[Page] [Page] The Description and Use OF TWO ARITHMETICK INSTRUMENTS. TOGETHER With a Short Treatise, explaining and Demonstrating the Ordinary Operations of ARITHMETICK. As likewise, A Perpetual ALMANACK, And several Useful TABLES.

Presented to His Most Excellent Majesty CHARLES II. King of Great Britain, France, and Ireland, &c.

By S. MORLAND.

LONDON, Printed, and are to be Sold by Moses Pitt at the White-Hart in Little-Britain, 1673.

[Page]

SAMVEL MORLANDUS Eques Auratus & Baronett us nec non CAMERAE PRIVATAE Generosus

[...]

A New, and most useful INSTRUMENT FOR ADDITION AND SUBSTRACTION OF Pounds, Shillings, Pence, and Farthings; Without charging the Memory, disturb­ing the Mind, or exposing the Opera­tor to any uncertainty; Which no Me­thod hitherto published, can justly pre­tend to.

Invented and Presented to His most Ex­cellent Majesty CHARLES II. King of Great Britain, France, and Ireland, &c, 1666.

By S. MORLAND.

And by the importunity of his very good friends, made publick 1672.

A
[Page 1]TO Set or Dispose THE INSTRUMENT FOR OPERATION: Which Instrument must be held in the Left hand, and the Index or blew Pin in the Right hand, as men usually hold a Pen, but something more upright or perpendicular.

IN those five upper Plates, (in every of which there are 10 small holes) if any of the Digits (suppose 3) ap­pear in any of the Windows, put the point of the Index into the hole over against that Figure in the Margent, which is the Complement to 10. (namely 7.) And then turn it under the Window, where (0) will appear. This done, put the point of the In­dex into the (0) of the small Plate above it; and then, if you would prepare the Instrument for Addition, turn about that small Plate, till 0 stand just under that Line or mark in the upper Plate which is on the Right­hand, thus 11/0 as in Fig. A. But if for Substraction, then turn 0 on under the Left-hand line or mark, thus 11/0, as in Fig. D.

After the very same manner are all the lower plates of Shillings, Pence, and Farthings to be disposed; Re­membring onely that in the place of Shillings, where (20) is the number, the Gomplement to (3) is (17) And in Pence, the Complement to (3) is (9) And in Farthings, the Complement to (3) is (1) And [Page]

B

[Page 2] thus are all the Plates fitted for Operation as in Fig. A.

This is likewise to be observed before the Instru­ment is set: If any of the Plates with Holes stiek, and will not turn about forwards, it is because neither (0) nor any Figure of the small Plate above it is under the Right-hand-line or mark. And if they will not turn backwards, it is because neither O, nor any Figure of the small Plate above it is under the Left-hand-line or mark. And lastly, if the small Plate above will not move, it is because no Hole in the lower Plate is di­rectly under the Window.

The Operation of Addition.

Let the Sums to be added together, be these.

l.sh.d.f.
714032
4811103
25217111
4337821
46461233

First I set on 7 l. that is, I put the point of the Index into the hole of the place of Ʋnites of Pounds, which is over against (7) in the Margent, and turn it under the Window. After that, I turn on (14) in the place of Shillings; (3) in the place of Pence; and (1) in the place of Farthings.

Again, as to the 2 Summe, I set on (48) thus, namely, (4) in the place of Tens, and (8) in the place of Ʋnites: As likewise 11 in the place of Shillings; (10) in the place of Pence; and (1) in the place of Farthings And having set on all the Summes in the manner and method aforesaid, they will appear on the Instrument as in Fig. B.

In fine, to perfect the Operation, I begin with the place of Farthings, and because the Figure (1) of the [Page]

C

[Page 3] Plates over head stands between the two Lines or Marks, therefore I turn on (1) in the place of Pence. And because (2) is between the two Lines in the place of Pence, I turn on (2) in the place of Sbillings: And because (2) is between the two Lines in the place of Shillings, I turn on (2) in the place of Ʋnites of Pounds. Thus, for (2) between the two Lines in the place of Ʋnites of Pounds, I turn on 2 in the place of Tens. And for (1) in the place of Tens, I turn on (1) in the place of Hundreds. Which done, the true Summe or Aggregat of all the Summes, appears in the several Windows, viz. 4646 l.—12 s.—3 d.—3 f. As in Fig. C. The same thing may be performed by ad­ding together, First the Farthings apart, then the Pence, next the Shillings, and lastly the Pounds.

If peradventure the Summes are so numerous, as that by often turning of the Plates with holes, the small Plates above them are like to be over-charged, That is, if the Figure (7) (8) or (9) of those small Plates come between the 2 Lines or Marks, then I dis­charge them, by turning on the same numbers in the next places: For example, if the Figure (7) of the small Plate above the place of Farthings, be under the Line, I turn on (7) in the place of Pence. If the Fi­gure (7) of the small Plate above the Pence be between the two Lines, I turn on (7) in the place of Shillings, and so to the end. And lastly, I turn O of each small Plate under its proper Line, And hen proceed (without setting the lower Plates to (0) or at all altering them otherwise than aforesaid) to set on the remaining Summes, to a Million of Pounds. (Or if it be desired, the Instrument may be made for a far greater Summe.) But the surest way, is to divide a long Page into two or three parts, and so to work them distinct­ly.

D
E

[Page 4] The Operation of Substraction.

Having prepared the Instrument for Substraction as is before directed, suppose I would from 327 l. - 12 s. - 07 d. deduct 39 l. - 14 s. - 3 d.

First, I set on 327 l. - 12 s. 7 d. in their proper places. Then I begin with 3 d. of the Summe to be deducted, and set it backward, That is, I put the point of the Index into the hole under the Window of Pence, and turn it till it stand over against (3) in the Margent, which will leave (4) in that Window. Thus I set 14 s. back­wards in the place of Shillings; And thus (39) in the Ʋnites and Tens of Pounds. And having so done, I ob­serve if any (0) of the small Plates over head be remo­ved from under its Line (as in Fig. D. in the place of Shillings, I find the small Plate over head moved) I put the point of the Index into that hole of the next place, that is under the Window, and turn it backwards under (1) in the Margent. After the same manner, because the small Plate over the place of Ʋnites of Pounds is re­moved out of its place, I set back for it (1) in the place of Tens; And so likewise, because the small Plate over the place of Tens is removed, I set back (1) in the place of Hundreds. Which done, the remaining Summe, viz. 287 l. - 18 s. - 04 d. is found in the re­spective Windows. As in Fig. E.

F

[Page 5]The Description and Use of an Additional Wheel, applicable to this New INSTRƲMENT, for all those who shall desire it, which renders it useful, beyond exception, for the longest Accompts, without either dividing the Page, or setting ☉ of any of the small Plates under their Lines a second time.

BƲt forasmuch as it may and will often happen, in the adding up long Pages of Accompt-Books, That the small Plates will be over-charged (and that more than once in a Page) which may be something trouble some to the Ope­rator; Therefore I have contrived an Additional Wheel, as in Fig. F. so that no one or two Pages of any Accompt-Book whatsoever can overcharge it: all the difference is on­ly this, That what Figure soever of this Wheel is found op­posite to the Line (or mark) above it, it must be set on (not in the next place, but) in the next place save one, in the manner following: That is to say, The Figure of the small Wheel above the Farthings, opposite to the Line above it, must be turned off in the place of Shillings. The Figure of the small Wheel above the Pence, opposite to the Line above it, must be turned off in the place of Unites of Pounds. And if any Figure of the small Wheel behalf a division beyond the said Line, there must be 10 turned off for it in the place of Shillings, (which is the great­est intrigue in the whole Operation, For,) The Figure of the small Wheel above the place of Shillings opposite to the Line, must be turned off in the place of Tens of Pounds. The Fi­gure of the small Wheel opposite to the Line, in the place of Unites of Pounds, must be turned off in the place of Hun­dreds. Lastly, the Figure of the small Wheel opposite to the Line, above the place of Tens of Pounds, must be turn­ed off in the place of Thousands; and so to the end, as in Fig. G. So that by the addition of this small Wheel, a [Page]

G

[Page 6] place more is gained, and the Instrument, if need were, would add up an Accompt of Ten Millions of Pounds, by the Example, compared with Fig. G. will more evidently appear.

EXAMPLE.

l.s.d.q.
7653413033
7653410112
8563714051
9379217102
7437909083
8546619102
7295410042
6111716083
5225216063
6557715053
4257315023
7257619092
6995517113
7277713091
8585504103
9888814062
9774416072
4575712043
7387916113
6454919082
9787214061
8567810101
7664408063
5627918032
4577215053
5657217102
6769914112
7157818082
8997919103
9797901063
9587808112
8678810091
9967813102
8948517062
4763216031
5741615093
6251714072
7452813112
4321512052
6774211083
3541809041
9226101093
4441512063
4332414031
3733802112
3451219042
5673517023
328376111061
A

[Page] Machina Nova CYCLOLOGICA Pro Multiplicatione. OR, A new Multiplying-INSTRUMENT: Invented, and humbly present­ed to the Kings most Ex­cellent MAJESTY CHARLES II.

THe Fabrick of this Instrument be­ing truly represented in Perspect­ive, there will be no need of any large Description of it.

The 5 upper Circles whereon are the Figures 0/9 ⅛ 2/7 3/6 ⅘ re­present 5 moveable Plates which lie always ready to be taken off the 5 Semi-circular Pinions (or Centers) S, T, Ʋ, W, X, and to be set on any of, or all the Centers (or Semicircular Pinions) a, e, m, n, o, p, interchangeably. And underneath each of those upper [Page]

B

[Page 9] Plates aforesaid,, are placed 5 other Plates of the same dimensions, and with the same Figures graved on both sides; so that of these upper Plates there are 30, that is to say, Six of each kind, besides 5 other Plates un­der the place marked Q / QQ which serve for extracting the Square, Cube, and Square-Square-Roots, of which in its place.

E. F. is a Line divided into 9 equal Parts, on which runs a small Black Pin or Index forwards or backwards at pleasure, being turned about by G. H. which is like to the Key of a Watch.

Lastly, PQ is a Plate opening with hinges, and shut­ting down upon the lower Circular Plates, and a little Bolt at R. locking it down, which Plate has 6 square holes (or foramina) through which are discovered such Figures onely, as are necessary for the operation.

K, L, M, is a long Rack on the backside of the In­strument, by which all the Plates are turned about on the Pinions a, e, m, n, e, The meaning of all which is this. The Instrument is to be supposed as it is represent­ed in Figure A save onely that there are as yet no Cir­cular Plates taken off the Semi-circular Pinions S, T, Ʋ, W, X, and that the number given to be multiplied is (1734.) Then one of the Plates marked ⅘ is first to be taken off the Pinion x and set on the lower Semi-circu­lar Pinion (r) which is the furthest on the right hand. Afterwards one of the Plates marked 3/6 is to be taken off the Pinion W, and set on the Pinion (o) next to the other.

