ae=½m + ½, the square of that is ¼mm + ½m + ¼, which added toge­ther make ¼nn + ¼mm + ½m + ¼, for the square of ma, to which adde the square of ah, that is p (for ax=p, and as=1, and conse­quently by Eucl. 6.13. the square of ah is equal [Page 174]to p) the aggregate is ¼nn + ¼mm + ½m + ¼ + p. equall to the square of Radius mh, but mh=mk. And therefore this aggregate is e­quall to the former aggregate & one may see that these quantities + ¼mm + ¼nn + ¼ + ½m. are comune to both; And therefore the residuals are equall.

Namely, + eeee − mee + ne=+ p,

That is in Des Cartes his form.

eeee=+ mee − ne + p,

Which was the aequation to be resolved, and therefore gk=e, which was to be proved. And therefore it gk be added to ½d, the sum is equall to a the quesititious root of the first aequation, and is equall to bo. So that the point o which was first sought, is hereby found; and the Probleme satisfied; which was to be done.

ADVERTISEMENT.

Now that the different ways of writing equations may cause no confusion, let it be supposed to be written ever thus. eeee, mee, ne, p=o. the signes + and − to be supplyed as the occasion requires. Then,

1 If it be − m, the center of the desired Cir­cle is within the Section, or at least below the Vertex.

2 If it be + m, the said center is above the vertex, and ½m is applied upon the Axis produ­ced, from the point c, which is always in the axis [Page 175]within the Section, distant from the vertex a by halfe the parameter; and therefore in this case the line m cannot be lesse then the parameter d, o­therwise ths point d, would still fall within the Section.

3 If it be + n, those perpendiculars (let fall from the severall intersections of the Parabola and Circle to the Axis) which are on the same side with the center are the false roots, and the o­ther the true roots, but if it be − n, then just contrary.

4 Lastly, if it be − p, then one auxiliary circle­will serve, as here it doth, but if it be + p, then there must be another also, the describing of both which is shewed in the former Chapter.

Probl. 2. Ʋpon a line given as a Base to describe an Isos­celes triangle, so that an inward parallel Base may cut off two segments of the sides betwixt the bases, so that either segment may be equal to the inwad base, the perpendicular from the vertex to the said inward Base being also by supposition given.

Let there be given the line cd, and the line bg.

And let it be required upon cd, as a base, to describe the Isosceles bcd, so as the line bg, fal­ling at right angles with fe, equidistant to cd, the lines df, ef, and ec, may be equall each to other.

Put gr=e, and fe=a, And cd=b and bg=d, Then d′, d + e″, a′ b″,

[mathematical diagram]

That is da + ea=db. Euclid. 6.16. which is the same aequation as was first in the former Probleme; and therefore if there be in the roome of da + ea=db substituted another equation like that in the former, such is the aequation.

a4 − 2daaa + ffaa + 2 dcca − ddcc=o.

And that purged from the second term, as be­fore, there ariseth a second equation.

+ eeee − mee + ne − p=o

And lastly, if d be a parameter, according to which a Parabola is described, the root e, and consequently the root a, may be found as in the former.

And thus having shewed the method general for all Aequations which attain but 3 or 4 dimensi­ons, and exemplified it by Problemes which lead to such aequations; I now say that was the end of my present businesse. And if any still desire a lon­ger reach, I referre him to Des Cartes, who hath proceeded to aequations of 5. and 6. dimensions; by which foure Mean proportionals, and quin­quisection of Angles, and other sursolid Pro­blemes may be found and effected.

Note 1.

The first Probleme of this Chapter, as it is more composed then trisection, so it comprehends it; as may be seen by (not only Pappus and others who applied it to that end but) the following Example. In which, let there be an arch of a cir­cle, namely bc, and let it be required to divide the arch bc, into three equal parts, or (which is as good) to find the third part of the arch bc.

Find out the center e, and describe the circle, and draw the Diameter bg, and produce it to p, or further, as is need, and make cr the right sine

[mathematical diagram]

of the arch bc, and from the center e; draw ex parallel to cr, and complete the rectangle cxer, and by the first Probleme, draw cp to cut ex, in f, so as fp, may be equal to bg, then I say that g [...] is equal to a third part of bc.

From o through e, draw oez.

Now because eo is Radius, fp the diameter, [Page 178]and the angle fepa right angle, therefore the lines fo, po, eo, are severally equal.

And the angles zeb=geo. and peo=epo.

And also foe=2peo. Euclid. 1.32.

Therefore also foe=2zeb.

But foe, that is coz, being in the peripherie, is measured by half the arch cz, Euclid. 3.20. wherefore bz which is the measure of half the angle coz, is a fourth part of the whole arch cz; and consequently a third part of bc, and therefore go which is equal to bz, is also equal to a third of bc, which was to be proved.

NOTE 2. Mechanically.

Seeing in this Scheme the line fo, is ever equal to the Semidiameter eg, if in commune practise there is bc, or any other arch (not greater then a quadrant) to be trisected, laying a thin Ruler to touch the point c, and cut the diameter bg, produced in p, the point required, the compasse being opened to the just length of Radius eg, setting one foot in ex, and shifting the Ruler till the other foot fall in the peripherie at o, the point o, shall always be distant from g, by a third of bc, the doing of which (although it must not be called Mathematical, yet) is very neer as easie, and as free from erring as from a point given to a point given to draw a streight line; or upon a center given with a distance given to describe a circle; & from a given point in it to set off an arch [Page 179]equal to an arch given: And therefore I recom­mend this as the most simple and short and safe way for Mechanick use.

NOTE. 3

If the arch to be trisected be greater then a quadrant, then trisect the complement thereof to a semicircle; and the third of this complement taken from 60 degrees (which is always a given arch) leaves the third of thé arch required.

Inscription of Chords in a Circle, and making aequicrurall triangles whose angle at the base shall be to the angle at the vertex in any given proportion, are the same thing: for to inscribe a figure of 3, 4, 5, 6, 7, 8, 9, 10, 11, sides &c, findes such triangles whose said angles shall be as 2/2, 3/2, 4/2, 5/2, 6/2, 7/2, 8/2, 9/2, 10/2, &c, as is easie to be seen by the operation.

Quadrature of the Circle is that in which (as yet) onely Archimedes hath laboured with any successe; he having demonstrated that the Cir­cumference is to the Diameter lesser then as 22, to 7, and greater then 21 70/71, to 7, within which strict limits, a French man many years since found it to be in whole Numbers. thus,

As the whole circumference, is to the perime­ter of the inscribed Square: so is 10, to 9, that is. Quadrant′. Subtense″. 10′.9″. which is easie from practise; and may be proved. thus,

Put the Diameter=7, the halfe=3½, of [Page 180]which the squar is=49/4, and doubled is 49/2, whose square root is the side of the inscribed square. The whole perimeter therefore is 4√49/2, and the whole circumference is found by this Ana­logisme.

9′.10″.4√49/2′.40/9√49/2″.

It rests to be proved that 40/9√49/2 is greater then 21 70/71, and lesser then 22.

Now 4,949, is somthing lesse then √49/2 which multiplied by 40, and divided by 9, the quotient it 21, 995, which yet is greater then 21 70/71. for 995/1000 > 70/71.

Again 4, 950. is somthing greater then √49/2, which multiplied by 40, and divided by 9, as be­fore, the quotient is 22, so that 40/9√49/2 < 22. and 40/9√49/2 > 21 70/71. which was to be proved

CHAP. XVI. Of Dialling.

Probl. 1. Ʋpon any declining Plane, to finde the height of the Stile, and place of the Substile, the Decli­nation being first known.

THe lines ab and ed, drawn to cut at right angles in c, and the angle cad, made the complement of Horizontal elevation: upon the center c, and space ca, describe the quadrant aq, [Page 181]and make qx the declination, and draw the de­clining Plane xcs, and on the center c, and space cd, describe the arch ds, cutting the decliner in s, from whence let fall to dc produced, a perpen­dicular sr. And make cy=rs, and draw the

[mathematical diagram]

line ay for the substile. Secondly make yg=cr, and the angle ayg a right angle, and by g, draw ag for the Stile.

Then if from the point y, to the stile, be let fall a perpendicular: and a line equal to it set off from y, towards a, it shall be the semidiameter of the aequator. Which aeqaator being described a line drawn from the center thereof & produced to the point where the tangent gy would intersect ac, [Page 182]produced shal cut it into two equal parts, whereof that semicircle which is neerer to the line yg di­vided into 12 equal parts beginning at the said point of intersection of gy, & ac, produced & rec­koning both ways as the Plane yq, shall admit, a Ruler layd from the center of the aequator to every one of those divisions, shall intersect the tangent to the aequator, that is gy, continued through which intersections lines drawn from a, (the center of the Diall) shall be the hour lines.

For seeing it is known already in Trigonome­trie that, as Radius to tangent complement of the Elevation, so is the sine of the Declination to the tangent of the substiles distance from the Meri­dian, and if ac be radius, cd, is the co-tangent of Elevation, if that co-tangent be made Radius, it is manifest that the sine of the declination sr, shall be made thereby the tangent of substilar distance, yc. In like sort it might be proved that, as Ra­dius cd, to cd the co-sine of the Elevation adc, so rc the co-sine of declination, to rc the sine of yag.

Probl. 2. Of Declining Recliners.

To finde the Meridian of the Place, the Meri­dian of the Plane, and height of the Stile.

Let the parallel lines qs, lg, represent the base of the declining reclining plane; and make the angle gxn equal to the reclination,

And let qs, and xw, cut at right angle in a.

Make the angle xab, equal to the declination.

And draw the lines xn, and ab.

And making xc=xn, draw cb, for the Me­ridian of the place; or hour of 12.

Then make am=ab, and the angle amz, equal to the latitude, and draw the line mz.

Make ak=ax, and ad=an.

Draw the line kd, and (where is need) pro­duce it.

[mathematical diagram]

From z, let fall a perpendicular to kd, in h, and from b, (the center of the Dial) to the verti­cal Meridian xw, apply br=kh, for the Sub­stile.

Lastly unto the line br, in the point r, raise a perpendicular re,=zh, and draw a line from, b to e, for the Stile.

So the line re, produced both ways, shall be a tangent to the Aequator, whose semidiameter shall be a line let fall from the point r, perpendi­cular to the line be. And the rest of the Dial may be finished like a Vertical decliner, as in Prob. 1

DEMONSTRATION.

First the Center b, is chosen at pleasure, in any place of the Horizontal Meridian ab, for the Pa­rallels qs, lg, might be put neerer or further off and not alter the work at all, onely the Diagran would be lesser or greater accordingly.

Now because xc=xn, the face of the Reclining plane xn, shall cut the Vertical Meridi­an xw, and let it cut it in the point c.