Thirdly, one of the Plates marked 7/2 is to be taken off the Pinion V and set on the Pinion (n.)

Lastly, one of the Plates marked ⅛ is to be taken off the Pinion T, and set on the Pinion (m) Which done, the Plate P Q. is to be shut down, and then will appear the number 1734 through the holes, as in Fig. B and this is all the trouble of the Instrument in this or any operation.

[Page]

C

[Page 10] The Figure B represents the very same posture of the Instrument in the foregoing Figure A save one­ly that the Plate P Q is now shut down and lockt upon the four lower circular Plates marked on the former Figure A with the large Figures ⅛ 7/2 3/6 ⅘ which was in effect, a preparation for the Multiplication of the Number (1234) which Number now ap­pears through the small holes or foramina of the said Plate P Q hiding all the other Figures from the Eye of the Operator. In which particular this In­strument far surpasses the Lord Napiers Lamina or Bones, which expose a great number of Figures to the Eye at the same instant of time (as well those that are not useful as those that are) besides that all the Figures in those Bones are placed Diagonally, which does very much strain and force the Eye of the Operator; Whereas in this they lie all in a straight Line, and as distinctly as can be desired.

The Instrument being disposed as aforesaid, sup­pose the given Number 1734, be to be multiplied by 24. I first set it down in Writing thus [...] then first I turn then Hand or Key G H till the Index point to 4 in the Line E F so have I the first Product (6936) given me in the Windows or Foramina of the Plate P Q, as in Fig. C which I set down thus

This done, I turn the Hand or Key back­ward, [...] till the Index point to the Figure (2) in the Line E F and in the Holes or Foramina of the Plate P Q (in Figure D.)

So have I given me the

[Page]

D

[Page 11] The Second Product, viz. 31468. which I set down under the first Product, and add all of them as in the ordinary Method of Multipli­cation, or else by the help of Napiers Bones.

[...]

And after this manner may any Number be Multi­plied by the help of this Instrument, which is capa­ble of being made for any Number of places.

This Instrument alone is also of excellent use in Di­vision; as likewise in extracting the Square, Cube, and Square-Square-Roots, for which the 6 Plates at the end of the Instrument on the right hand are extreme ready and serviceable, and are to be used after the very same manner as are the Lord Napiers Bones, and therefore need no further explication.

But if any person have the Curiosity, and is willing to goe to the Expence; the Adding Instrument being joyned to the Multiplying Instrument, performs Addition, Subtraction, Multiplication, and Division; as likewise the Extraction of the Square, and Cube-Roots, &c. without the help of Pen and Ink, or exposing the Operator to any difficulty or uncertainty.

SOLI DEO GLORIA.

A CAUTION To all who desire to make Use of either of these INSTRUMENTS.

IF any person desire to have either of these Instruments exactly made, and so as it may be serviceable for many years. He may bespeak it of Mr. Humphry Adamson, living at pressent at the House of Jonas Moor, Esq in the Tower, who is the onely Work­man that ever as yet could be found by the Author to perform the said In­strument, with that exactness that is absolutely necessary for such Operati­ons.

FINIS.

A Short DISCOURSE Concerning the ANTIENT, & COMMON WAY OF NUMBRING.

BUt for the better understanding of these Arithmetical Instruments, I shall en­deavor so to explain and demonstrate the reason of the Operations of Addi­tion, Subtraction, Multiplication, Division, and Extraction of the Square, and Cube-Roots, as to render them plain and obvious to the meanest capacities.

The way of Numbring in use with the Antients, was by the Letters of their respective Alphabets; For example: with the Romans C signified 100. D 500. M 1000. ↃMC 10000. CC.I.ↃↃ 20000. CqↃ 100000. &c.

So among the Grecians, Α or α. signified 1.▪ Β or β 2. Τ or τ. 3. Ι ι. 10. Ρ ρ. 100. α, 1000. δ▪ 4000. ζ. 7000. &c.

But note that the most common way of expressing the value of Numbers, is by the Arabick Notes or Charact­ers called Ziphers, by the Hebrews, Sephers; and by Us, Cyphers; and they are ten, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The way of placing those Characters, is from the right hand to the left, after the manner of the Hebrews, in their Writings, as Gemma Frisius and others observe.

[Page 14] The Progression of them is Decimal, or by Tens; for every place to the left, is ten times the value of the next place to the right, as in the following Table.

The Third PeriodThe Second PeriodThe first Period
Hund. of MillionsTens of MillionsUnites of MillionsHund. of Thous.Tens of Thous.Unites of Thous.Hund. of UnitesTens of UnitesUnites of Unites
CXICXICXI
MMMMMmuuu
986157432
Nine H. Eighty six Milli.One H. fifty seven Thous.Four H. thirty two Unites.

In this Table, the first Period is of Ʋnites, the second of Thousands, the third of Millions, Those who would proceed to more Periods then what are here set down, may give them what ap­pellations they please, as Billons, Triltions, Quadrillians, Quintillians, &c. as Tacquet and others have done before us.

But the best and surest way of placing and distinguishing great Numbers, is to put a Comma, or other mark of distinction betwixt every three of them, thus, 12, 345, 769.

CHAP. 1. The Precept for ADDITION of Integers in Plain Numbers.

HAving placed the Unites of the respective Pro­gressions in Ranks and Files; then begin and add together the Unites of the right-hand-File, setting down the sum underneath, if it be under 10. but if just ten, set down 0. and carry 1. to the next place; and if above 10. set down the excess in the first place, for every 10. an Unite.

Examples.

[...]

CHAP. II. The Precept for SUBTRACTION of In­tegers in Plain Numbers.

HAving placed the less number under the greater, according to the respective places or Progressions, begin at the right hand, Substracting the lower Figure out of that above it, and setting the Remainder under­neath. But if the Figure chance to be the less, (as oft it falls out) then there must be an Unite borrowed, and brought from the next Progression, to supply the defect, which must be again repaid, by adding an Unite to the next lower Figure on the left, which is the same [Page 16] thing as if the Figure above it were diminished by an Unite. And for a proof of the operation, the number Subtracted, and the remainde, must evermore equal that out of which it was Subtracted.

For out of a number A C let a less number A B be de­ducted, Then by the Hypothesis, A B together with the remainder B C are equal to the whole number A C For­asmuch as The parts united, are always equal to the whole.

[...]

CHAP III. The Precept for MLTIPLICATION of Integers in Plain Numbers.

HAving placed the Numbers one under the other as in Addition, Multiply the last right-hand Figure of the Multiplicand, by the last right-hand Figure of the Multiplicator, and set the Product (if less then 10) un­derneath; but if greater, carry the excess (that is for every 10, an Unite) to the next place. And if the Mul­tiplicator have more places then one, set down the first Figure of each respective Product under that Figure of the Multiplicator by which it was made, and so on to the left: observing Ranks and Fyles.

[Page 71] For example, 7 times 6 is 42, that is setting down 2 in the place of Unites and carrying 4 in the mind to the next place. Again 7 times 2 is 14, and 4 that was car­ried in mind makes 18, that is setting down 8, and carry­ing 1 to the next. Again 7 times 4 is 28, and 1 is 29.

[...]

And so in the second Pro­duct twice 6 is 12, that is 2 carrying 1 to the next, and so on to the end.

The reason of which operation is plain by the follow­ing Table.

[Page 18]

 CMXMMCXU [...]. 
    426Multiplicand.
    327Multiplicator.
For 7 times 6 is 42; and 7 times 20 is 140; and 7 times 400 is 2800, or two thousand eight hundred.    42First Product
   140Second Prod.
  2800Third Product
Again, 20 times 6 is 120; and 20 times 20 is 400; and 20 times 400 is 8000 or Eight thousand.   120Fourth Prod.
   400Fifth Product
  8000Sixth Product
Thirdly, 300 times 6 is 1800; 300 ttimes 20 is 6000; & 300 times 400 is 120000, or One hundred and twenty thousand.  1800Seventh Prod.
  6000Eighth Prod.
120000Ninth Prod.
The sum total of which Products amounts to One Hun. thirty nine Th. three Hund. and thirty two Unites.139332Summe total of all nine Products.

CHAP: IV. The Precept for DIVISION in Plain Numbers.

DIVISION, is in effect nothing else, but the de­ducting of a less number as oft as may be out of a Greater, and so finding at last the number, by whose U­nites that less number being repeated, makes a number equal to the Greater.

Now the greater of these numbers is Vulgarly called the Dividend, the less the Divisor, and the last the Quo­tus or Quotient.

The method of this Operation is thus;

1. Set the Figures of the Divisor under an equal num­ber of Figures of the Dividend on the left hand, if those Figures of the Dividend be of greater, or at least of equal value with those of the Divisor: Otherwise you must place the first Figure of the Divisor under the second Fi­gure of the Dividend. And having set the Divisor right, put pricks over the Figures of the Dividend, from the Unite place of the Divisor, inclusivè. And the number of pricks denote the number of places in the Quotient.

2. You must evermore prepare such a Tariffa (or Ta­ble of Multiplication) for the Divisor, as is here set down on one side of the Operation, and is of excellent use, making the work ten times more easie and certain.

3. You must find by the Tariffa how many times the Divisor is found in those Figures of the Dividend under which they are placed, and the answer to that, is the first Figure of the Quotient; by which you have multiplied the Divisor in the Tariffa, then deduct the product out of those upper Figures of the Dividend, and what remains must be considered in the next operation, if there be more pla­ces then one in the Divisor.

4. The next Figure of the Dividend must be taken down and set next to the Remainder, if there be any. And the Divisor must be again set under it, if the value [Page 20] of the upper Figures be sufficient; if not, there must a Null or (0) be set in the Quotient, and then the next Figure of the Dividend taken down, and the very same Operation repeated, till the work be at an end. But one Example in things of this nature clearly and distinctly set down, is better then a thousand verbal directions.

Let the Dividend be that Number, which was last found by Multiplying (426) by (327)

That is to say, Let the Dividend be And the Divisor be [...]

Having pointed the Dividend, and pla­ced the Divisor un­der (1393;) look for 1393 (or the nearest number to it) in the Tariffa, which is 1308. wherefore I set that down; and sub­tracting it from 1393, there remains 85; then (having set down 4 for the first Figure of the Quotient) I take down the next Figure, or Cypher of the Dividend, viz. (0) which makes it (850) In this (327) the Divisor by the aforesaid method is found twice; wherefore I set (2) in the Quotient, and then deduct the Product, viz. (654) out of it, and there remains (196) to which in the last place, I take down (2) the last Figure of the Dividend, and make it (1962) in which (327) is found 6 times, and so the work is at end.

[Page 21] The reason of this Operation is plain in the subsequent Table.

 C M X MMCXU 
 139302Dividend.
  327  Divisor, which is advanced 3 places, and is not now 327. but 32700.
Quot. 400.1308  This Product is not 1308, but 130800, which is 4 times 32700, that is 420 times 327, which is the Quotient in the Margent.
   85  That is 32700 being deducted out of 130800, the remainder is 8500, and so is ended the first Operation.
   850 Here begins a second Operati­on. And this is not 850, but 8500.
   327 This is not 327 but 3270, which being doubled makes 6540, which in effect is 20 times 327, as is exprest by the Quotient in the Margent.
Quot. 20.  654 
   196 That is, 6540, being deducted out of 8500, there remains 1960, and so ends the second Operation.
   1962 
    327Here 327 is brought down to its own value again.
Quot. 6.  1962That is 6 times 327 is just e­qual to 1962, and so the work is at end.
   0000 

[Page 22] But now if this Dividend had been greater by 20 Unites, that is, if it had been (139322) the work had been the same, and the Quotient had been the same num­ber of Integers, but there had been found remaining a broken part of Fraction of 20. which must have been be set thus 20/327

CHAP V. Notation of FRACTIONS.

A Broken Number, (otherwise called a Fraction) is part of an Integer; For example, A Foot in length contains 12 Inches. One pound 20 shillings, One shilling, 12 pence; One penny, 4 Farthings, &c.

The Parts of a Fraction.

A Fraction consists of 2 parts, The Numerator and the Denominator, which are placed one above the other, and separated by a little line; For example, If I would ex­press three quarters of a yard, it must be set thus ¾ Numerator Denominator.

F. Inch.

Three Foot and nine Inches, thus. [...]

Or if a Foot be divided in­to an hundred parts, thus [...]

Or if a Foot be divided onely into four parts, thus [...]

And so are all other broken Parts or Fractions exprest of what kind soever.

CHAP. VI. The Reason of Translating Fractions from one Denomination to another, as likewise of reducing them to their least Tearms, and the truth of the Operations demonstrated from several Propositions of Euclids Ele­ments.

HAving thus explained the nature of Fractions, I shall in the next place proceed to demonstrate the Reason of Translating Fractions from one Denomi­nation to another, as likewise the Reason of depressing or reducing them to their lowest and least Terms. All which is necessary to be known by those who desire to be Masters of Fractions.

First of Translating Fractions from one Denominations to another.

Eucl. l. 7. Theor. 16. Prop. 18. [...], &c.

If two numbers multiplying any number, produce other numbers, the numbers produced of them, shall be in the same proportion that the numbers multiplying are, vid. Fig. E.

For the applying of this Proposition to the matter in hand, I lay before me these two Fractions, namely ⅔ and ¾ as in Fig. F. And first I single out ⅔ and Multiply them by (4) the Denominator of the next Fraction and they become 8/12 for 4 times 2 is (8) and 4 times 3 is (12) as may be seen in Fig. G.

[Page]

[figure]

[Page 24] This done, I say, that by thenfore-mentioned Propo­sition, this Multiplication of the two numbers (2) and (3) by a 3d number (4) has altered no proportion, but the products have the same proportion one to another, as the numbers (2) and (3) have, by which they are multipli­ed, that is to say, as (2) is to (3) so is (8) to (12).

Again, I single out the other Fraction, viz: ¾ and I multiply these two by a third Number, namely by (3) the first Denominator. And the Products are 9/12 as in Fig. H. Neither hath this Multiplication altered any proportion, by the same reason with the former. For as (3) is to (4) so is (9) to (12). By which means I have two new Fractions, viz. 8/12 and 9/12 which are the same in effect with the first two, viz. ⅔ and ¾. And whereas they were before of different Denominations, they are now brought under one and the same Denomination.

And thus is that 18th Prop. of the 7th Book of Euclid, the true ground and reason of altering and translating Fractions from one Denomination to another; And from divers and different Denominations, to one and the same. Which Proposition being once throughly digested, and imprinted in the mind and memory, all other Operati­ons which relate to Fractions, (as Adding, Substracting, Multiplying, and Dividing them, as also extracting their Square and Cube Roots) will admit of very few or no difficulties.

Secondly, The way of reducing Fractions to their least Terms.

But forasmuch as it often happens in the multiplying and translating of Fractions that the swell into too great numbers, which are not so tractable as smaller numbers are. It will be proper in the next place, to shew the way of reducing them to their least terms, either before, or after they are thus multiplied or translated, as the practitioner shall see occasion. For the better ef­fecting of which, he is desired to consider some few Definitions and Problems of the 7th Book of Euclid's Ele­ments,

Eucl. l. 7. Defin. XI. [...], &c.

A prime number is that which is measured onely by an Unite.

That is to say 2, 5, 7, 11, 13, &c are prime numbers, because neither of them can possibly be divided into e­qual parts by any thing less then an Unite.

Defin: XII. [...], &c.

Numbers prime the one to the other, are such as only an Unity doth measure, being their common measure.

And such are (9) and (14) for these two numbers can­not be divided into less then Unite parts, so as that which measures one, may measure the other.

For though (3) will measure [...] 9, that is to say, it is found 3 [...] times in (9) yet it will not measure (14) that is, after it has been found as many times as it can be found in (14) there will be two odd Unites left.

Again though (2) do measure (14), that is to say, is found just 7 times in (14) yet it cannot be found any certain number of times in (9) but there will be an odd Unite or Unites left.

Defin. XIII. [...], &c.

A composed number is that which some certain num­ber measureth.

So is (15) a composed number, because (3) multiply­ed by (5) makes (15). And so is (20) a composed number; because 4 multiplyed by (5) makes a Product of (20).

Defin. XIV. [...], &c.

Numbers composed, one to the other, are they, which some number, being a common measure to them both, doth measure.

And thus are (8) and (12) composed Numbers one to another, because there is a certain number, viz. (4) which being repeated, or added to it self a certain num­ber of times, composes both the one and the other of these numbers. That is to say, (4) being repeated (or added to it self) twice, composes (8) the one of these numbers, and the same (4) being repeated (or added to it self) 3 times, composes (12) the other of these numbers. And this (4) is called the common measure of (8) and 12. as in the following Table.

Two composed Numbers8 and 12
Divisors of these 2 composed Numbers.2 3
Their common measure.4
Three composed Numbers6 & 8 & 12
Divisors of these 3 eomposed Numbers3 4 6
Their common measure2

Probl. 1. Prop. 2. [...], &c.

Two Numbers being given (not prime the one to the other) to find out their greatest common Measure.

For example, let the Numbers given be (A=15) and (B=9)

I take the less Number [...] (B=6) from the greatest (A=15) as oft as I can, which is once: If nothing remain then (B=9) is the greatest common measure. If something? Let it be (C=6), and then I [Page 27] take (C=6) out of (B=9) as oft as I can, and if there remains nothing, then (C=6) is the greatest common measure; and if there remain something, let it be (D=3) then I take (D=3) twice out of (C=6) and because nothing remains, therefore (D=3) mea­sures (C=6) and therefore it measures (B=9) and al­so (A=15) And is likewise the greatest Number that measures B and A, that is to say their greatest common measure. And this will be in all other Numbers (not prime one to another) before I can come to an Unity.

Probl. III. Prop. 35. [...], &c,

How many Numbers soever being given, to find the least Numbers, that have the same proportion with them.

Let the Numbers propounded be (A=6) (B=8) (C=12) either these are Prime Numbers, or else there is a certain Number which being repeated, (or added to it self) a certain Number of times, composes either of them, and is therefore their common measure.

If they be Prime Numbers, I have granted what I desire.

[...]If not, let then their greatest com­mon measure be D. And look how many times D is found in A, let E have so many Unites. And let F have as many Unites as D is found times in B. And G as many Unites as D is round times in C. That is, (D=2) multiplied by (E=3) produces (A=6). And (D=2) multiplied by (F=4) produ­ces (B=8). And (D=2) multiplied by (G=6) pro­duces (C=12).

Then E, F, and G, are the least Numbers that have the same proportion with the first Numbers, A, B, and C.

CHAP. VII. Addition of FRACTIONS.

LEt the given Fractions to be added together be ⅔ and ¾.

First I reduce them by the 6 Chapter, to 8/12 and 9/1

Then I add 8 to 9, which makes it 17/12 or 1 5/12.

If they be more then two. First reduce two of them to one Denomination. And then the sum of those two, and the next, &c.

For example, Let the Fractions be ⅔.¾.⅘. First, ⅔ and ¾ make 17/12. Then 17/2 and ⅘ being added as be­fore, make 36/60 and 85/60. which make 121/60 or 2 1/60

CHAP. VIII. Subtraction of FRACTIONS.

FRom ¾ are to be deducted ⅔. That is by the forego­ing Rules from 9/12 are to be deducted 8/12. There­fore there remains 1/12.

CHAP. IX. Multiplication of FRACTIONS.

THis is onely to Multiply the Numerators one by an­other, for a New Numerator, and then the Denomi­nators one by another for a New Denominator.

[Page 29] For example. To multiply ⅔ by ¾ is to multiply 2 by 3. (6) for a new Numerator, and then 3 by 4 (12) for a new Denominator. This is ⅔ by ¾ give 6/12 or ½ for the Product.

But to make all more plain, and to present both the Operation and the Reason of it, to the Readers eye, at one and the same instant; let him consider the less Ob­long (6) in Fig. L. (made by multiplying the Numera­tor (2) by the Numerator (3) included in the greater Oblong (12) made by multiplying the Denominator (3) by the Denominator (4) then which nothing can possibly be more plain, or satisfactory.

[Page]

L

CHAP. X. Division of FRACTIONS.

THe Rule for Division of Fractions, is this,

Multiply the Denominator of the Divisor, by the Nu­merator of the Dividend, for a new Numerator; And the Numerator of the Divisor, by the Deno­minator of the Dividend, for a new Denominator. And the new Fraction is the Quotient.

For example, if ¾ be to be divided by ⅔

Then is ¾ the Dividend.

And ⅔ the Divisor.

  • Then / 3 into 3 make a new Numerator, viz 9/8
  • And 2/ into / 4 make a new Denominator, viz. 9/8

that is 9/8

That is to say ⅔ is contained in ¾ once, and one eighth part. 1 1/8

The Reason of this operation is this.

First, I turn the two Fractions ¾ and ⅔ into two new Fractions, viz. 9/12 8/12 (as before has been shown). Then, as if they were plain Numbers, I divide (9) by (8) and the Quotient is (1 1/8). That is to say, the Di­vidend (¾) is to the Divisor (⅔), as (9/12) to (8/12). But now (9) is the Product of the Divisors Denomina­tor Multiplied by the Dividends Numerator, and is the Numerator of the Quotient, And (8) is the Product of the Divisors Numerator Multiplyed by the Dividends Denominator, and is the Denominator of the Quotient; which is consonant to the foregoing Rule, and that which was to be demonstrated.

[Page 31] Thus if 6/12 be to be divided by ¾.

/4 into 6/ make a new Numerator 24 And 3/ into / 12 make a new De­nominator 36 viz. 24/36

Which being reduced to its least terms, is ⅔

CHAP XI. Of Decimal FRACTIONS.