And because the line bc, subtends both the Meridians ba, and ac, and is drawn upon the Plane from the Center, it shall represent the Me­ridian of the Place: for the Sun enlightening the point c, at 12 of clocke in the vertical xw, and the point b, at the same time in the horizontal meridian ab, it shall at the same time illuminate the whole line bc, so as the Stile of the Dial shall shadow the same justly at the same time.

Again, because the triangles anx, akd, are aequiangular and equal, if the point w, be the Ze­nith, the point x supposed to be laid upon the point k, the lines kd=xn, and the angle akd equal to the reclination, kd shall then truly re­present the reclining Plane.

Moreover am being equal to ab by constructi­on, and the angle amz equal to the latitude or elevation, the point z represents a Pole of the Aequator, because the Axis mz, and the Meridi­an az meet there.

[mathematical diagram]

If therefore from the Pole z, a meridian zh fall upon the Plane kd, (or the back side thereof) at right angles, it shall fall upon a point of the sub­stile in h, which point h therefore doth limit the substile on that part.

But the horizontal line in which b, the center of the dial is taken must limit it on the other part, to wit in the point x, or k, so that hk is the just length of the substile upon the Plane.

And because the substile must both passe through the Center b, and incline to the vertical xw (to [Page 186]which the plane it self inclines) the line br=kh being so placed is the substile.

Lastly, because zh is the neerest distance be­twixt the Pole and the Plane, er being equal thereunto, and perpendicular to the Substile, it shal be the length of the Stile, that is (where the Sub­stile br, is Radius) the tangent of the height of the Pole above the Plane, which is all which was to be proved.

This Plane declining 30 degrees and reclining 20 degrees, may serve for an example of planes reclining and declining which lie between the Zenith and the Pole.

The distance of the Meridian from the Hori­zon is the angle gbc=78.50′.

NOTE.

Unto any Declination given may be found a Reclination, so as that the plane composed shall have the Aequator in the Zenith: that is the Poles shall have no height above the Plane, onely by making an=az in the former figure, and drawing xn; the angle gxn is the thing sought.

Or if the Reclination gxn, were given, a De­clination might be found to doe the same, onely by making still az=an; and the angle azm equal to the height of the Aequator above the Ho­rizon; and ab=am; and drawing the line ab, the angle xab, shall be the declination sought for.

And those Planes so made are called meridi­onal [Page 187]Planes, because, they are parallel to the Axis of the Aequator. The Dial upon any of them is like the Polar Plane, having the same reason for the parallelisme of the hours; but because the Sub­stile may chance to fall neither upon our 12, nor 6, but any where else according to several decli­nations, the place of that must be found.

Which is done by finding a certain angle comprehended betwixt the Meridian of the place and the Meridian of the Plane, which angle is commonly called the angle of inclination of Me­ridians, and may be found as follows.

Describe the following Circle wnes for the Horizon, and draw ns for the Meridian, and we,

[mathematical diagram]

for the Prime Vertical, qgx, for the declining Plane; pr, the Meridian upon it, and pn the ele­vation of the Pole above the Horizon. Then in the [Page 188]triangle pgz right angled at g, it is evident by the first Axiome of Spherical triangles, that if you work by the artificial sines & tangents, & also chuse pz for the middle part, the Aequation will be this.

s.c.pz + Radius=tc.gzp + tc.gpz.

And therefore adding the sine of the elevation to Radius, and from the aggregate subtracting the tangent complement of the declination, the re­maine will be the tangent complement of the an­gle of inclination of Meridians, which angle sought is zpr for gpr is 90 d.

Example.

Let the Declination wzq be 60 degrees.

The Reclination zg, so much as may cause the Plane qgx to passe through the Pole at p, and let pr be supposed to passe by the pole of the circle qgx, that so the angle rpg, may be a right angle: and choose pz for the middle part in the triangle bgz, right angled at g, because in the triangle gxz, the sides xz and xg are quadrants, then in latitude 51. 32′.

The sine compl. pz, is the sine of the elvation.

That is sine 51 32′ log. 9893745

To which add Radius, sum is 19893745

From which take tc.gzp that is, t.30d.9761439

Remains tc.gpz or tang. 53d. 36′. = 10132306

The angle fpz, therefore is 53. 36′. Which is the angle of inclination of meridians sought for, which being divided by 15 (the number of deg. of [Page 189]the Aequator accompted for every hour) the quo­tient is 3 57/100, or rather 3 172/300. which shews the hour from Noon over which the stile must hang that is in the after noon 3 172/300 of clock if the de­clination be westerly, or 8 128/300. if easterly, this last the morning hour, the other afternoon.

Now this fraction being rectified (which every man that hath any skil in Arithmetique knows how to doe) and the intire hours of 8 and 9, or 4 and 3 being assigned, the rest may be found by tangents of arches encreasing stil from the substile by the quantity of 15 d. for every hour. And so the Dial may be made on paper.

But to place it right after it is made, the angle comprehended betwixt the substile and the hori­zontal line which is here the line lg upon the Reclining Plane, must also be found out: And may thus (working stil by Logarithmes) from the sine of the latitude plus Radius, take the sine complement of the Reclination, there shall remain the sine of an angle, which angle is the true di­stance of the substile from the Horizon. And must be set off from the horizontal line at the cen­ter of the dial westward if the declination (as here it is) be eastward: Or else eastward if the de­clination be westward. And so the dial shall be rightly made and situate.

Now though all this be most easie to all that know how to use the Logarithmes; yet that this may not depend thereon, the same things may be [Page 190]found out Geometrically by describing any Cir­cle at pleasure.

For first, Cotangent Declination′. Radius″. sine latitude′. tangent Inclination of Meridi­ans″.

Secondly, Cosine Reclination′, sine Lati­tude″; Radius′. sine distance of substile from Horizon″.

Which is enough, any circle will afford these naturally; for such as affect not the artificial, and the former Scheme will demonstrate this easily.

Prob. 3. To find the same in those called South Declining Inclining Planes.

Put the parallel lines p, q, for the horizontal Base of the inclining Plane.

The Declination, angle abc=60 d.

The Inclination pcd=58 d.

Make yz perpendicular to the line q in b.

Make also ce=cd.

And bdz equal to the Complement of the ele­vation.

Joyne b and z. And make bk=bz.

Through k (the line ab being first produced so far) draw ku parallel to zy, cutting the line qd in s.

And make bl=sk.

[mathematical diagram]

From l let fall a perpendicular to cd to wit lo.

Make hg=co.

Then draw eg for the substile. And,

Make gn=lo, and at right angles with eg.

And draw en for the Stile.

And ea for the houre of 12,

So the Dial may be finished like a Declining Verticall.

DEMONSTRATION.

Because abc=kbz, is equal to the decli­nation, therefore the line kba, is the horizontal Meridian upon the base pq. And bk=bz by construction, and the triangles dbk and dbz are equiangle and equal, having one commune side db; and a commune angle bdz, and bk=bz, therefore a right line passing from k, in the ho­rizontal Meridian to d, or e in the vertical Me­ridian zy, shall represent the Axis of the Aequa­tor; for the angle bdk equal to bdz, is the com­plement of the elevation by Construction; and dbk a right angle.

And therefore whensoever the poin k, is either shadowed, or inlightened, the point a is the same; and the point e also, because it is in the same Axis with k, is at the same time so affected; wherefore the center of the dial being at e, a line drawn from thence upon the Plane to a, shall be the hour of 12.

And because the Hypothenusa dk, or ek, is the Axis passing from e, in the inclining Plane by k, in the horizontal Meridian; and the point k being in the line kshg, a perpendicular let fall from thence to the Plane shall fall in the same line kshg.

Make ex=hg=co. Then a perpendicular from l to x, is the same with that from l to o, namely the line lo.

[mathematical diagram]

And because bl=sk. and cx=hg, there­fore a perpendicular let fall from k, to g, shall be equal to lx, or lo.

And because k (as hath been shewed) is a point in the Axis, and g, a point in the Dial Plane, the perpendicular kg, or lo, shall be the height, or (as some call it) the length of the Stile; or more properly (making eg, radius) the tangent of that height whereby the Pole is elevated above the Plane.

And the point g, where it cuts the Dial plane at right angles, shall be a point in the substile.

But e, or d, is the Center of the Dial, as hath been shewed already.

Therefore a line drawn from e, to g, that is the line eg, is the Substilar.

And because gn=lo, and the angle egn, is a right angle, therefore the line en, drawn by the points e, and n, is the Stile.

Enough is already written to shew how to find the Meridian, Substile, and Stile, in all de­clining verticals, and declining Horizontals, I mean declining and reclining Planes; In which there is this of brevity, that not onely the things, that is their Magnitude, but their places and Si­tuations upon the Planes, are all obvious together in the very working; or with a little transpositi­on made so.

I meant not to be general in this subject which is the reason why I have forborn to say any thing of Horizontals, Prime Verticals, Equinoctials, and Polar Dials. Yet because the book shall not be rendred (to some persons) uselesse for want of these; at the end hereof I purpose to append a Table (out of Kercherus) by which the Horizon­tal and Prime Verticals all over Europe especial­ly, may be made by the quantities of their Arches set down in the table and to be set off from the Meridian of the plane upon the Dial Plane, and may be measured upon any Circle there described at pleasure.

Aequinoctial Planes (I mean such as are so de­nominated from the planes not the Poles) are such as have one of the Poles for Zenith, upon [Page 195]these, a circle divided into 24 equal parts gives the hours; and a pin perpendicular to the center, (of any length) is the Stile.

Polar Planes have the Aequator in the Zenith; where these are proper Horizons, the Substile and Meridian of the Place are all one, the hours pa­rallel to it; their distances from it measured by tangents of 15, 30, 45, 60, 75, 90, &c, degrees according to any Radius; provided that Radius be the same with the height of the stile; which is a pin set upright in the center of the Circle to which the tangents belong.

East and West Planes in any latitude are of this kinde, differing onely in longitude, 90. de­grees; by which it comes to passe that in these the houre of 6. is the substile, and the rest of the worke (leaving out unnecessary hours) the same as in the former.

To place these right upon the plane draw a line parallel to the horizon; and in any conve­nient place from the horizontal line in an arch of some circle set off the latitude; included between the horizontal line & another, that other line, cut by a third line at right angles, the third line shall be the Aequator, the second the substile, the rest like that before. Prob. 1.

Probl. 4. To draw any Verticall Dial by help of an Ho­zontal dial, without any Aequator.

Making the Center x, and at any distance de­scribe [Page 196]a circle, on which (having made xa the Meridian) set off the horizontal Arches proper to the latitude (taken out of the table hereafter following, or any other way;) from the Meridian making thereby markes in the circle, by which, and the center x draw the horizontal lines x 1. x 2. x 3. &c. on both sides the Meridian to cut the declining plane (which in this example is the line rt, declining from the Prime Vertical os, as­much as is the angle oar that is almost seven de­grees) in the points 1, 2, 3, &c, on the one side; and 8, 9, 10, and 11, or as many of them as the plane will receive on the other side of the Meri­dian: to which points lines drawn from the cen­ter of the dial, that is z shall be the hour lines, the angle axp, is the elevation.

The said center z, is found out by making, az=ap, the said ap being always the sine of the latitude, where the line ax is the sine of the complement thereof, that is, having made xa the meridian, and os, the prime vertical, and the an­gle axp equal to the latitude, the point p, and the line ap, are thereby given, then having made za, perpendicular to rt, make za equal to ap.

Now to find the stile and substile, it is already shewed at first in the vertical decliner.

Concerning Azimuths, Almicanters, and other such things, I shall not say much, because where there needs most Art to describe them, [Page 197]

[mathematical diagram]

there they are most uselesse; serving but to make a Dial more blind, which looked too much a squint before. And although the Dialler have all prompt in his head; yet very seldom doth his hand so concurre herein as to inscribe these things in their right places, especially in oblique Planes.

Before I medle at all with these, it will be ne­cessary to proportion the Perpendicular Stile to the Plane.

CHAP. XVII. Of the Perpendicular Stile.

THis must be perpendicular to the substile, and the top thereof determined in the Stile, or axis.

If the Plane be smal, consider whether it be direct, or declining, and much declining.

If direct, the substile may be placed in the mid­dest, if declining, then on the part opposite to the declination.

The substile well placed, (and roome left for the figures) divide it into two parts, so as that part next the center of the Dial may be the tan­gent complement of the height of the Pole above the plane; and the other part the tangent com­plement of the Sun's meridional height in the be­ginning of that Tropique which is to be more re­mote from the center of the Dial.

And the Radius proper to these tangents shall be the perpendicular stile, to be placed in the point of Division in the substile, perpendicular there­unto.

Of the Signes, or Parallels.

A Signe is a twelfth part of the Ecliptique, and containes therefore 30 degrees.

A Parallel, according to the vulgar sense, is the Suns diurnal Motion day by day: And because there are 47 degrees from Tropique to Tropique, there may be so many Parallels, that is, circles which the Sun describes every 24 houres supposed Parallel to the Aequator though not exactly so; and although there are 47 of these yet in our lati­tude of 51.32′. we accompt but 9. viz. those which are the day from Sun to Sun when it is 8, [Page 199]9, 10, 11, 12, 13, 14, 15, or 16. hours long. The Description of these parallels and of the signes is made the same way: onely due respect must be had to the quantity of the Suns declination, for in all direct horizontals, the perpendicular stile being Radius, the tangent complement of the Suns height, in any signe or parallel, at any hour of of the day, set off from the foot of the said stile, and extended to the hour line, gives a marke, by which the parallel of that day shall passe. So that this Worke repeated so often as the number of paral­lels to be inscribed, and the hour lines require, shall give respective points enough in each hour to draw each parallel by.

Example.

In the latitude 51. 32′. the Sun being in Pisces (the beginning thereof) the degrees of the Suns height above the horizon at every hour being as followeth, that is, 25. 37′. at one of clock, 21. 49′. at two; 15. 57′. at three; 8. 32′. at foure, and the same for eight, nine, ten, and eleven re­spectively, if the perpendicular stile being Radius, the tangents of the complements of 25. 37′. 21. 49′. 15. 57′. 8. 32′. be applyed from the foot of the stile to the respective hour, that is, the co­tangent of 25. 37′. from the foot of the stile to the hours of 1. and 11. and so the others, they shall give points in every hour-line one, by which a line being drawn with an even hand shall be the [Page 200]Parallel at the beginning of Pisces. And the like of all the rest.

And therefore generally in verticals, as also in all recliners that is to say upon all planes what­soever, draw a horizontal dial proper to the plane, and inscribe the signes or parallels upon it, by set­ting off from the foot of the perpendicular stile, the tangents complements of the Suns height at every hour in the beginning of every such signe, above that plane taken as an horizon, the perpen­dicular stile being ever Radius; and at the end of these tangents so set off, upon every respective hour-line, will be a point, by which points, lines drawn with an even hand, shall give the parallels desired. This horizontal Dial being drawn in obscure lines, the Dial for the plane may be drawn afterwards. The Parallels serving which were drawn before.

Example.

Suppose (as M. Wells doth pag. 185) a plane declining 30 degrees, and reclining 55 degrees; the height of the Pole above the plane 19 degrees 25 minutes; the Suns height at the beginning

  • Of Taurus to be at the hours of 12h. 82d. 5′.
  • Of Taurus to be at the hours of 1 73 30
  • Of Taurus to be at the hours of 2 60 3
  • Of Taurus to be at the hours of 3 46 1
  • Of Taurus to be at the hours of 4 31 53
  • Of Taurus to be at the hours of 5 17 47

The tangents of the complements of 82, 5′. and 73, 30′. and 60, 3′. &c. set off from the foot of the perpendicular stile (the said stile being the Radius to those tangents) to the obscure horizon­tal hours of 12, 1, 2, &c. give the true distances between the foot of the stile and those auxiliary hours for the parallel of Taurus. And so the o­ther Parallels may be found.

It is true, the height of the Sun at every hour of the day, at the beginning of every signe in any latitude is not easily found out without Trigono­metrical Calculation by Logarithms of the sines & Tangents, or by trusting to Tables already Cal­culated, if any happen to be done for that latitude already, the way of making a table shall be shew­ed towards the end.

Of the Vertical Circles.

These are vulgarly called Azimuths; and are great Circles whose Poles lye in the horizon, and intersecting one another in the Zenith and Na­dir of the Place.

The whole Horizon being divided into 32 parts equal, these circles shewing those divisions are called points of the Compasse, and marked S.SbE.SSE. &c. Every one distant from o­ther by, 11¼ degrees.

But the better way of accompting them is 10, 20, 30, 40, 50, 60, 70, &c. degrees from the Meridian.

1 In all horizontal dials, the Perpendicular stile being chosen, making the foot thereof the center, at any convenient distance describe a cir­cle, and accompt from the meridian both ways arches equal to 10, 20, 30, &c. degrees, from which divisions right lines drawn to the-foot of the stile aforesaid, shall represent those Azimuths upon that dial.

2 Upon a Prime Vertical (or South) Dial, through the foot of the perpendicular stile draw a right line parallel to the horizon, and making the said stile radius, upon the parallel line set off both ways from the Meridian tangents of 10, 20, 30, 40, &c. degrees, through which divisions right lines drawn all at right angles with the parallel line shall be the Azimuths.

3 Upon any declining Vertical the same being done shall give the Azimuths of 10, 20, 30, &c. from the meridian of the plane, or from the Me­ridian of the place, just allowance being made for the distance of Meridians.

4 In South Declining Reclining Planes, the perpendicular stile being chosen, and made the Radius, the tangent complement of the Reclina­tion applyed from the foot of the said stile to the meridian of the place, shall determine the Zenith of the place, through which, and the foot of the stile, that is the Zenith of the plane, a right line drawn shall be a perpendicular to the horizontal line, which shall concurre with the aequator in the [Page 203]hour of 6, and the therefore if from the foot of the stile upon the said perpendicular towards the North (for the former application is made to­wards the South) be set off the tangent of the re­clination, a line drawn from the end thereof at right angles with it, shall be the horizontal line: upon which the tangents of 10, 20, 30, &c, (the secant of the reclination being now made Radi­us) set from the said right angle, lines drawn from them to the Zenith of the place shall be the A­zimuths.

5 The distance betwixt the Meridians being known, upon the horizontal line, the Azimuths which were accompted from the meridian of the plane may be fitted for accompt from the meridi­an of the place with easie.

For example, let that distance be the tangent of 20 deg. then that Azimuth which is 10 from the one, is 10 from the other also, and that which is 30 on the same side of the substile, is 10 on the o­ther side of the Meridian of the place, the like Method serves for any distance.

Note 1.

It may be noted, that although I have shewed the construction of a South reclining plane at the beginning hereof in a figure proper only to those planes which recline not further then the Pole, whereas in those that doe, and although there be some variation of the Scheme as you may see by [Page 204]comparing this with the former (at the first be­ginning of this subject) for the point h, which

[mathematical diagram]

there fell on that side of the vertical meridian zx towards q, here falls on the other side towards s, likewise the hour of 12, that is bc, did there fall betwixt the axis and the substile, but falls here be­twixt the substile and the horizontal meridian ba: yet this notwithstanding the construction is the very same in both.

NOTE. 2.

It may be further Noted, that as the Reclinati­on may increase, the points n, c, r, all approach still neerer to a, and when the reclination is 90 [Page 205]they are all coincident, and this vanisheth into an horizontal dial whose substile will be ba.

Also if the Declination be still increased, at last the points b, and m, will be coincident, and the dial plane will be parallel to the prime verti­cal, and the work a South dial whose substile is or may be za.

But whether the declination increase or not, if the reclination amount to 70d. or above, then the substilar line applyed (as before) from b, to the vertical xz, shall fall between the points a and x, upon the base of the plane. By which means the former method is rendered insufficient, or (at least) inconvenient in this case. Wherefore ha­ving described any semicircle on paper, and made therein the angle dab, equal to the declination

[mathematical diagram]

(which let be 50d.) and the angle fab equal to the reclination (which let be 75) make ba′ fo″. bc′. q″. (having first drawn the sines fo, and de, [Page 206]and the tangents bc,) so q becomes a line known let be br=q, and draw ra cutting the circle in n, draw also nm, for the sine of nb, and againe make. ba′. ma″. oa′. x″. x therefore being a line known, may be a sine, as let it be the sine of the arch gl, and let gp be the latitude 51. 32′. and ls the sine of lp: make again, ma′. ls″. ea′. z″. make th=z, the sine of the arch pt and make ha′. ta″. sa′. y″. Draw radius aw per­pendicular to bg, Lastly make ku=y, the sine of the arch gu: then shall the arch wn be the di­stance of the meridian of the place from the ho­rizon; and the arch pt the height of the stile, and the arch gu the substilar distance, which are all that were sought.

This, being according to the common way trodden in Trigonometry, I shall not need to prove.

In like sort when any such inconveniency shall happen in South declining incliners, they that would doe it without Logarithmes may work by these Analogismes.

1 Radius′. s inclination″. t declination′. t. b″.

2 s. declination′. radius″. s. b′. s. c″.

Then, c + complement of latitude taken from 180d. let the rest be called d.

3 s. c′. s. inclination″. s. d′. s. f″.

4 t. inclination′. s. b″. t. f′. s. g″.

Then b, is an arch whose complement to 90, [Page 207]is the distance betwixt the meridian and horizon.