THese Decimal Fractions are of all Fractions the most natural: For the truth is, all plain Num­bers as they are exprest by the Arabick Notes of 1, 2, 3, &c. are nothing else in effect, but Decimal Fractions. For example, Let any Number be given (432) The last Figure (2) is really 2/10, and so are the last two Fi­gures (32), truly and properly 32/100. And all three are a Fraction of 432/1000. For as the very progression of these and all other plain Numbers is Decimal; that is, each Figure on the left hand, is ten times the value of the same number placed in the next place on the Right. Thus in (333) the last (3) on the left is ten times the value of the (3) next to it on the right; And the (3) in the middle, is ten times the last (3) on the right, and but the tenth part of that (3) on the left. And therefore all, or any of them may be pro libitu, either Fractions or Integers: If I would have them Integers, I set them down without any Line drawn under them; But if I would have the two last a Fraction, I put a se­parating Comma between them, thus, 3, 33, that is 3 Integers, and 33/100 for the Denominator is here to be un­derstood to be an Unite of the next place or Denomina­tion, as was before explained. And thus 5270 is an Integer or whole number, and the same number with a se­parating [Page 32] Comma, thus 5, 270 is an Integer (5) with a Fraction of 270/1000. Or the whole is a Fraction of 5270/10000.

CHAP. XII. Of Addition and Substraction of Decimals.

THe Operations of Addition and Substraction in De­cimals, Integers, and Fractions, is the very same with that of plain Integers, only the careful setting the Unites of all the Integers in one File, and if there be any void places, they are to be imagined to be filled up with Cyphers.

Examples of Addition.

[...]

Examples of Subtraction.

[...]

CHAP. XIII. Multiplication of Decimals.

THis is likewise the very same Operation with that of Multiplying plain Integers, save onely, when [Page 33] all the work is ended, there must be as many places of Parts, or Fractions in the Product, as there were places both in the number multiplied, and in the number multi­plying. As in the following Example, there are two places cut off in the number multiplied, and one in the number multiplying; Therefore there are three places cut off in the Product. And in the second Example, be­cause there are three places cut off in the Multiplicand, and two in the Multiplicator: therefore there are five cut off in the Product.

  • 1. Example. [...]
  • 2 Example. [...]
  • 3 Example. [...]
  • 4 Example. [...]
  • 5 Example. [...]
  • 5 Example. [...]

In the 4 Example, because there are no Integers, the Product is 783/1000

In the 5 Example, because there do not arise but five places in the Product, viz., 01875 and yet by the Rule there ought to be six places cut off, therefore two Cy­phers [Page 34] must be prefixt to make up the number of six pla­ces, viz. 001875/1000000

CHAP. XIV. Division of Decimal FRACTIONS.

1. IF the Dividend be greater then the Divisor, the Quo­tient will be either a whole number, or a mixt; but when the Dividend is less then the Divisor, the Quo­tient must be a Fraction.

2. Whatever the Diuidend be, if need require, there must be a competent number of Cyphers added to it, so make room for the Divisor to stand under it, and then it is no other but ordinary Division, as will appear by the following Example.

Dividend 172, 5

Divisor 3, 746

which the Dividend being supplied with Cyphers, stands thus

[...]

CHAP. XV. The Vulgar Precept for extracting the Square Root or side of any Plain Number.

HAving first pointed the given Number, (suppose 625) that is to say, sett a prick or point over every other Figure, beginning with the last. Whereby I know that the Root has two places and no more. First, I enquire whether (6) the last pointed Figure or Figures, be a true square Number. If it be, then I set down the Root (which is easily found by the following Table, and indeed ought to be retained in the memory) some where on the right hand of the work, and so is the first Figure of the Squar-Root found. But if it be not a true Square Number, then I take the Root of the Square next to it, as in this case, I see (6) is not a true Square, I therefore take the Root of (4) which is the nearest to it, and make that the first Figure of the Root. And then Subtracting (4 the Square thereof) out of (6) I have left remaining (2) which I set underneath, and so is the first Operation ended.

[...]

This done, I take down the Figures of the next Square Number, viz. (25.) Then I set the double of the first Figure of the Root under the first Figure of the second Square Number; and I seek out a Digit, which being multiplied into it self, together with the double of the Quotient (or first Figure) may take a­way the remaining Figures of the given Number, viz. (225) or at least as mvch as may be; which Digit, by as king how many times (4) the double of the Root is sound in the Figures standing over it, as in the nature of a Dividend (namely 22) for I find that (4) will be found in (22) 5 times, and enough over and above to [Page 36] multiply that 5 into it self also. Therefore I set down (5) for the second Figure of the Root, and I also set the same (5) under the last Figure of the second Square; and then I multiply the double of the first Figure, augmented by that second Figure (5) that is, I multiply (45) by that (5) and if the Product be ei­ther equal to the Figures (225) standing above them, or so much less then (225) as the value of any Num­ber under (45) then the second Figure is rightly chosen otherwise the Figure next less must be taken. But in this example, the Product happens to be just equal And so the work is ended.

[...]

Now in case this (625) had been (655) which is a greater Number then (625) by 30 Unites; the Integers of the Square Root had still been (25;) onely there had been left (30) which had been the Numerator of a Fracti­on, whose Denominator must evermore be the double of the Root, augmented by an Unite, and then the Operation had been, as here you see.

[...]

CHAP. XVI. The Reason and Demonstration of the Vulgar Operation of Extracting the Square-Root.

ANd after this very manner and method may the Square-Root of any Plain Number in Integers be [Page 37] extracted, though never so great; but that this and all other Operations of the same nature may be also performed with understanding, and satisfaction, it will be necessary to make some reflections upon the nature and genesis of a Square-Number, and in order there­unto, the Practitioner is desired to consider the fol­lowing Definition of a Square-Number.

Eucl. l. 7. Defin. XVIII. [...], &c.

A Square-Number is that which is equally equal, or, which is contained under two equal Numbers.

THus the Square-Number (4) is contained under two equal Numbers, viz. (2) and (2) and the Square-Num­ber (9) is contained under two equal Numbers, namely (3) and (3) and so on as in the following Table.

A Table of Squares with their Genetive equal Number.

Equal MumberSquare
1 into 11
2 into 24
3 into 39
4 into 416
5 into 525
6 into 636
7 into 749
8 into 864
9 into 981
10 into 10100

Thus the Square-Number (625) is contained under two equal Numbers, viz. (25) and (25) That is to say, [Page 38]

Sectio QUADRATI (625) in quatuor Plana, à duobus Lateris (25) Segmentis, viz. (A=20) & (B=5) effecta; quorum tria ordinatim sumpta, sunt continuè proportionalia, nimirum

  • 1 A quadratus=400=maximus Proportionalis
  • 2 A in B=100=medius Proportionalis
  • 3 B quadratus=25=minimus Proportionalis

[Page 39]

One of the equal Numbers25
being multiplyed by the other equal Number.25
Makes the Product a Square Number, viz.625

Eucl. l. 2. Theor. 4. Prop. 4. [...], &c.

If a right Line be cut any wise into two parts, the Square made of the whole Line, is equal both to the Squares made of the Segments, and to twice a Rectangle made of the Parts.

THis holds good likewise in Numbers. For example, Let (25) be the Number of a Right Line, and that di­vided into two parts: viz. (20) the greater, and (5) the lesser. I say,

The Square of the whole (25) is equal to the two Squares of (20) and (5) more by the two Oblongs made of (20) multiplied by (5) as in the opposite Fi­gure may be plainly seen.

For let (20) be calledA
And (5) be calledB
A q: (or 20 multiplyed by 20) makes400
B q: (or 5 multiplied by 5) makes25
A, Multiplied by B, that is (20) multiplied by (5) makes100
A, Multiplied by B, that is (20) multiplied by (5) makes100
That is to say, these are the parts, which being united are equal to the whole Square625

These things being premised, I say, that whereas the vulgar Rule directs (after the pointing of the Num­ber 625 whose Square Root is to be extracted) to find out the Square-Root of the last pointed Figure on the left hand, that is (6) and not finding (6) a true Square Number, to take the next to it, viz. (4) whose Square-Root is (2.) I say, the meaning is this; That [Page 40] (6) is in effect (600) which is not a true Square-Number, and the nearest to it is (400) whose Square-Root is (20.)

Again, whereas the vulgar Rule directs to subtract the Square-Number (4) out of (6) and so to find out how many times the double of the first Root is con­tained in the Remainder and the first Figure of the next Square, viz. (22) that is 5 times, But with this provision that there may remain a Number equal to the Square of that (5) as in the example.

[...]

The meaning is this, Having Subtracted A q, or the Square-Number (400) out of the Number (625) there remains (225) which is (A=20) into (B=5) or (100) for one of the Oblongs, and (A=20) into (B=5) for the other Oblong, and B q or (25) for the lesser Square. All which are the very parts of a Square, expressed in the foregoing Proposi­tion of Euclid. Namely.

(A q) The Square of the greater Segment which is equal to—400
(B q) The Square of the lesser Segment which is equal to—25
(A) Into (B) one of the Rectangles—made of the Segments, and equal to100
(A) Into (B) the other Rectangle made of the Segments and equal to—100
[Page 41]The Square made of the whole (A▪ [...]) [...] or (25) multiplied by (25) is [...] [...]

And be the Square-Number never so great, both the manner of the Operation, and the Reason of the Ana­lysis, or extraction of its Square-Root▪ is the very same.

But for the better understanding of all that haha been said, let the Practitioner consider w [...]ll the Figure Where he may evidently see, how the Root of every lesser Square being doubled, and an Unite added to it, makes up the next greater Square

Thus twice (3) or the double Root of the next less Square, more by (1) being added to that Square (9) makes up the next greater Square, viz. (16) And twice (5) or the double Root of the next less Square (25) more by (1) that is to say (11) being added to that (25) makes it (36) which is the next greater Square.

CHAP XVII. The way and Reason of extracting the Square-Root of FRACTIONS.

HE who would rightly comprehend the Nature and Reason of extracting the Square Root of any Fraction, let him consider the Numerator to be a lesser Square-Number, and the Denominator to be a greater Square-Number, and so the lesser to be included in the greater; as for example, if it were demanded to ex­tract the Square-Root of 9/25 the Numerator (9) is to be considered as a lesser Square-Number, and the De­nominator (25) as a greater Square-Number, including or comprehending that lesser or (9 And then all that is to do, is onely to extract the Square-Root of the Nu­merator (9 for a new Numerator, viz. (3) and then the Square-Root of the Denominator (25) for a new De­nominator, [Page 42] viz. (5) As if they were Plain Numbers; That is to say, the Square-Root of 9/25 is ⅗

But now, if neither the Numerator or Denominator be Pure-Square-Numbers, the Operation is somewhat more intricate, though the Reason of it be still the same. As for example, if it were demanded to extract the Square-Root of ⅗ still the Numerator (3) is to be considered as a lesser Square, included in a greater, viz. (5)

That is to say, The Square Root of (3) is [...]