Also c, is an arch which being added to the complement of latitude, and the aggregate taken from a Semicircle, the residue, namely d, is an arch composed to find f, which is an arch equal to the height of the stile, or Pole above the Plane.

Lastly, g or the complement thereof to 90 deg. is an arch equal to the distance of the substile from the meridian of the place.

And these are enough for any man that is but indifferently skilled, to finish the dial with, which being deduced from M. Wells, in his Chap. 20. I shall not need to prove. This is for such planes as incline more then the distance betwixt the Ze­nith and the Aequator.

Almicanters

Are lesser Circles of the Sphaere, and may be called the Parallels of declination from the ho­rizon; having in all respects the same relation and habitude to the Azimuths, that the Signes have to the meridians, although these are accompted by 15d. and those usually by 10d. and therefore as in the description of the Signes an horizontal dial proper to the plane being first (obscurely) de­lineated, it was shewed that the points through which the signes or Parallels must passe upon e­very hour, might be had by applying the tangents of the complements of the Suns height at those hours in those Paralles, from the foot of the per­pendicular [Page 208]stile to the respective hours; so here making use of that Azimuth which is perpendi­cular to the Plane; (which in all planes is that which passeth through the foot of the perpendi­cular stile) the rest of the Azimuths being also in­scribed, the tangents complements of the Suns height above the plane, when he is in any Azi­muth applyed from the foot of the stile to the said Azimuth gives a point, through which that height, or almicanter upon that Azimuth must passe.

Example, in the Triangle pzs, let there be given the complement of Elevation pz, the com­plement of the Declination ps, and the Azi­muth pzs, to find the complement of the Sun's height zs.

[mathematical diagram]

Suppose the side sz continued till a Meridian from p, cut it at right angles in c.

Then first it is tc. pz′. Radius″. s. c. pzc′ t. zc″.

Secondly, s. c. pz′. s. c. ps″. s. c. zc′. s. c. sc″.

So ze and se, being severally found, the diffe­rence betwixt them namely se − ze, is the com­plement of the Suns height above the Horizon.

Then find how high the Sun is above the plane of the dial at the same time, the tangent comple­ment of that height applyed, from the stiles foot to the Azimuth representing the angle pzs, gives upon it the Almicanters point, or passage.

Or because s, ps′. s, pzs″. s, zs′. s, zps″. the hour from Noon, that is the angle zps, is found, which will crosse the azimuth aforesaid in the same point also.

Which hour if it be uneven, and unfit to re­main with the rest, may be drawn obscurely.

Of the Jewish, Babylonish and Italian hours.

The Babylonish are accompted equal hours from Sun rising, and may be inscribed upon any Plane by help of those two parallels, which shew the longest, and shortest day consisting of intire hours, as here 16 and 8 hours, and of the Aequator; for a line drawn through the hour of 5 in the first, 7 in the aequator, and 9 in the other, is the hour of 1 from Sun rising.

Likewise in the same order, through 6. 8. and 10 shall passe the hour of 2, the like order in all.

In Winter when the parallel of 8 hours shall faile, the other two points will serve; because the hours to be drawn are right lines.

But after the first six hours are inscribed, the Ae­quator [Page 210]failing also, some other diurnal arch as of 9 or 10 hours must be described to supply that want.

The Italian hours are accompted 1, 2, 3, &c. from Sun setting; to inscribe these the same diur­nall arches will serve, and a line drawn through them in the hours 9. 7. and 5. after noon, (obser­ving the same order as before) shall be the hour of 1. the like through 7. 5. and 3. shall be the hour 23, the night hours of 9, 10, &c. are the morning hours produced.

The Jewish hours are reckoned like the Baby­lonish, from Sun-rising, but unequally, their sixt hour being noon; and every hour a twelfth part of the artificial day, of what length soever that be.

The vulgar hours proper to the Plane being first drawn, and the Diurnal arches of 15. 12. and 9. (if it may be) divide the degrees in each by 12. and the quotients by 15. or else (which is all one) divide the said arches by 180, the three quotients shall give the just times in hours or usual parts of hours from 12 of clock upon the two Parallels and the aequator; through which lines drawn by a Ruler shall be the Jewish hours desired.

Example, in latitude 51. 32′, the diurnal arch of 15 hours, is in degrees 225, which divided by 180 the quotient is 1¼. and so much the Jewish hours of 5 and 7 are distant from noon, an hour and quarter being a twelfth part of the diurnal arch of 15 hours, which hour and quarter being doubled, gives the place of 4 and 8, tripled the [Page 211]place of 3 and 9, &c. from noon, upon that paral­lel of 15 hours.

In like manner the diurnal arch of 9 hours, is 135 d. which divided by 180, quotient is 135/180 that is ¾ of an hour, which shews the place of the Jew­ish 7 and 5, to be three quarters after or before noon, and doubled is 1½. which gives the place of 8 and 4, all one with our 1½. and 10½. and so tripling and quadrupling and quintupling of ¾ gives the places of the other hours on this paral­lel of 9 degrees.

And these parts doubled and tripled as is said, will always (in this parallel and the former) fall upon even hours halves or quarters of our hours, which is the onely reason why these two parallels of 15 and 9 are preferred; there being no necessity of using them more then the tropiques or other parallels, onely this conveniency of even parts.

Lastly, in the diurnal arch of 12. that is, the Aequator the equall and unequal hours con­curre, that is, the Jewish hours of 5 and 7, with our hours of 11 and 1, so their 4 and 8, with our 2 and 10, &c. so that a line drawn from 1¼. in the arch of 15 to 1, in the aequator, and from thence to ¾ in the arch of 9 is their 7, &c.

The Circles of Position I omit, not for that the inscription of them in any plane is difficult, but where the labour is not much, and the use of the thing not any, I hold that labour too much.

The way to describe dials upon Rings, Qua­drants [Page 212]and Cilinders, as also Globes, and Con­cave-Hemispheres, I also passe over: not for the same reason as the other; for of all these there is much use, and pleasure in using; but because eve­ry man that shall have travailed through dialling, on planes, with the dresses thereto belonging, cannot possibly want so much ingeniosity, as may direct him to doe these without book.

CHAP. XVIII.

IT remaines onely to say somthing of the form of the Parallels upon Planes; which (when they are not circles, (as under the Poles) are all and always Conique Sections, as is shewed by Mydorgius lib. 4. Prop. 34. And here shall be shewed how at any time or place it may be known what Section it is; although this is not necessary, for the Dialler to know; because with­out knowing them he may draw them upon the plane, as hath been shewed already.

Take therefore these three briefe Rules, which by Aguilonius lib. 6. Prop. 83. are proved.

Rule. 1.

When the Sun is in any Parallel, if the plane of the dial be parallel, to a great Circle of the Sphere which toucheth the parallel, and the op­posite thereto, the projection of the shadow is a Parabola.

Rule. 2.

If the Dial plane be equidistant, to a great Circle which cuts the parallel, and the opposite, the shadow runs in an Hyperbola.

Rule 3

If the dial plane be equidistant, to a great Circle which neither toucheth, nor cutteth the Parallel, the shadow (of the perpendicular stile, for so we mean all this while) gives an Ellipsis.

Example.

Let the Poles elevation be (in the Scheme fol­low,)

  • the arch ep.
  • The Horizon we.
  • The Meridian wnes.
  • The Aequator aet.
  • cl the Ecliptique.
  • cy, xl the Tropiques.
  • bd any other Parallel.
  • n, s, Zenith and Nadir.

Now therefore it is clear, that the horizontal dial in latitude 51. 32. being equidistant to the great circle we, which cuts the tropiques, and all the parallels between them, as bd, or any other, is (according to that which hath been said) such, as that the shadow thereupon all the year long shall describe Hyperbolaes; but of different kinds, as it shall cut several parallels more or lesse unequally.

[mathematical diagram]

But if a dial were made parallel to the Eclip­tique cl, which toucheth the Tropiques, the sha­dows thereon when the Sun is in the Tropiques would be Parabola's.

Lastly, if the plane were equidistant, not to the aequator aet, but to some other plane mk, whose great circle neither touches any of the parallels, nor cuts them, the shadow there shall always trace some Ellipsis; not always the same, but lesser, as the Sun, or his parallel approacheth towards the aequator: but greater in those horizons which make more acute angles with the aequator; until at last the horizon and the aequator being coinci­dent, [Page 215]the projecture of the shadow shall be a Circle.

Likewise the horizon howsoever situate, if the Sun be in no Parallel, the Projecture is a right line.

It shall not need to bring hither the demon­strations, which Aguilonius useth to prove all this; for the whole matter with smal adoe may be manifested thus.

The Sun being in the southern signes, suppose the darke Cone cay, in North latitude to be cut by a plane cay, through the Vertex a, perpen­dicular to the center of the base, it gives the tri­angle cay, for the flat and under superficies of the semicone cay. And let ro, be the horizon, or dial plane, (for every dial plane is parallel to some horizon) and let it be equidistant to ca; then the commune section ro, (which while the Sun is in the tropique x, distinguisheth, or rather separateth the light from the darknesse) is by the Definition general, of Chap. 11. A Semiparabola, in like sort might be proved, if the Sun were in any other parallel, as suppose bd, for supposing the pricked line ro, to represent the dial plane, parallel to ba, the same conclusion follows by the said De­finition.

Secondly, let the dial plane be R ☉, parallel to the great circle we, which cuts the parallel cy, in ☉, then by the said Definition general, the commune section R ☉, which that day separates [Page 216]the light and darknesse upon the Plane, is a se­mi-hyperbola: or if instead of the semicone, one conceive the section of the Cone through the ver­tex and axis, to be the plane triangle cay, and R ☉ as a right line, to be onely the diameter of the section, the thing is the same, and the section by the second of the third definitions of Mydorgi­us, an Hyperbola.

Lastly, (to avoid confusion of lines) let the Sun be in the northern signes, and xal the darke Cone, and zl the dial plane in South latitude e­quidistant to the great circle mk, which neither toucheth nor cutteth the parallels, it is evident that zl, or any line equidistant to mk, shall cut the triangle xal, made by a section from the ver­tex, as before, in both the sides xa, and la, and is therefore by the third of the third Definitions of Mygorgius the diameter of an Ellipsis, or con­ceiving all by the general definition of Chap. 11. as before, it is half an Ellipsis.

And so any other line parallel to zl, and grea­ter then xl, is the diameter of a greater Ellipsis, or rather an Ellipsis of a greater Cone, which might be made by producing the lines ax, and al at pleasure.

And this is proofe enough, for all that hath been said of this matter; and enough hath been said (I hope) to make it intelligible.

I will now shew how the Suns height for every hour of the day, in the beginning of any signe, within any latitude may be found.

Let the Latitude of the Place, or the dial plane, be 19. 25′. and the signe the beginning of Tau­rus, declining 11. 30. as in Chap. 17.

And let it be required to find the Suns height, at any hour that day in that place.

Suppose any spherical triangle fit for the pur­pose, (no matter for exact delineation, so there be somthing to help the fancy,) as here the trian­gle zps, wherein let p be the pole of the Aequa­tor, z the Zenith of the place, and s the Sun; and

[mathematical diagram]

zps the distance betwixt the meridian of the Sun sp, and the meridian of the place zp, that is, the hour from noon.

The easiest way to work, is by Logarithmes of the sines and tangents, because so Addition and Subtraction supplies for Multiplication and Di­vision.

The side pz, is the complement of the latitude, and is 70. 35′. likewise the side ps, is the com­plement of the declination, and is 78. 30′. and the side sz, is the complement of the Suns height, [Page 218]or the complement of the thing required. Now then suppose the perpendicular zr, to cut ps in r at right angles; also suppose the hour required to be 4 or 8 of clock, and 4 an afternoon hour, then zps=60. which is given, to find sz, or rather the complement of sz.

In the right angled triangle zrp, choose the angle zpr, for the middle part. Then it will be s.c. 60 + radius, whose log. are 196989700

Equal t.c. pz + t. pr subst. t.c. pz log. 95471377

Remaines the tan. of pr=t. 54. 49. log. 101518323

Take 54. 49′. from 78. 30′. the rest 23. 41′. is equal to sr. But it is known in Trigonometrie.

That s. c. rp′. s. c. rs″. s. c. pz′. s. c. sz″.

Adde the two middle most.

That is, to s. c. 23. 41′. 99617909

Adde s. c. 70. 35′. 95217074

Summe is = 194834983

From which take s. c. rp (54. 49′.) 97605692

Remaines s. c. sz=31. 53. 97229291

Which is the height of the Sun above that Ho­rizon, at 4 after noon or 8 before noon. And so at two operations, may any altitude, for any hour given, above this, or any other horizon be cer­tainly found. If any like the natural sines and tangents better, he hath three things given to find a fourth, for first.

t. c, pz′. s. c. rpz″. Radius′ t. pr″. Secondly s. c. pr′. s. c. sr″. s. c. pz′. sc. sz″. as before.

The perpendicular zr, shall ever (for the hour betwixt 6 and noon) fall within the triangles, be­cause the angles zsp, and zps are both acute, as in Note 2. of Definition 2.

At 6 of clock the angle zps, is 90d. and

s. c. zs + r=s. c. pz + s. c. ps.

I put r, for Radius, always for abridgment.

At the hours after 6 at evening, or before in morning the said angle zps, is obtuse; then the same triangle remaining.

It is, s. c. pr′. s. c. pr + ps″. s. c. zp′. s. c. zs″.

The arch pr, being first found, as before. Or it is the same in hours equidistant from 6.

A Table of semidiurnal arches, in the beginning of e­very signe for 32 diverse Latitudes.
Po.. alt ♋♊ ♌♈ ♎♓ ♍♒ ♐♑
D.d. ′d. ′d. ′d.d. ′d. ′d. ′
35107 44104 5698 119081 4975 472 16
36108 25105 3098 309081 3074 3071 35
37109 8106 698 499081 1173 5470 52
38109 52106 4299 99080 5173 1870 8
39110 37107 2099 299080 3172 4069 23
40111 24107 5999 509080 1072 168 38
41112 12108 39100 119079 4971 2167 48
42113 3109 21100 339079 2770 3966 57
43113 55110 4100 569079 469 5666 5
44114 50110 49101 249078 4069 1165 10
45115 46111 35101 449078 1668 2564 14
46116 46112 24102 109077 5067 3663 14
47117 48113 14102 369077 2066 4662 12
48118 53114 7103 49076 5665 5361 7
49120 1115 2103 329076 2864 5859 59
50121 13116 0104 29075 5864 058 47
51122 29117 1104 339075 2762 5957 31
52123 49118 6105 69074 5461 5456 11
53125 15119 14105 409074 2060 4654 45
54126 46120 26106 169073 4459 3453 14
55128 23121 42106 539073 758 1851 37
56130 8123 3107 339072 2756 5749 52
57132 2124 31108 159071 4555 2947 58
58134 6126 4109 09070 0053 5645 54
59136 21127 46109 479070 1352 1443 39
60138 52129 35110 389069 2250 2541 8
61141 40131 39111 329068 1848 2538 20
62144 52133 47112 309067 3046 1335 8
63148 35136 13113 329066 2843 4731 25
64153 3138 58114 399065 2141 226 57
65158 49142 6115 529064 837 5421 11
66167 35145 44117 119062 4934 1612 25
A Table of the amplitude in the beginning of every signe, for 27. Elevations of the Pole.
Eleva. of the Pole♋♊ ♌♉ ♍♈ ♎
    In these 2 signes there is no ampl.
D.d. ′d. ′d. ′
3629 2925 1314 15In the other op­posite signes ♏, ♓; ♐, ♒ and ♑, the same Table may serve, giving to op­posites equal ampli­tude as to ♑, in each degree the same as to ♋, to ♐ and ♒, the same as to ♊ and ♌, lastly to ♏ and ♓, the same as to ♍, and ♉.
3729 5525 3414 26
3830 2125 5714 38
3930 4926 2014 51
4031 1926 4815 4
4131 5127 1115 18
4232 2427 3815 32
4332 5928 715 48
4433 3728 3816 4
4534 1629 1116 21
4634 5929 4516 39
4735 4330 2216 58
4836 3131 117 19
4937 2231 4217 40
5038 1732 2618 3
5139 1533 1318 27
5240 1834 318 52
5341 2634 5719 19
5442 3935 5519 48
5543 5836 5720 19
5645 2438 420 51
5746 5939 1621 26
5848 4340 3522 4
5950 3842 122 44
6052 4743 3523 28
6155 1345 2024 15
6258 147 1525 5
A Table of arches of the Horizon intercepted betwixt the Meridian, and each hour line upon the Diall Plane, for Horizontall and Verticall Dials, Calculated for 21 Elevati­ons of the Pole.
The Altitude of the Pole for Horizontals.1211.1.10.2.9.3.8.4.7.5.6.The Altitude of the Pole in Verticals.
 d. ′d. ′d. ′d. ′d. ′d. ′d. 
35 8 4318 1829 4944 4964 359055
36 8 5718 4630 3245 3065 29 54
37 9 1019 931 246 1166 0 53
38 9 2219 3431 3746 5066 29 52
39 9 3319 5832 1147 2866 55 51
40 9 4520 2132 4448 767 21 50
41 9 5720 4433 1648 3967 47 49
42 10 1021 733 4649 1268 11 48
43 10 2221 2934 1849 4468 33 47
44 10 3221 5134 4750 1668 54 46
45 10 4322 1235 1750 4669 15 45
46 10 5422 3335 4451 1569 35 44
47 11 522 5336 1151 4269 53 43
48 11 1723 1336 3752 970 11 42
49 11 2523 3337 352 3570 28 41
50 11 3523 5237 2853 070 43 40
51 11 4524 937 5253 2470 59 39
52 11 5524 2738 1553 4671 13 38
53 12 524 4338 3754 871 28 37
54 12 1325 238 5854 2971 41 36
55 12 2225 1839 1954 4971 54 35

These Tables need no explanation, the use of them being evident. But if they prove not sa­tisfactorie, for want of calculation, for further degrees of Elevation; or for want of halves, and quarters of degrees, or the like of hours, they are as I had them out of Kercherus his Ars Magna. Neverthelesse I will shew the making of them, whereby any man may fit them for his own pur­pose, and for his place, (if it happen without these limits) by his own calculation, as followeth.

First for the Table A.

Radius′ t. c. Elevation″. t. c′. Declination′. b″.

This b, is the sine complement of the angle at the Pole, which shews the hour from Noon, in Winter; and the hour from Midnight in Summer, wherein the Sun riseth, having declination, which declination is ready in tables, the making of which shall be shewed anon and also the table of the Suns declination at the end hereof shall ensue.

The angle at the Pole, so found being divided by 15. shewes in Winter the semidiurnal arch in hours, which was had by the first working in de­grees and minutes.

And in Summer the seminocturnal; whose complement to 180 degrees or to 12 hours, is the thing required, here all the time from the vernal to the Autumnal Aequinoctial, is called Summer.

Secondly, for the Table B.

s. c. Elevation′, s. Declination″, Radius′. s. c. Azimuth″, which Azimuth being compared with 90 d. difference is the Amplitude.

Example, for Elevation 40. initio ♊

To Radius log. 10000000

Adde Sine declination 20. 13′. log. 09538537

Summe 19538537

Subtract sine complement elevation,

Remaines Sine complement Azimuth, which complement here is the Amplitude 9654283

The arch belonging to sine 9654283. being sought in the Canon, is 26. 48′. which is the Amplitude required, where the Elevation is 40 degrees.

Thirdly, for the Table C.

First, to find any horizontal arch for any hour, as for Example 3, or 9.

Radius′. t. of the hour in degrees, that is here t. 45″.

Sine Elevation′. t. of the arch required″.

Or else, t. c. 45′. r″. s. Elevation′. t. arch re­quired″.

Secondly, in a Prime Vertical.

Radius′. t. hour″. s. c. Elevation′. t. of the arch″.

It is at first in the Symbols, Chap. 1. advertised [Page 225]that the letters s. and t. or s. c. and t c. signifie the sine, and tangent; or sine complement and tan­gent complement of an arch or angle.

And working by the Logarithmes of the sines and tangents, the former Analogismes happen not but in their stead certain Aequalities, or Aequa­tions, as follow.

For the Table A.

tc. Elevation + tc. declination=Rad: + b.

Which b, is the co-sine of the thing required, that is, of the angle at the Pole, which divided by 15, gives the time.

For the Table B.

Rad: + s. declination=sc. elevation + sc. Azimuth.

For the Table C.

s. Elevation + t. the hour=Radius + t. the arch, or more readily thus, tc. hour′ r″ s. Elevat′ t. of arch from the substile″.

It must still be remembred that r, stands for Radius.

The Elevation is always taken for the height of the Pole above the horizon, which horizon is the dial plane.

In other planes, as the Prime Vertical, and all other verticals, the height of the Pole above the plane must be used, having therefore found that, call it p, or else call the declination q, then s.q + s. elevat.=r + s. p.

And, r + s, p,=tc. the hour + t, the arch of that hours distance from the substile.

So after still till all the hours be found, this la­ter work must be repeated.

Whereas we use tc. the hour, and t. the arch, by the hour is always meant the angle at the Pole, or the space there included between any hour, and the substile; as 15d. for 1, 30d. for 2, &c. The arch is the distance of any hour from the substile measured in the arch of a circle, whose center is the center of the dial, when it is projected upon the plane.