And the Square-Root of (5) is [...]

First, I bring the two Fractions ⅔ and ⅗ to two Fra­ctions of the same Denomination, by the Rule laid down Chap. 8. Pag. 34. and they are 10/15 and 9/15. So then

The Square-Root of (3) is 1 10/15 or 25/15

And the Square-Root of (5) is 2 9/15 or 39/15

CHAP. XVIII. The Vulgar method of Extracting the Cube-Root.

FOr example, let 15625 be the Number whose Root is to be extracted.

First, it is to be pointed, beginning with the last Fi­gure on the Right hand, and from thence to the 4th from the right hand inclusive, and so to the end, pointing one and leaving two. Thus 15̇625̇ Where Note that the Number of pricks, or points signify the places of the Root sought.

Secondly, the Cube-Root is to be enquired of the last prickt Number on the left hand, (be they three, two, or one) As here the Cube-Root of 15 is to be demand­ed, and being found by the Table of Cube-Numbers Pag. 45. to be 2. I set it in the Quotient for the first [Page 43] Number of the Cube-Root, and then setting the Cube of that (2) viz. (8) under (15) and subtracting the first out of the last, there remains (7) And this is the first step, and the work stands thus.

[...]

In the next place having set (6) the triple of (2 the first found Figure of the Root) under the last Figure save one of the next prickt Number, I square the same (2) and then it is (4) and then I triple that Square (4) and make it (12) and then I set that (12) under the Remainder 762 just as in Division, and this done I ask how many times, (12) is found in the Figures 76 Iust above them, and because it may be found 5 times I set (5) at all adventures for the second Figure of the Root, and then the work stands thus.

[...]

This done, I multiply (the Triple of the Root mul­tiplied by the first Figure of the Root) by this second Figure; That is, I multiply (12) by (5) and set it just under (76).

Again, I multiply the Square of the last Figure, by the Triple of the first, that is (25) by (6) and set it underneath likewise.

In the next place, I Cube that last Figure, (that is I Cube 5) and set that Cube, viz. (125) under the second Cube-number, that is, setting the Unite place of the Cube (125) under the second prickt [Page 44] Figure. And add together those Products, and then the work stands thus.

[...]

Lastly, I compare this (7625) with the Remainder that was left after the deduction of the first Cube-num­ber, 8 and finding them just equal, I turn it off and there remains nothing, the work is at an end, and (5) is the second and the last Figure of the Cube Root.

But now, if the aforesaid Numbers 6000, 1500, & 150, had been greater Numbers, and consequently, being added together, had made a Number greater then 7625) then I must have taken a lesser Number, and consequent­ly the second Figure of the Cube-Root had been not 5) but 4).

Again, if the Numbers 6000, 1500 and 150 being added together, had not made a Number first equal to (7625) That is to say, suppose the Cube-Number given had been 15645, which is more by (20) consequent­ly after the work ended, there had remained 20) which 20 had been the Numerator of a Fraction, whose Denominator is the difference of the Cube of 25 and (26 and that Fraction had been set down thus, 20/1951

And then the answer had been thus, the Cube-Root of (15645) is 25 20/1591

CHAP. XIX. The Reason and Demonstration of Extract­ing the Cube-Root.

THough the practick of this Extraction may at first sight seem something difficult, yet the Reason and Demonstration of it, will I doubt not, make re­compence.

And in order thereunto I shall first desire the Practi­tioner to consider the Definition of a Cube, as also the Sections of a Cube into its Analytical parts, according to the Rule of Perspective.

Eucl. l. 7. Defin. XIX [...], [...], &c.

A Cube is that Number which is equally equal, or, which is contained under three equal Numbers.

THus (8) is a Cube which is contained under (2) and (2) and (2).

A Table of Cubes with their Gene­tive equal Numbers.
Three equal NumbersCube,
1into1into11
2into2into28
3into3into127
4into5into464
5into5into5125
6into6into6216
7into7into7347
8into8into8512
9into9into9729

[Page 46]

The Section of a CUBE (125) by a Binomial Root (A+B) or (3+2)

Sectio CUBI in Octo Solida, a duobus Lateribus {Act B} i. e. {3 et 2} effecta

Quorum Quatuor ordinatim sumpta, sunt continue Proportionalia; nimirum

  • A cubus = 27
  • A quadr in B = 18
  • A in B quadr = 12
  • B cubus = 8

[Page 47] Thus (3) and (3) and (3) make the Cube (27) for (3) multiplied by (3) makes the Square (9) and the Square (9) multiplied again by (3) makes the Cube (27) as is more clear by the foregoing Table.

And so doe (5) and (5) and (5) make the Cube (125) That is, (5) into (5) makes the Square (25) and the Square (25) into (5) makes the Cube (125) &c.

Wherefore to Extract a Cube-Root, is nothing else then to find out a number, which being first multipli­ed into it self, and then into the Product, produces the given Cube-Number. Thus to extract the Cube-Root of (15̇625̇) is to find out the number (25) which be­ing first multiplied into its self (makes 625) and then multiplied into that (625) makes the given Number (15̇625̇)

Now because this construction of the Cube from a sin­gle Root, contributes nothing towards the finding out that Root from a given Cube-Number, therefore was found out by the Antients, that admirable Art of cutting or dividing the Root into two parts, which they therefore called a Binomial Root; and from those two parts they erected 8 solid numbers, whereof the great­est and the least are always two pure Cube numbers, of those two distinct parts, and of the other six Paral­lelepipedons, by which I mean solid Numbers made by multiplying the Square of one Number into another Number, in imitation of the Geometrical Parallelepi­pedons defined by Euclid. lib. 11. Defin. 30. to be a solid Figure contained under six Equilateral Figures, whereof those which are opposite are Parallel.

The three greatest Parallelepipedons are equal one to another, and each of them made by multiplying the Square of the greatest part of the Binomial-Root into the lesser part.

And the three lesser Parallelepipedons are equal to one another, and each of them made, by multiplying the Square of the lesser of the Binomial Root into the great­er part.

Thus in the opposite Figure the whole Root is (5) and divided or cut into a greater part A= 3 and a lesser part B=2

[Page 50]

The Cube of the greater A=3 is equal to27
The Cube of the lesser B=2 is equal to8
One of the greater Parallelepipedons, or (Aq into B) is equal to18
One of the lesser Parallelepipedons or (A into Bq) is equal to12

Then To those two Cubes, namelyThe greater or Ac27
The lesser or Bc8
Adding Three of the greater Paralle­lepipedons, viz.(Aq) into (B)18
(Aq) into (B)18
(Aq) into (B)18
And Three of the lesser Parallele­pipedons, viz.(A) into (Bq)12
(A) into (Bq)12
(A) into (Bq)12
The total summe is the entire Cube Num­ber of the Binomial-Root A=3 more by B=2 that is to say (5) and amounteth to125

Now the Practitioner is to conceive the Unites of the Cube-Number (125) to be as so many Dice, or Cu­bical Unites, and 27 of these being piled one upon another orderly and equally to make up (Ac) and 8 of them to make (Bc) and 18 of them to make one of the greater Parallelepipedons (Aq into B) and 12 of them to make one of the lesser Parallelepipedons (A into Bq) and then all these eight solid Numbers being orderly put together, to make up the entire Cube or (125) And this is the Genesis of a Cube. vid. Fig.

After the very same manner, let the Root (25) be made Binomial and cut into two parts, viz. (20) and (5) and the greater called A, and the lesser B

Then

The Cube of the greater A=20 is equal to8000
The Cube of the lesser B=5 is equal to125
The greater Parallelepipedon, or (Aq into B) is e­qual to2000
The lesser Parallelepipedon, or (A into Bq) is equal to500
[Page 49]Then to these two Cubes, namelyThe greater or (Ac) 8000
The lesser, or (Bc) 125
Adding Three of the greater Parallelepi­pedons, viz.(Aq) into (B) 2000
(Aq) into (B)2000
(Aq) into (B)2000
And Three of the lesser Parallelepipe­dons, viz.(A) into (Bq) 500
(A) into (Bq)500
(A) into (Bq)500
Summe15625
The total Summe of all the eight Solids, is the Summe of the entire Cube which amounteth to15625

All which is consonant to that Theorem of Ramus, (which is in imitation of that of Euclid concerning a Square Number.

P. RAMI Geometr. Lib. XXIIII, De Cubo pag. 135.

IF a right Line be cut into two Segments, the Cube of the whole shall be equal to the Cubes of the Se­gments, and thrice the double Solid made of the Square of one Segment.

Thus in the last Example, the Square of the greater Segement, multiplied by the lesser, and the Square of the lesser Segment multiplied by the greater Seg­ment, (for that is meant by the double Solid made of the Square of the one Segment into the other Segment) is (2500)

And,

The triple of that (2500) is7500
To which adding the two Cubes, or8125
The total is the entire Cube, viz.15625

After all this, it will be easie for the meanest ca­pacity to conceive, that while by the vulgar Rules of Extracting the Cube-Root of 15625, the first Cube of (8) is subtracted out of (5) it is indeed (8000 or Ac [...] that is the Cube of the greater Seg­ment subtracted out of (15625) whose Cube-Root is [Page 48] (2) or in appearance, but in reality is (20) or the greater Segment (A)

Again, whereas in the next place the vulgar Rule di­rects, to set down the triple of the first found Figure (2) under the last Figure save one of the next Cube-Number, viz. (2) and then to square that (2) and so make it (4 and then to triple that (4) and so make it (12). And then to find how many times that (12) can be found in the remainder, or (7625) which suppose to be 5 (times) with this provision, that after (12) has been deducted thus 5 times out of (7625) there still remain a Number, either greater or equal to the Square of that (5) that is (25) multiplied by the triple of the first-found Root (2) more by the Cube of 5) that is (125) and if so, the work stands good, and the Cube Root is (25).

The true meaning of all is this.
  • 1. The first found Root (2) is Really (20) and is the greater Segment (A)
  • 2. The Square of that (20) is (400) and is (A) Square.
  • 3. The Triple of that (400) is 1200) and is (A) Square thrice.
  • 4. This (1200) is found in (7625) 5 times, which (5) is the lesser Segment (B)
  • 5. This (1200) multiplied by (5) or (B) makes (6000) and is thrice (A) Square into (B)
  • 6. The Square of that 5) is (25 or (B) Square.
  • 7. This (25) or (B) Square multipled by (6) or the triple of (A) makes (1500) and is (A) into (B Square thrice.
  • 8. The Cube of this (5) is (125) or (B) Cube. All these being added together, makes up the en­tire Cube.