To find the Declination of a Place.

The declination is an arch of a great circle passing through the Poles of the World, and the center of the place whose declination is sought, intercepted between the said center and the Ae­quator.

If the place have no latitude, that is, if it be in the Ecliptique, the neerest distance from Aries or Libra being given, call it b.

Then r′ s. 23d. 32 m″ s. b′ s. e″ and e, is the declination required: working by the Natural Sines.

Secondly, if the place have latitude, that being given, or found in Tables, and the right angle which the circle of latitude makes with the Eclip­tique, (for all circles of latitude do so, as the cir­cles of declination do with the Aequator) and the [Page 227]next distance to Aries or Libra being also given.

1 Then if the place lie betwixt the Ecliptique and the aequator, call the neerest distance to ♈ or ♎, b, as before.

And the latitude given c. It is Logarichmical­ly s. c + r=s, b + s, a & a, is an angle, which being taken out of 23. 32′. leaves an angle, which angle call d, then s, b + s, d=r + s, e.

And e, is the declination required.

2 If the place lie betwixt the Ecliptique and the Pole, the angle a, found as before, must be added to 23. 32′. and call the sum f, then s, b + s, f=r + s, e, &c.

3 Lastly, let the place lie betwixt the aequator and the other Pole, then s, c + r=s, b + s, a, and from a, subtracting 23. 32′. call the rest g.

Then s, b + s, g=r + s, e, and e, the decli­nation.

To find the right Ascension of a Place.

If it be in the Ecliptique, as the Sun is, let the neerest distance from Aries be called still b. And working by Logarithms, it is r + sc, 23. 32′.==tc, b + t, a.

And a is the right Ascension.

2 If the place have latitude, call it still c, and let the declination found with latitude, as before, be called, q.

Then sc, b, + r=sc, q + sc, a and a, is the right ascension; or (between ♋ and ♎) the complement of it.

To find the Ascensional Difference.

Thus t. elevation + t. declination=r + s.y.

And y, is the Ascensional difference.

To find the, Oblique Ascension.

In the southern signes adde the Ascensional dif­ference to the right Ascension: or in the northern signes subtract the same from the right Ascension, the sum in Winter, and the remainer in Summer, is the oblique Ascension.

That which hath been last said concerning the Declination, Right Ascension, Ascensional diffe­ence and oblique Ascension, may be illustrated, and demonstrated also out of this figure, at least with some small variationr.

[mathematical diagram]

In which let ho, be the horizon, aeq, the aequator, [Page 229] pp the Axis of the World, ec the Ecliptique, mn, the parallel in which the Sun is at d.

po the Elevation, z the points ♈ and ♎, bz. the neerest distance from one of those points.

Which is supposed given, or known, dr the de­clination.

zr the right Ascension of the Sun: or some­times the complement of it.

The angle xpo, is the complement of the As­censional difference, yq the measure of it.

Therefore here in Winter ry, is the oblique ascension, but the Sun being now supposed to be in d, that is 20 degrees of Taurus, by that which hath been said before, rz − zy, is the oblique ascension.

All these may be found (enough being given) as first in the triangle drz right angled at r, are given the side dz=5 cd. the angle dzr=23.32′. and the angle r 90 d. to find the declination dr. Working by the Artificial sines and tangents, put br, for the middle part, therefore, s. dr + radius,=s. dz + s. dzr, which resolved into propor­tionalty by the 14. of the 6. of Euclide, will be.

Radius′ s.dz″ s.dzr′ s. dr″ or,

Alternately, Radius′ s.dzr″ s.dz′ s.dr″, the thing sought. And the Analogisme the very same with that shewed before for finding the de­clination.

Then for the right ascension, zr make dzr the middle part, it is sc. dzr + Radius=tc. dz + t. zr.

[...]
[...]

And resolved, tc.dz′ Rad″: sc. dzr′ t. zr″.

That is Radius′ t. dz″ sc. dzr′ t. zr″. which is the same Analogisme with that before, for finding the right ascension.

Secondly for the ascensional difference in the triangle xpo, right angled at o, put the angle xpo, for the middle part.

Then sc. xpo + Radius=t.po + tc.xp but tc. xp=t. zy.

That is, Rad′ t.po″ t.xy. (or br′) sc.xpo″

And the complement of xpo, is xpz.

Whose measure is zy, the ascensional diffe­rence sought for.

Example of all, first for the declination rd, put rd for the middle part.

Unto, s. dz, 50. log. 9884254

Adde, s. dzr, 2 3 ½. log. 9601570

Summe is 19485824

And subtracting Radius remaines.

Which is the sine of 17.49′. for the declina­tion.

Then for the right Ascension rz, put that for the middle part, and

Unto, t. dr, 17.49. log. 9507027

Adde, tc. dzr, 66.28 log. 10361007

Summe 19868034

Whence subtracting Radius, remain. 9868034

The sine of 47.33′, for rz: the right ascension.

Secondly, for the ascensional difference zpx, in the triangle xpo, putting the angle xpo, for the middle part,

Unto t. po, 51.32′. log. 10099913

Adde, tc.px, that is t.br, 17.49′. log. 09507027

Summe is 19606940

From which taking Radius, rests 09606940

The Sine complement of xpo, or the sine of xpz=23.51′. the Ascensional difference.

Which being taken from rz, the right ascension 47 33. remaines rz − zy=23.42. for the ob­lique ascension.

Note 1.

The complement of the ascensional difference is equal to the quantity of hours and parts be­twixt midnight and sunrising.

NOTE. 2.

The oblique ascension of the Sunne being taken, from the right ascension in Summer, the residue is equal to the excesse, whereby the semi­diurnal arch is more then 6 hours: also the right ascension taken from the oblique ascension in Winter, the rest is the defect whereby the semi­diurnal arch is lesse then 6 hours. And the right and oblique ascension are neerer to aequality, as the Sun attaineth neer either ♈ or ♎.

If the place be not in the Ecliptique, but hath [Page 232]latitude, as the asterisme ⚹ in the last figure, if there be given that latitude d⚹, and the distance from the next aequinoctial point z ⚹, these with the right angle at d, are sufficient to find out all, in manner as hath been shewed before.

HEre follow two Tables of Interest, whereof the first sheweth how much 100 li. with all its increase by meanes of Compound Interest at the several rates of 5, 6, 7, 8, 9, and 10 per. Cent. amounts to annually for 31 years: The se­cond sheweth the like increase for 100 li. An­nuity or yearly rent, at the like rates, and for the same terme. In both which the first Columne to­wards the left hand shews the number of years suc­cessively to 31, the second gives the increase to­gether with the Principal in intire pounds ster­lin; the third hath the Numerators of the fracti­ons of a pound to be added to their respective In­tegers. Onely in the rate for 6 per. Cent. (which is now of more frequent use) the fractions are re­duced to shillings and pence, (ommitting lesse then pence) as may easily be seen by the table. And the fractions in the other rates (whose common denominator is 1000000) may be easily enough, either so reduced, or very neerly guessed, by such as are but moderately versed in Arithmetique.

ye.At 5pe. Cent.ye.At 6p.Ce.
11050000001106  
2110250000211272
3115762500311920
41215506254126411
51276281565133165
61340095636141170
7140710041715073
8147745543815978
915513282091681811
101628894611017918
1117103393411189167
121795856301220144
1318856491113213510
141979931561422619
1520789281315239131
162182874531625408
172292018251726956
182406619151828588
1925269506219302112
2026532984520320142
2127859633721339189
222925261542236070
2330715246123381195
24322510084244041710
2533863558825429311
26355567367264541811
2737334573527482410
283920130212851137
2941161367229541170
304321943553057472
3145380407231608165
[...]
[...]
ye.At 7pe. Cent.ye.At 8p. Cent.
11070000001108000000
21144900002116640000
31225043003125971200
41310796014136048894
51402551735146932807
61500730356158687431
71605781477171382355
81718186178185092943
91838459209199900378
1019671513410215892408
1121048519311233163800
1222521915612251816904
1324098449713271962256
1425785341214293719236
1527590315115317216774
1629521637116342594116
1731588151917370001645
1833799322518399601776
1936165275019431569818
2038696844220466095403
2141405623121503383035
2244304016722543653677
2347405297923587145971
2450723668724634117648
2554274425525684847059
2658073635326739634823
2762137389827798805608
2866487007128862710056
2971141097629931726860
30761209744301006265009
31814494426311086766210
ye.At 9p. Cent.ye.At 10p. Cent.
11090000001110000000
21188100002121000000
31295029003133100000
41411581614146410000
51538623955161051000
61677100106177156100
71828039117194871710
81992562838214358881
92171893489235794769
1023673638910259374246
1125804266411285311671
1228126650412313842838
1330658048913345227122
1433417273314379749834
1536424827915417724817
1639703062416459497299
1743276338017505447029
1847171208418555991732
1951416617319611590905
2056044112720672749995
2161088082821740024994
2266586010322814027493
2372578751223895430242
2479110838824984973266
2586230814325108347053
26939915876261191817652
271024508305271310999417
281116714052281442099359
291217218317291586309295
301326767966301744940225
311446177082311919424247

The second Table for 100 li. year rent, or Annuity.

ye.At 5p. Cent.ye.At 6p. Cent.
11000000001100  
22050000002206  
3315250000331872
4431012500443787
55526756255563136
66803094066697911
78143248767839611
89550411198989142
9110279317491149111
101257932832101318010
11142082947311149726
121591870946121686190
13177146449313188834
14196003771714210192
1521580396021523271011
16236594158216256740
17248423866117282147
182813450594183090101
193054123123193375188
203306829279203678910
21357217074221399941
22385077927922433931
234142318242234699100
24444943415424508195
25477190586125548672
2651105011542659151010
27546602621127637096
285839327521286852140
296231293897297363173
306642858591307905149
31707500152031848009
[...]
[...]
ye.At 7p. Cent.ye.At 8p. Cent.
1100 1100 
2207 2208 
33214900003324640000
44439943004449611200
55750739015585580096
67153290746732426503
78654016097891019623
8102597972181062301192
9119779830191247285287
101381644182101446068109
111578359274111661753557
121788844423121893693941
132014063532132145189456
142255047979142416804612
152512901337152710148980
162788804430163026960898
173084020740173369117769
183399902191183738647190
193737895344194137738965
204099548018204568758082
214486516379215034258738
224900572525225536999437
235343612601236079959391
245817665483246666356142
256324902066257299664633
266867645210267983637803
277448380374278722328827
288069767000289520115133
2987346506902910381724346
3094460762383011312262290
31102073015743112317243213
ye.At 9p. Cent.ye.At 10p. Cent.
1100 1100 
2209 2210 
33278100003331 
44573129004462100000
55984710165608310000
67523334566769141000
79200434677946055100
8110284737981140660600
9130210364391354726600
101509292970101590199200
111745129337111849219100
121992190977122134141000
132271488165132447575100
142575922099142792310610
152907755087153171541671
163209453045163588695838
173598303819174047565421
184022150662184552321925
194484144223165107554159
204987717203205718309575
215536611751216390140532
226134906809227129154585
236787048420237942070043
247497882777248836277047
258272692226259819904751
2691172345262610901895226
2799321863482712092084748
28109321863482813401293222
29120160831102914841422544
30131975305893016425564798
31144853083423118168121277

The use of these Tables is thus.

If it be asked how much 100 li. put forth to use comes to in 17 years, at 6 per cent. compound interest? Look for the title of 6 per cent. at the top of the leafe, and in the first Table, then also looke down in the column entituled Years, till you finde the number 17, just over against 17, to­wards the right hand you shall finde 269 li. 5 s. 6 d. which is the thing required.

Or if it be asked how much 100 li. per Annum in rent amounts to in 13 years at 8 per cent. com­pound Interest? Look in the second Table for 13 years, and under 8 per cent. you shall see 2145 li. and 189456/1000000 of a pound, that is reduced, 2145 li. 3 s. 9 d. and something more, which more, being lesse then a peny, I omit, as (in this case) not considerable.

NOTE.

Although I say it amounts to so much, yet I do not say it is worth so much; for who would part with 2145 li. 3 s. 9 d. presently, in hope to get it up again in 13 years by 100 li. per annum? When money was at 8 per cent. a Lease of 21 years was accounted by some worth 9½ years purchase, by others worth 10 years purchase: so that 100 li. per annum rent at the most, is worth but 1000 li. [Page 241]in 21 years: that is (by the rule of three) 619 li. 11 d. for 13 years.

Note 2.

It is also fit to be known, that proportionally as Money is lesse valued, land is more, et contrà. So that according to 10 years, purchase for Rent Charges or Annuities for years, when Money was at 8. per. Cent. the money being now at 6. per. Cent. the purchase must be 13 years 4 months rent of the land, &c.

So likewise, Money being at 8. per. Cent. land for ever used to be sold for 20 years Rent, but now (if no external accident hinder) it ought to be sold for 26⅔. times the yearly rent thereof.

For 6′ 8″. 20′ 26⅔. Also, 6′ 8″. 10′ 13⅔″. And the like Analogisme will serve being used in other rates. As if money were at 5 or 7. per. Cent. then 5′ 8″. 20′ 32″. or 7′ 8″. 20′ 22 6/7″.

Also 5′ 8″. 10′ 16″. Or 7′ 8″. 10′ 11 6/14″. And so of any.

This, considering the largenesse and clearenesse of the Tables, is all I mean to say concerning that compound interest which is called Direct or Pro­fitable.

There is another sort of Interst which gives the yearly decrease of 100 li. or any other Summe. And this is called Compound Interest Rebated, or Damageable.

Which Decrease is orderly made by subtract­ing the Interest from the principall yearly, as the increase of 100 li. in the former Tables was caus­ed by adding the yearly interest.

Example.

If 100 give 6, then at the end of the first year the 100 li. is increased, and become 106, and so again, if 100 give 6, what 106? it makes 6 36/100 which adde to 106, then the said 100 li. is at the end of two years hereby increased to 112 36/100, and so the first Table is made for every year.

But now if 100 give 6, what shall 94 give? it is 5 84/100, which taken from 94, rests 88 16/100, so the 100 li. at the end of the first year is decreased to 94 li. and at the end of the second to 88 16/100.

But because the composing Tables for this is much labour as that which hath been done alrea­dy, also for varieties sake, I will adde a third Ta­ble (which I take out of Simon Stevens Practical Arithmetick) consisting of artificiall numbers, which wil serve as wel for direct as rebating inte­rest. And when I have shewed the way to make and use those Tables, and put a few Problemes re­quisite and difficult, and adjoyned a Table of the Suns Declination, I mean to conclude this Trea­tise.

Here followeth the third Table.

 The Table for 5. p. C.for 6. per. Cent.for 7. ptr. Cent.The Table for 8 p. C.for 9. per. Cent.for 10. p. Cent.
1952381094339629345794925925991743129090909
2907029588999648734387857338684168008264463
3863837683961928162979793832277218357513148
4822702579209367628952735029870842526830135
5783526274725817129862680583164993146209214
6746215470496056663422630169559626735644740
7710681366505716227497583490354703425131582
8676839362741245820091540268850186624665075
9644608959189855439337500248946042774240977
10613913255839485083493463193442241073855434
11584679252678754750928428882838753283504940
12556837349698934440120397113735553473186309
13530321246883904149645367697932617862896645
14505067844230093878173340461029924642633314
15481017041726503624461315241727453802393922
16458111439364623387347291890525186972176293
17436296637136433165745270269023107311978448
18415520635034372958640250249121199371798589
19395733933051292765084231712119448961635081
20376889431180462584191214548217843081486837
21358942329415532415132198655716369801351306
22341849827750502257133183940515018171228460
23325571226179722109470170315313778141116782
24310067824697851971467157699412640501015256
25295302723299861842493146018011596790922960
26281240721981001721956135201910639260839055
27267848320736791609305125186909760790762777
28255093619563011504023115913808954850693434
29242946318455671405629107327608215460630395
30231377417411011313672099377407537120573086

The construction of these Tables, or any other the like is as followeth.

Having made choise of some great Decimall number, I mean, so it may consist of all Ciphers, except unity towards the left hand, as in these it is 10000000 (which shall be called the Radius of the Tables) this Radius being multiplyed by 100 (which is the principall) and divided by the prin­cipall plus the Interest, the quotient is the num­ber in the Table for the first year, which quotient being again multiplyed by 100, and the product divided by principall plus interest as before, the quotient shall be the number in the Tables for the second year, and so may every years respective number be found, as was the second.

Example in Interest 7 per cent.

First, 10000000 into 100 gives 1000000000 which divided by 100 + 7, that is, by 107, the quotient is 9345794, which is the number an­swering to the first year in the Table of 7 per cent. For although there be a remain, after the division, of 42, yet because 42/107 < ½ it is here neglected: But if the remain had happened 51/107 > ½, then 1 being added to the quotient, it is 9345795, and is so much neerer the thing required.

Secondly, 9345794 into 100 gives 934579400 which divided still by 107. quotient is 8734386, and 98 remaining, but 98/107 > ½, therefore ad­ding 1 to the quotient, it is 8734387, for the number answering to the second year in the Table [Page 247]of 7 per cent. After this manner are all the Ta­bles made.

Ʋse of the Tables.

This shall be shewed in a few Examples.

Example 1. Interest profitable.

If 100 li. give 6 li. for one year, what shall 500 li. give for 17 years, principall and Interest?

The Rule.

Multiply the Principall by the Radius, the Product is 5000000000, which divide by 3713643 (which is the number answering to 17 years in the Table of 6 per cent.) the quotient will be 1346 1444522/3713643 of Pounds. Which is the just sum of 500 l. with all its compound interest, at 6 per cent. for 17 years, and reduced it is 1346 li. 7 s. 8½ d. the like way of working will effect any question of this nature, which exceeds not the Tables in time or rates.

NOTE.

It may here be noted, that if 1346 li. 7 s. 8 ½ d. were due to be received 17 years hence, it is, or may be called equivalēt to the receiving of 500 li. in hand, that is, such a reversion is worth 500 li. in ready money.

And therefore by inversion of the former Rule, may the Rebatement, or Interest damageable be found.

Example 2. Interest damageable.

If there shall be 1000 li. due at the end of 21 years, and money run at 8 per cent. to be accor­dingly rebated, how much is this worth in ready money?

The Rule.

In the Table for 8 per cent. finde the number an­swering to 21 years, which is 1986557, multiply this by the principal, the product is 1986557000, which divided by radius the quotient is 198 6557/10000. pounds.

That is, 198 li. 13 s. 1 ½ d. which is the thing required.

Probl. 1.

If 1000 li. be to be paid at the end of 7 years and 500 li. more 2 years after that, what shall both these be worth, for borne till the end of 12 years? at 6 per. Cent.

See (by the Rule belonging to Example 2.) First, what each of them is worth at the end of their proper termes,

The first is, 665,0571000

The second 295,9492500

In all 961,0063500

Which is all that both are worth in ready mo­ney.

The secondly, seeke (as before) what 961, 0063500 li. ready money is worth 12 years hence, rebating interest of 6. per. Cent. It will come to about 478, 4778100 li. for the thing required.

Probl. 2.

If there be due in ready money 500 li. which at the end of 20 years will increase, and be 2330, 477015 li. what is the rate of the interest here?

Take the fist part of the Number, (because the tables are made for 100 li.) which is 466,095403. and in the first table look for it against the number 20. it will be found in the rate of 8. per. Cent. and such is the interest.

Or in the third table.

Say, 2330, 477015′ 10000000″ 500′ 2145482″.

Which last viz. 2145482. being looked for in the third table, will be found over against the year 20 under the title of 8. per Cent. which shews a­gaine that the rate of the interest is 8. per. Cent.

And herein the Probleme is not onely cleared; but the use of both tables exemplified.

Prob. 3.

In like sort, if the interest, years, and totall in­crease be given, to find the principal. As if one re­ceive 1000 li. for compound interest at the rate of 10. per. Cent. for 7 years; how much was the principal summe?

See in the third table in the rate of 10. per. Cent. what number answers to 7 years; it will be found 5131582. which being subtracted from 10000000 there resis 4868418. And then say, 4868418′ 5131582″ 1000′ 1054, 269428″. That is, the principal was 1054 li. 1 s. 1 d. and a little more, which we omit.

And the same sum will be found if one use the first Table, where the interest of 100 li. for 7 years at 10 per cent. is 94,871710, for then 94,871710′ 100″ 1000′ 1054 li. 1 s. 1 d′. and a little more as before.

Probl. 4.

Or the rest being given, and the time required, As if there be 1000 li. due at the end of some years, and the Creditor instead of it takes 100 li. ready money, rebating compound interest at 8 per cent. at the end of what years was this at first payable?

Say 1000′ 10000000″ 100′ 1000000″. Which fourth proportionall number being (as neer as may be) sought for in the third Table un­der 8 per cent. will be found to fall neer 30 years: that is, the time here required is almost 30 years.

Prob. 5.

If 1000 li. be due at the end of 4 years, and the parties agree to have it paid at four yearly pay­mēts, that is 250 li. (rebating 6 per cent.) at the end of every year, how much is to be paid at each time?

1 Look in the third Table under 6 per cent. for 3 years, against that the number 8396192 stands, which last being multiplyed by 250, and after the product divided by Radius, that is by 10000000, the quotient is 209,9048000 li. that is, 209 li. 18 s. 1 d. for the first yearly payment.