That is to say,

A, Cube, or the Cube of the greater Segment8000
2. A Square into B, or the triple of the solid Number made of the Square of the great­er Segment multiplied by the lesser Seg­ment6000
[Page 49]3. A, into B Square, or the triple of the solid Number made of the Square of the lesser Segment multiplied by the greater Segment1500
B Cube, or the Cube of the lesser Segment125
The total sum is the intire Cube15625
The given Cube
Num. 
15625AB the Binomial 25 Root, viz. A=20
8(A) Cube.
7625Remainder
 60(A) thrice
1200(A) Square thrice
Aggregate of the three for­mer sums and equal to the Remainder a­bove said.
6000(A) Square into (B) thrice, or the triple of the greater Parallel.
1400(A) into (B) Square thrice, or the triple of the lesser Parallel.
7125(B) Cube.
 625 

CHAP. XX. A plain and easie Method of extracting the Square-Root of any Number, (how great soever) without the help of either Mul­tiplication, or Division.

For Example. Let the Square number be—

2616
6843456
4   
284  
276  
 834 
 521 
 31356
 31356
 00000

THere are therefore 4 Figures in the Root.

The first, viz. (2) is found by inspection.

The second is had by the Tariffa (A) thus,

Take the double of the first Fi­gure (2) viz. 4, and make a Mul­tiplication Table of it, mingling with the respective Products, the Squares of the 9 Digits as in A. So shall you by inspection find that 276 being the nearest Number to 284, stands over against 6 in the Margint, and therefore (6) is the second Figure of the Root.

[Page 51]

A
1 41
2 84
3129
416 
 16
520 
 25
624 
 36
728 
 49
832 
 64
936 
 81

B
1 521
21044
31569
4208 
  16
 260 
   25
     
     
     
     
     
     
     
     

C
1 5221
210444
315669
420886
   1 
52610 
   25
63132 
   36
73654 
    9
      
      
      
      

In the next place, I offer to make the Tariffa B, for the double of the two first Figures 2 and 6, viz 52. But by inspection, I find that it is need­less to fill up the spaces, the very second Product being too great for the Number 834. Therefore (1) must be the next Figure of the Root.

Lastly, I make a Tariffa for the double of the first 3 Figures of the Root, 2, 6, and 1, viz. 522. And by that time I come to the seventh place, I find that 6 is the last Figure, for it gives me 31356, which is the very Number I sought for.

This Method is the most certain, and has the least of difficulty in it of any Methods I ever yet saw.

CHAP. XXI. Of Proportions Arithmetical, Geometrical, and Musical.

1. Arithmetical Proportion or Habitude, is an Equality of Differences, That is to say, when several Numbers have one and the same Diffe­rence: And this Habitude is two fold.

1. COntinued: When of several Numbers, the second exceeds the first by the very same Unity or Number of Unities, as the third doth the second, and as the fourth doth the third; and so in Infinitum, Thus, 1, 2, 3, 4, 5, 6, 7, 8, &c. differ by an Unit. And 1, 3, 5, 7, 9, &c. have their equal Difference (2.) The orderly proceeding of which Numbers from the lesser to the greater in a Scalal-way, is that which is properly called Arithmetical Progression.

2. Disjunct, when the second exceeds the first by the same Number of Units, as the fourth doth the third, but not as the third doth the second. As for Example, 1, 3, 7, 9, are four Disjunct Arith­metical Proportionals. For (3) exceeds (1) by the same Number of Units as 9) doth (7), but not as (7) doth (3). And thus, 2, 7, 10, 15, are four Proportionals of the same kind, for (7) ex­ceeds (2) by 5, Units, and so doth (15) exceed (10).

[Page 53] 2. Geometrical Proportion or Habitude, is the Equality of Ratio's, that is to say, it is that which shews what part or parts one Number is of another.

Thus, (1, 2, 4, 8) (2, 4, 8, 16,) (3, 6, 12, 24) (4, 8, 16, 32, 64) are Geometrical Proportionals: For in the first Example, as (1) is the half of (2), so is (2) the half of (4), and again, (4) the half of (8), as may be seen in the following Tables.

3. Musical Proportion or Habitude, is when the first Number hath the same Proportion to the third, which the Difference between the first and the second hath to the Difference between the second and the third.

As in (3, 4, 6), (3) is the half of (6), and so is (1) or the Difference between (3) and (4) the half of (2), or the Difference between (4) and (6); and so in (6, 8, 12). But of this in its proper place.

That which is at present to be handled, is the Nature, Properties, and Similitude of the two first kinds of Proportion, Namely Arithmetical and Geometrical, which may be viewed in their several Progressions, by the following Tables.

[Page 54]

123456789101112
24681012141618202224
369121518212427303336
4812162024283236404448
51015202530354045505560
61218243036424854606672
71421283542495663707784
81624324048566472808896
918273645546372819099108
102030405060708090100110120
112233445566778899110121132
1224364860728496108120132144
13263952657891104117130143156
14284256708498112126140154168
153045607590105120135150165180
163248648096112128144160176192
1734516885102119136153170187204
1836547290108126144162180198216
1938577695114133152171190209228
20406080100120140160180200220240
21426384105126147168189210231252
22446688110132154176198220242264
23466992115138161184207230253276
24487296120144168192216240264288
255075100125150175200225250275300
265278104130156182208234260286312
275481108135162189216243270297324
285684112140168196224252280308336
295887116145174203232261290319348
306090120150180210240270300330360
316293124155186217248279310341372
326496128160192224256288320352384
336699132165198231264297330363396
3468102136170204238272306340374408
3570105140175210245280315350385420
3672108144180216252288324360396432
3774111148185222259296333370407444
3876114152190228266304342380418456
3978117156195234273312351390429468
4080120160200240280320360400440480

[Page 55]

 1 1 1
 2 3 4
 4 9 16
 8 27 64
 16 81 256
 32 243 1024
 64 729 4096
 128 2187 16384
 256 6561 65536
 512 19683 262144
 1 1 1
 5 6 7
 25 36 49
 125 216 343
 625 1296 2401
 3125 7756 16807
 15625 46536 117649
 68125 279216 823543
 340625 1675296 5764801
 1703125 10051 [...]76 40353607
 1 1 1
 8 9 10
 64 81 100
 512 729 1000
 4096 6561 10000
 32768 59049 100000
 262144 531441 1000000
 2097152 4782969 10000000
 16777216 43046721 100000000
 134217728 3874204891000000000
 1 1 1
 11 12 13
 121 144 169
 1331 1728 2197
 14641 20736 28561
 161051 248832 371293
 1771561 2985984 4826809
 19487171 35831808 62748517
 214358881 429981696 815730721
2357947691515978035210604499373

[Page 56]

Geometrical Progression.Arithmetical Progression.
 11
 33
 95
 277
 819
 24311
 72913
 218715
 656117
 1968319
 5904921
 17714723
 53144125
 159432325
 478296929
 1434890731
 4304672133
 12914016335
 38742048937
 116226146739
 348678440141
 1046035320343
 3138105960945
 9414317882747
 28242953648149
 84728860944351
 254136582832953
 762559748498755
 2287679245496157
 6863037736488359
 20589113209464961
 61767339628394763
 185302018885184165
 555906056655552367
 1667718169966656969
 5003154509899970771

Geometrical Progression.Arithmetical Progression.
 11
 22
 43
 84
 165
 326
 647
 1288
 2569
 51210
 102411
 204812
 409613
 819214
 1638415
 3276816
 6553617
 13107218
 26214419
 52428820
104857621
209715222
419430423
838860824
1677721625
3355443226
6710886427
13421772828
26843545629

[Page 57]

Geometrical Progression.Arithmetical Progression.
 101
 1002
 10003
 100004
 1000005
 10000006
 100000007
 1000000008
 10000000009
 1000000000010
 10000000000011
 100000000000012
 1000000000000013
 10000000000000014
 100000000000000015
 1000000000000000016
 10000000000000000017
 100000000000000000018
 1000000000000000000019
 10000000000000000000020
 100000000000000000000021
 1000000000000000000000022
 10000000000000000000000023
100000000000000000000000024
1000000000000000000000000025

In this Progression it is more visible then in any other, how Addition and Substraction in Arithmeti­cal Progression answers to Multiplication and Divi­sion in Geometrical Progression. For, as in Geome­trical Progression, 1000 Multiplied by 100, 000 produce 100, 000, 000; So in Arithmetical Progres­sion, the Number answering to 1000, and 100, 000. viz. (3) and (5) being added together make (8), which answers to the Product 100, 000, 000. Again, as in Geometrical Progression (100, 000 000) being divided by (100, 000) the Quotient is (1000) So in Arithmetical Progression, it from the greatest of their Corespondent Numbers, viz. (8) you Sub­stract any one of the other, viz. (5.) the remainder is (3) and answereth to (1000.)

I. Reflection. [...]. Prop. Tertia.

β.α..γ..δ..ε. B.A..G..D..E.

[...], &c. If Numbers (how many soever they be) exceed one another by an equal Interval, then the Interval between the great­est, and the least, is Multiplex of that equal Interval, according to the Multitude of the Numbers propound­ed, less by one.

Let the Numbers propounded be four, viz. |BA1| |BG3| |BD5| |BE7| whose common In­terval is equal to |AG2|

By the Hypothesis |AE6| is the Interval between the greatest |BE7| and the least |BA1|: And likewise the three Numbers |AG2| |GD2| |DE2| and every of them equal to the Common Difference, and equal the one to the other. And the multitude of them equal to the Multitude of the Numbers given less by one (viz. the least.) And lastly, the Aggregate of these three Numbers is equal to the Interval |AE6| (the parts united being equal to the whole.)

That is to say, |AE6| or the Interval between the greatest |BE7| and the least |BA1| is Multiplex of the common Interval

I. Reflection

IF Numbers (how many soever they be) contain the one the other by an equal Ratio; Then the greatest of those Numbers is Multiplex of the Powers of the Denomination of that equal Ratio Multiplyed by the least, according to the Multitude of the Num­bers given, less by one

This Reflection I have framed for Geometrical Proportionals, in imitation of that Diophantus for Arithmetical Proportionals.

Let the Numbers given be Four, viz. 2, 6, 18, 54.

And let the Denominator of the Ratio be 3.

Then by the Hypothesis, the first Multiplyed by (3) is equal to the second; and the second Multi­plyed by (3) is equal to the third; and the third Multiplyed by (3) is equal to the fourth: And so in Infinitum.

That is to say,

  • 1. The first Term (2) is equal to 2
  • 2. The 2d Term (6) is equal to 2 into 3 1st power of the Ratio, or the single Ratio.
  • 3. The 3d Term (18) is equal to 2 into 3 into 3 2d power of the Ratio, or the Ratio Squared.
  • 4. The 4th Term (54) is equal to 2 into 3 into 3 into 3 3d power of the Ratio, or the Ratio Cubed.

That is to say, in a Symbolical way.

Let there be any Number of Proportional Num­bers, as, A, B, C, D, E, F, &c. And the Denomi­nation of the Ratio be R.

[Page 60] |AG2| according to the Multitude of the Numbers propounded less by one, Which was to be Demon­strated.