2 In the same Tables against 2 years is found the number 8899964 which used in all respects like that against 3 years already done, the quoti­ent [Page 251]wil be 222,499 1000, that is, 222 li. 9 s. 11 d. for the second.

3 And against 1 year is 9433962, which be­ing multiplyed like the two former, the quotient is 235, 8490500, or 235 li. 16 s. 10 d. for the third payment.

4 Lastly, Must be the full fourth, viz. 250 li. for that being not paid otherwise then in due time, suffers no Rebatement.

Prob. 6.

If there be a Reversion of a Lease or Annuity of 100 li. per Annū, and for 11 years to come at the end of 14 years, what is this worth in ready mo­ney? money being at 6 per cent.

1 Adde 11 to 14, it makes 25 years, and look in the second Table for 6 per cent. against 25 years there stands 5486, 7, 2 d. likewise against 14 years is 2101, 9, 2 d. the difference is 3384, 18, 0 d. which cannot be received till the end of 25 years. Therefore in the third Table for 6 per cent. against 25 years, finding the borrowed num­ber 2329986, multiply it (as hath been lately shewed) by 3384 li. 18 s. 0 d. that is by 3384 9/10 li. the product divided by 10000000 is 789,4672624 That is reduced 789 li. 9 s. 4 d. And so much it is worth in ready money, and the Probl. is solved.

All cases cannot be instanced, if the question happen to be like none of these, yet the Reader by his own judgement may (without doubt) resolve it by some of these Tables, to illustrate the use of which, was chiefly my end in putting and re­solving the precedent demands

A Table of the Suns Declination for the yeare 1654.
DayesJanu.Febru.Mar.AprilMayJunJulyAug.Septē.Octob.Novē.Decē.
 SouthSouthSouthNorthNorthNorthNorthNorthNorthSouthSouthSouth
121 4913 563 3508 2617 5823 [...]22 1015 174 3007 0917 3623 08
221 3913 363 1108 4818 1323 [...]22 0214 594 0707 3217 5223 12
321 2913 162 4809 0918 2823 [...]21 5314 403 4407 5518 0823 16
421 1812 552 2409 3118 4323 [...]21 4414 223 2108 1718 2423 20
521 0712 352 0009 5318 5723 [...]21 3514 032 5808 3918 4023 23
620 5612 141 3710 1419 1123 [...]21 2513 442 3409 0218 5523 26
720 4411 531 1310 3519 2823 [...]21 1413 252 1109 2419 0923 28
820 3211 320 4910 5619 3823 [...]21 0413 051 4809 4619 2423 29
920 1911 100 2611 1719 5123 [...]20 5412 461 2410 0819 3823 30
1020 0610 490 0211 3720 0423 [...]20 4312 261 0110 2919 5223 31
1119 5310 270 N2211 5820 1623 [...]20 3112 060 3710 5120 0523 32
1219 3910 050 4612 1820 2823 [...]20 1911 460 1411 1220 1823 31
1319 2509 431 0912 3820 4023 [...]20 0711 260 S1011 3420 3123 31
1419 1009 211 3312 5820 5123 [...]19 5411 050 3311 5520 4323 29
1518 5508 581 5613 1721 0223 [...]19 4110 440 5712 1520 5523 28
1618 4008 362 2013 3721 1223 [...]19 2810 231 2012 3621 0623 26
1718 2508 142 4313 5621 2323 [...]19 1510 021 4412 5721 1723 23
1818 0507 513 0714 1521 3323 [...]19 0109 412 0713 1721 2823 20
1917 5307 283 3014 3421 4223 [...]18 4709 202 3113 3721 3823 17
2017 3607 053 5314 5221 5123 [...]18 3308 582 5413 5721 4823 13
2117 1906 424 1715 1022 0023 [...]18 1808 363 1814 1621 5723 02
2217 0206 194 4015 2822 0823 0 [...]18 0308 153 4114 3622 0623 03
2316 4505 565 0315 4622 1623 0 [...]17 4707 534 0414 5522 1522 58
2416 2705 325 2616 0322 2422 5 [...]17 3207 314 2815 1422 2322 52
2516 0905 095 4916 2122 3122 5 [...]17 1607 094 5115 3322 3122 46
2615 5104 466 1216 3822 3822 4717 0006 465 1415 5122 3822 39
2715 3204 226 3416 5422 4422 4 [...]16 4306 245 3816 0922 4522 32
2815 1403 596 5717 1122 5022 3 [...]16 2606 016 0016 2722 5122 24
2914 55 7 1917 2722 5622 2716 0905 386 2316 4522 5722 16
3014 35 7 4217 4323 0122 2 [...]15 5205 166 4617 0223 0322 07
3114 16 8 04 23 06 15 3404 53 17 19 21 58

Postscript.

THe reason why in Chap. 15. I did not (as Des Cartes) continue the method to Sursolid Problemes, is because the de­scription of such curve lines, is not only difficult and laborious, but (as he confesseth) in­commodious: And although he saith it is easie to finde a thousand other sorts of wayes, amongst which some might be better, yet I conceive it is easier for any man to believe that Des Cartes ha­ving found one way might be allowed to say this, then for that other men to finde any one other way better and easier. And Des Cartes gives but only one Example, which is to finde four Means, and omits quinquisection of angles, which might have been of some use, whereas the other is but of little.

For the principall use of two Means seems to be in doubling the Cube, or in making rectangle Parallelepipedons retaining any proportion gi­ven, whose three dimensions also shall be pro­portionall: or the like of Solid Rhombi, or ob­lique [Page 256]parallelepipedons, all which are bodies to be seen and handled, whereas four or more means to be so employed require bodies of four or more dimensions, of which we have yet no fancie.

Besides, Galileo in his Book called Systema Mundi (the beginning almost) seems to prove that there can be but three dimensions in nature. Neverthelesse, if any one happen to discover the Biquadraticall Body, he may then as well de­monstrate that there can be but foure.

And yet more, seeing some Aequations of 6. dimensions require a circle to touch or cut a curve line in 6 points, which cannot be done but very obliquely, the method therefore here growes un­usefull: and for septisection, and Aequations of 8 or more dimensions, it will be unsufficient.

If for all this, any one hath a minde to surso­lids, his aequation, by some rules going before, or here following, must be reduced to this forme.

a6 − ba5 + ca4 − da3 + faa − ga + h=o

In which it behoveth that the quantity called c, be greater then the square of ½b.

In the section 6, of the first Rule of Chap. 4. it is shewed how all the false roots in any aequation may be made true.

Also in Rule the second of that Chap. it was shewed how to free the aequation from the second terme. There remaines three other Rules now of some use.

RƲLE 1.

To cause the known quantity of the third term to be greater then the square of the like in the se­cond, and also to change the false roots to true ones, without causing the true ones to become false. Encrease the true Roots by a quantity greater then any of the false Roots. For sure it is possible to guesse such a quantity, although the false roots be unknown.

Example, in the aequation + aaaa + ba3 − ccaa − ddda + ffff=o

Put e − b=a Then it will be [...]

That is [...]

And making 5bb + cc=gg & b3 + 2bcc + d3=h3 And lastly, 2b4 + bbcc + f4 − bd3=llll

Then it is + e4 − 3bc3 + ggee − h3e + llll=o In which aequation all the Roots are true by Sect. 5. Rule 1. Chap. 4.

And secondly, it is manifest that gg, which is the known quantity of the third terme, being e­quall to 5bb + cc is greater then 9/4bb, which is the square of halfe the known quantity of the se­cond term.

I have instanced in an Aequation of but 4 di­mensions, for brevity sake, the work is true, or may be so, in those of six dimensions, which they that resolve sursolid Aequations will be put to.

RƲLE 2.

If yet any term of the Aequation be wanting, Encrease the Root e never so little, and thereby all the places will be filled: As practise will shew.

RƲLE 3.

If the Aequation have but 5 dimensions, it must be brought up to 6 as followeth. Let it be + a5 − ba4 − dddaa + f4a − g5=o

In stead of it wrie + a6 − ba5 − d3aaa + f4aa − g5a=o

And make e − q=a, and the thing will be effected which was desired:

So if between two lines given b & c, and c > b, it be required to finde 4 meanes, putting a for the lesser mean the Aequation will be + a5==bbbbc, that is + a5=cb4, And the places which are empty may be filled up by the second Rule: and brought to 6 dimensions (if yet it be not so) by the 3d. Rule.

NOTE

All Aequations whose dimensions are expres­sed [Page 259] [...] numbers, can by no meanes be brought do [...] [...] fewer dimensions by any artifice that can be used: and for this reason the invention of any even number of means is much harder then to finde an odd number of the like, so here where to finde foure meanes runs to an Aequation aaaaa=bbbbc, the Probleme is absolutely Sursolid: but to find five meanes is as easie almost, as to finde two, for the worke brings us to the Ae­quation a•=cb5 which by the 5 Chapter may be brought downe to the Equation. aaa=cdf. or to aaa=ddd if the Probleme be plaine, that is if it be c′ d″ f′″ otherwise by making df=gg it may be aaa=ggc. and the root a. found by a portion of a Parabola, as in the case of two means, Chap. 14.

Lastly, for quinquisection, the Radius being unity, put the whole subtense b. The subtense of the fift part required a. Then + a5 − 5aaa + 5a − b=o. As is demonstrated by Pitiscus, in his making the sines, Probl. 7. and shall here need no proofe.

It will require some labour (as may be seen by Chap. 15. Probl. 1.) to bring this Equation to the forme required, for which purpose, first in stead of unity call Radius, r. And then it will be + a5 − 5rra3 + 5rrrra=5rrrrb, Afterward by Rule 3 it may be done.

And when this is done, the rend [...] [...] work, as well Construction as Demon [...], according to the method proposed by Des Cartes may be done also, but the doing will be both tedi­ous and intricate.

And therefore I shall no further prosecute the said Method, & for use, sectiō of angles in generall may be done by the Rule of False Positions, as the said Pitiscus made his Canon of Sines. But the Canon of Sines, the late Prince of Mathemati­cians VIETA, vouchsafes to call The Mathe­maticall Canon. Neverthelesse, if the industri­ous Reader desires more exactnesse (I mean in Theorie) according to the former Method, or any other which he shall finde better for his purpose, he may proceed at pleasure,

A Rule for squaring binomiall surds, which should have been in Chap. 6. pag. 76.

Multiply the quantity to which the signe √ be­longs, into the square of the Coefficient, and the product is the square required.

Example in Numbers.

If the square of 3 √ 7 be required, multiply 7 into 9, the product is 63, the square required.

Or, If the square of 4 √ 9 be demanded, 9 in­to 16 produceth 144, which is the square de­manded: the like of all others. This needs no proof. See Page 77. And this shews that all such Surds are commensurable in power.

Deo Gloria.

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