In a Symbolical way, Thus,

Then let the Terms be A, B, C, D, and the Common Difference E.

Thus, [...]

Or thus, [...]

In plain English thus,
  • 1st Term (1) is equal to 1
  • 2d Term (3) is equal to 1 more by 2.
  • 3d Term (5) is equal to 1 more by 2 more by 2.
  • 4th Term (7) is equal to 1 more by 2 more by 2 more by 2.

That is to say, by the Hypothesis, the greatest Term is equal to the least, and as many Differences as there are more Terms besides the least; there­fore the greatest Term less by the least is Multiplex of the Difference according to the Number of Terms less by one.

As was to be Demonstrated.

  • [Page 61]A=A
  • B=A into R
  • C=A into R into R
  • D=A into R into R into R
  • E=A into R into R into R into R
  • F=A into R into R into R into R into R
  • G=A into R into R into R into R into R into R

Or thus,

 Powers of R.
A=A 
B=A into R1
C=A into R Square2
D=A into R Cube3
E=A into R Squared Square4
F=A into R Squared Cube5
G=A into R Cubick Cube6

As this Reflection carryes its evidence along with it, so is it of admirable Speculation and Use, as hereafter will appear.

II. Reflection.

WHere three Numbers are three Arithmetical Proportionals, the Sum of the two Extreams is equal to the Double of the Mean.

Let the three Numbers be (2, 4, 6)

And the Common Difference be (2)

Then by the First Reflection.

The First Term is 2

The second Term is 2 more by the Difference once.

The third Term is 2 more by the Difference twice.

Whence it is evident and obvious, That twice (2) more by twice the Difference, is both the sum of the first and third Terms, and also the double of the second or Mean.

Which was to be Demonstrated.

Vid. Fig.

II. Reflection. Eucl. Lib. 7. Prop. 20.

[...], &c. If there be three Numbers in proportion, the Number contained under the Extreams, is equal to the Square made of the Mean; And if the Number contained under the Extreams be equal to the Square of the Mean, those three Numbers shall be in Proportion.

Let the three Numbers be (2, 6, 18)

And the Equal Ratio (3.)

Then by the first Reflection.

The first Term is 2 A

The second Term is 2 into (3) A × R

The third Term is 2 into (the Square of 3) A × R × R

Whence it is evident and obvious in the first place, that the first Term drawn into the third is equal to 2 into 2, into the Square of 3.

Again it is evident, That the Square of the second Term is equal to the Square of 2 into the Square of 3.

But these two are equal; That is to say, (2) into (2) into (the Square of 3) = 36 (the Square of 2) into (the Square of 3) = 36 Ergo,

The Product of the First Multiplyed by the Third, is equal to the Square of the Second or Mean. Which was to be demonstrated.

Vid. Fig.

III. Reflection.

WHere four Numbers are four Arithmetical Proportionals, the Sum of the first and fourth, is equal to the Sum of the second and third.

First, let them be four continued Proportionals, viz. (4, 12, 20, 28)

And their Common Difference (8)

Then by the second Reflection.

  • The first Term is 4
  • The 2d Term is 4 more by once the Difference.
  • The 3d Term is 4 more by twice the Difference.
  • The 4th Term is 4 more by thrice the Difference.

Secondly, let them be four discontinued Propor­tionals, viz. (4, 12, 30, 38.)

And their Common Difference (8)

Then by the second Reflection, and the Hy­pothesis.

  • The first Term is 4
  • The 2d Term is 4 more by once the Difference.
  • The 3d Term is 30
  • The 4th Term is 30 more by once the Difference.

III. Reflection. Eucl. Lib. 7. Prop. 19.

[...], &c. If there be four Numbers in proportion, the Number produced of the first and fourth, is Equal to the Number which is pro­duced of the second and third. And if the Number produced of the first and fourth be Equal to that pro­duced of the second and third: Those four Numbers shall be in proportion.

First let them be four continued Proportionals. viz. (2, 6, 18, 54) and the Equal Ratio (3).

Then by the Second Reflection.
  • The first Term is 2.
  • The second Term is 2 into (3).
  • The third Term is 2 into the Square of (3).
  • The fourth Term is 2 into the Cube of (3).

Secondly, let them be four Disjoynt Proportionals. viz. (2, 6, 54, 162) and the Equal Ratio (3).

Then by the second Reflection, and the Hypothesis.
  • The first Term is 2.
  • The second Term is 2 into (3).
  • The third Term is 54.
  • The fourth Term is 54 into (3).

[Page 66] In the first Example it is evident,

That twice (4) more by thrice the Difference (24), is both the Sum of the Extreams, and also the Sum of the second and third, viz. 32.

In the second Example it is as evident,

That (4) more by (30) more by once the Dif­ference, is both the Sum of the Extreams, and also of the second and third Terms, viz. 42.

Vid. Fig.

[Page 67] In the first Example it is evident by the Multi­plication of Powers.

That the Square of (2) Multiplyed by the Cube of (3) is both the Product of the Extreams, and also of the two Means.

In the second Example it is as evident,

That (2 into 3) Multiplyed by (54) is both the Product of the Extreams, and also of the Means.

Vid. Fig.

IV. Reflection.

IN all Continued Arithmetical Progressions, (how many soever the Terms be) the Sum of the Ex­treams is equal to the Sum of any two of the other Terms equidistant from the Extreams, and to the double of the Middle Term, in case the Number of Terms be odd.

Let the Number of Terms be Seven.

viz. 3, 6, 9, 12, 15, 18, 21, and the Common Difference (3).

First by the Hypothesis.

The four Numbers (3, 6, 18, 21) are four Pro­portionals.

Therefore by the fourth Reflection.

Their two sums, viz

  • 3 more by 21
  • 6 more by 18

are Equal.

Secondly, by the same Hypothesis.

The four Numbers (6, 9, 15, 18,) are four Proportionals.

Therefore by the fourth Reflection.

Their two sums, viz.

  • 6 more by 18
  • 9 more by 15

are Equal.

Lastly, by the same Hypothesis.

The three Numbers (9, 12, 15) are three Proportionals.

IV. Reflection.

IN all Continued Geometrical Progressions (how many soever the Terms he) the Product of the Extreams is equal to the Product of any two of the other Terms equidistant from the Extreams, and to the Square of the Middle Term, in case the Number of Terms be odd.

Let the Number of Terms be Seven.

Viz. (2, 6, 18, 54, 162, 486, 1458); and the equal Ratio (3).

First, by the Hypothesis.

The four Numbers (2, 6, 486, 1458) are four Proportionals.

Therefore by the fourth Reflection.

Their two Pro­ducts, viz.

  • 2 into 1458
  • 6 into 486

are Equal.

Secondly, by the same Hypothesis.

The four Numbers (6, 18, 162, 486) are four Proportionals.

Therefore by the fourth Reflection.

Their two Pro­ducts, viz.

  • 6 into 486
  • 18 into 162

are Equal.

Lastly, by the same Hypothesis:

The Three Numbers (18, 54, 162) are three Proportionals.

[Page 70] Therefore by the third Reflection.

  • 9 more by 15
  • 12 doubled

are Equal.

Therefore by the 1st Ax. Euclid.

  • 3 more by 21
  • 6 more by 18
  • 9 more by 15
  • 12 doubled

are all Equal one to a­nother. As was to be Demonstrated.

Vid. Fig.

[Page 71] Therefore by the third Reflection. 18 into 162 and 54 Squared, are Equal.

Wherefore,

  • 2 into 1458
  • 6 into 486
  • 18 into 162
  • 54 Squared

do make 2916, and con­sequently are all Equal one to another: Which was to be Demonstrated.

Vid. Fig.

The foregoing Reflections applyed to the Golden Rule, or Rule of Three.

The Rule of Three, or the Golden Rule.

THis Rule is either Single, or Compound;

1. The Single Rule of Three.

The Single Rule of Three, is when three Num­bers are Given, and a fourth Proportional De­manded.

Now this Single Rule, is either Direct, or Inverse.

1. Direct.

The Single Rule of Three Direct, is when three Numbers are Given, and a fourth is Demanded, which bears the same proportion to the Third, as the Second bears to the First.

Example.

If 4 Acres of Ground cost 80 Pound, what will 8 Acres of the same Ground cost?

To understand this Operation, look back to the third Reflection upon Geometrical Proportion, and you shall find it Demonstrated, that if there be four Proportionals, the Product of the First and Fourth is equal to the Product of the Second and Third.

Wherefore in this Example.

If the Product of the Second and Third Term, [Page 73] viz. (640) be Divided by the First, namely (4) The Quotient, viz. (160) is the fourth Proportional sought.

And for a Proof of this,

Let the fourth found Term (160) being Mul­tiplyed by the first (4); the Product will be the same with the Product of the third Multiplyed by the second, viz. 640.

[figure]

The Argumentation is plainly thus,

If the Product of (80 by 8) be equal to the Product of (4 by the unknown Number.)

Then,

The Quotient will be the very same, whether I Divide the Product of (80 by 8,) or whether I Divide the Product of (4 by the unknown Num­ber) by 4. For either of the Products being 640, the Quotient must needs be 160.

But now it is manifest, that if

I Multiply 4 by 160, and Divide the Product back again by 4, it will give 160 for the Quotient. Because, whatever Multiplication doth, is again undone by Division. And this is the true and genuine Reason of the Operation in this Rule of Three.

[Page 74] That is,

As (4) is found in (80) just 20 times, so is (8) found in (160) just 20 times.

Or thus,

As 4 Multiplyed by 20 makes 80, so 8 Multi­plyed by 20 makes 160.

2. Example.

If 80 l. will buy 4 Acres, what will 160 l. buy?

The Answer will be found as before.

[figure]

By Dividing the Product of (4 into 160) viz. (640) by (80) For then the Quotient will be 8.

For as 80 contains 4, twenty times, so 160 be­ing Divided by 20, the Quotient is 8.

2. Inverse.

The Single Rule of Three Inverse is when there are three Numbers given, and a fourth demanded, which bears the same Proportion to the second, as the third doth to the first.

Example.

If a quantity of Hey will keep 8 Horses 12 Days, How many Days will the same quantity keep 16 Horses.

[Page 75] Here it is observable, that most of those who have hitherto Treated of this Rule of Three, puzzle both young Learners and themselves, with this distinction of Direct and Inverse, as though they were two distinct and different things, whereas in truth they are the very same thing, only care must be taken how to place them. For the true un­folding of this Question, is this,

Look what proportion 16 Horses bear to 8 Horses, so do 12 Days bear to a fourth Number of Days. And they ought to be placed thus,

[figure]

And then the operation is the very same as be­fore; for the Product of 8 by 12 is (96) which being Divided by 16, the Quotient is (6), which the fourth Number sought.

[figure]

2. The Double Rule of Three.

And this is

  • 1. Direct.
  • 2. Inverse.

1. Direct.

THe Double Rule of Three is when more then three Terms are given; as the ordinary Books of Arithmetick teach us.

1. Example.

If 4 Men spend 19 Pound in three Months; how many Pounds will 8 Men spend in nine Months?

The Resolution of this Question is thus per­formed.

1. If four Men spend 19. what will 8 Men spend.

[figure]

[Page 77] 2. If 38 l. be spent by any Number of Men in 3 Months, how many Pounds will be spent by such a Number of Men in 9 Months?

[figure]

2. Example.

If 9 Bushels of Oats serve 8 Horses 12 Days, how many Days will 24 Bushels last 16 Horses.

Read it thus,

1. If 9 Bushels last 12 Days, how many Days will 24 Bushels last?

[figure]

[Page 78] 2. As 16 Horses is to 8 Horses, so is 32 Days to 16 Days.

[figure]

And this is the whole Intrigue of the Golden Rule, or the Rule of Three. The which being rightly understood, (together with the foregoing Reflections on Arithmetical and Geometrical Pro­portion) those other Rules of Fellowship, and Alligation, as likewise the Rule of False, will not be at all difficult. And therefore I think it needless to multiply Examples.

The Diameter of any Circle being given in Integers, to find the Peripherie, and the Square-Root of the Area, in infinitum, without the help of either Multiplication, Division, or Extraction of the Root.

FOr Example, Let the given Diameter of a Circle be 351. And let it be required to find the Pe­ripherie (or Circumference.) First I set down (300,) under that (50,) and beneath that 1, as in the fol­lowing Operation. This done, I first seek the Peri­pherie of a Diameter of (3) in the subsequent Table, viz. 9, 42477795, and because it is the place of Hundreds, I add thereto on the right hand two Ci­phers (00.) Then, under that Sum I set the Number of the Peripherie of (5.) adding to it one Cypher, because it is the place of Tens (viz. for 50.) Lastly, I place under the two Sums abovesaid the Peripherie of (1.) without adding any Cypher, because it is the place of Ʋnits. And adding these 3 Sums together, (evermore distinguishing the Integers from the Fractions, after the 8th place from the right hand, with a Comma, or Line of Seperation) the Sum, or Aggregate (viz. 1102, 6990215,) is the true Peri­pherie of that Circle, whose Diameter is 351.

300—9424777, 9500
50—1570796, 3250
1—31415, 9265
351 sum [...] 11026990, 2015

After the very same manner, if it were required to give the Square-root of the Area of that very Circle, whose Diameter is 351. I add the 3 following Sums, and the Aggregate is what I desire. For Example.

300—2658680, 7700
50—443113, 4620
1—08862, 2692
351 sum 3110656, 5012

If the first place of the given Diameter be of Thousands, there must be 3 Cyphers; if of Tens of Thousands, there must 4 Cyphers be added on the right hand, and so in Infinitum.

[Page]

Diamet.Peripherie.Square-root of the Area
131415, 9265 8862, 2692
262831, 853017724, 5385
394247, 779526586, 8077
4125663, 706035449, 0770
5157079, 632544311, 3462
6188495, 559053173, 6155
7219911, 485562035, 8847
8251327, 412070898, 1540
9282743, 338579760, 4232

But in case of Fractions, as for Example, if the Diameter had been given 351 ¼, and the Peripherie were demanded: The way is, to take ¼ of 3, 1415, 9265. (viz. 0, 7853, 9816) and to add it to 1102, 6990, 2015, which makes 1103, 4844, 1831 for the Peripherie of that Circle whose Diameter is is 351 ¼. And the same method is to be used for any Fraction whatsoever.

The Circumference of a Circle being given to find the Diameter, and the Square-root of the Area, in In­finitum, without the help of either Multiplication, Division, or Extraction of the Root.
Circum.Diameter.Square-root of the Area
1 3183, 0988 2820, 9479
2 6366, 1977 5641, 8958
3 9549, 2965 8462, 8437
412732, 395411283, 7916
515915, 494214104, 7395
619098, 593116925, 6875
722281, 692019746, 6354
825464, 790822567, 5833
928643, 889725388, 5312

This Table is to be used after the very same man­ner with the former.

A Perpetuall Almanack invented by S. Morland 1650.

To find the Dom: Letter for ever.
CEGBDFA
BDFACEG
ACEGBDF
GFBADCFEAGCBED
1000120014001600110013001500
1616202428323640
44485256606468
72768084889296
17004812162024
28323640444852
2400190021002300180020002200
To find what day of the Week each Month begins.
ABCDEFG
Octob: 31May. 31August 31March 31Iune. 30Septem 30Aprill. 30
Ianu: 31  Novem 30 Decem: 31Iuly: 31
   Febru: 2 8/9   
To find the day of ye Month
 18152229 
29162330
310172431
4111825 
5121926 
6132027 
7142128 

AN EXPLANATION OF THE Perpetual Almanack.

THis Almanack was first intend­ed to be as short and compen­dious as was possible, and to be Graven on a small plate of Silver, about the breadth of a Shilling, and so portable toge­ther with Money: but having now design'd it for a small pocket Book, I have thought it more convenient to divide it into three distinct Tables, which are much more plain and easie, that so the use thereof may be obvious to the meanest capacity. And when the three following Tables and their use are throughly understood, This Almanack will need no explanation, and may be used sometimes as well as the other three Tables.

The use of the Table in Page 6.

By the Table in Page 6. you may immediately find the Dominical Letter from the first year of our Lord, to the year 3400, &c. so long as the world shall last.

For example, For any year of Our Lord under 100. they are exprest in the Marginal Columnes of Numbers, and are answered in the first Columne of Letters. Thus for the years of Our Lord 28, 56, and 84. The Domi­nical Letters are DC. For the years 1, 29, 57, and 85. the Dominical Letter is B. For the years 11, 39, 67, and 95. the Dominical Letter is D.

[Page 2] Now for all the even Hundreds or Thousands from 100. to 3400. (which are all Leap-years) the Domi­nical Letters are to be found above them in the 1, 2, 3, 4, 5, 6, and 7th Columne.

And for any other Numbers, the Rule is for ever this; Where the Marginal Number (equal to the Fracti­on or part of an Hundred sought) meets with the Columne where that Number of Hundreds or Thousands (which is the other part of the Number sought) is exprest; That is the Dominical Letter (or Letters) for that year.

For example, Let the Dominical Letter be demanded for the year of Our Lord 1672. The number of Hun­dreds is, 1600. and is found in the third Columne. And the broken part (72) among the Marginal Numbers meets with that third Columne in the Letters GF. which are the Dominical Letters for that year, namely the first of them (G.) from the first of January to S. Matthias Eve, the twenty fourth of February, and the other Letter F. to the years end, which is a constant Rule for all Leap-years.

If the Dominical Letter be demanded for 1481. The Hundreds are 1400. and are found in the first Co­lumne, the broken part of an Hundred (81) meets with that Columne in G. which is the Dominical Letter for that year.

So if the Dominical Letter be demanded for 1349, 1300. is found in the 7 Columne, where the broken part (49) among the Marginal Numbers; meets in the Letter D. which is the Dominical Letter for that year.

Again, Let the Dominical Letter be demanded for the year 1729. 1700. is found in the 4 Columne, and 29. meets with that Columne in E.

And so for any year past or to come.

The use of the Table in page 7.

The use of this Table is to find readily, both upon what day of the week any moneth of the year begins for ever. But also how many days each month con­tains, [Page 3] which is to be sound under the name of each re­spective Moneth.

For example, I desire to know upon what day of the week the moneth of April begins, in the year 1623.

First I find by the Table in page 6. that the Domini­cal Letter for that year is E. Then I guide my Eye down that Columne of the Table in page 7. where E. is at the top, and at the same time observe where April is in the Margent, and where the Line of April meets with the Columne E. there I find (Tuesday) which gives me to understand that April in the year 1623. begins on a Tuesday.

By the same Rule

The moneth of May in the year 1615. begins on a Munday.

The same moneth in the year 1616. began on a Wednesday; where it is to be noted that forasmuch as 1616 is a Leap year, I make use of the 2 Letter (F.) for the Table in page 7. which must always be ob­served in Leap-years.

The moneth of September in the year 1537. began on a Saturday.

The moneth of July in the year 1471, began on a Monday.

The Moneth of December in the year 1642. began on a Thursday.

The moneth of August in the year 1781. will begin on a Sunday.

And so in all the rest.

The Use of the Table in Page 8.

The use of this Table (having first found upon what day of the week any Moneth begins) is to know readily the day of the Moneth.

Suppose it to be Thursday, and the beginning of October in the year 1672.

By the two former Tables I find, that the Moneth of October 1672. begins on a Tuesday; wherefore I [Page 4] look among the Tables in page 8. till I find a Ta­ble that begins with a Tuesday; and finding that the Thursdays of that Moneth are 3, 10, 17, 24, and 31. and that it is yet but the beginning of the Moneth, I conclude that it must needs be the 3 day of the Moneth.

Where it is to be observed, that if I cannot by some Marks or Circumstances help my memory, to know at least what week of the Moneth it is, no Alma­nack in the World can inform or tell me what day of the Moneth it is.

Two excellent Uses of the Ta­ble, in Page 8.

1. By it may be readily found what day of the week, any day of the Month was, or will be, of any year past, or to come.

For example; Such a man was born, or such an acti­on was done, or such a Letter was written the fourth of September in the year 1618. and it is demanded what day of the week it was?

1. By the Table in Page 6. I find the Dominical Let­ter for that year to be D.

2. By the Table in Page 7. I find that September in that year began on a Tuesday.

3. And by the Table in page 8. I find that the fourth of September was the first Friday of that Moneth. And so may any other day of any week, of any other Month, of any other year past, or to come, be known.

[Page 5] 2. By it may be readily found what day of the Month, was or will be, of any day, of any week, of any month, of any year past, or to come.

For example; Suppose such a Man was born, or such an Action done, or such a Letter written, upon the first Friday of September, in the year 1618.

Having found by the foregoing directions, that Sep­tember in that year began on a Tuesday, I find that the first Friday of that Moneth, was the fourth day of the same Moneth.

And so may be readily found what day of the Month, was or will be, any day of any week, of any month, of any year, past or to come.

Al which are of excellent and daily use for all Ministers of State, and for all Merchants in their correspondencies.

[Page 6]

A Table shewing the Dominical Letter from the first year of Our Lord to the year 3400 & may be continued for ever
    1234567
    DCEDFEGFAGBACB
    000100200300400500600
    7008009001000110012001300
    1400150016001700180019002000
    2100220023002400250026002700
    2800290030003100320033003400
0285684DCEDFEGFAGBACB
1295785BCDEFGA
2305886ABCDEFG
3315987GABCDEF
4326088FEGFAGBACBDCED
5336189DEFGABC
6346290CDEFGAB
7356391BCDEFGA
8366492AGBACBDCEDFEGF
9376593FGABCDE
10386694EFGABCD
11396795DEFGABC
12406896CBDCEDFEGFAGBA
13416997ABCDEFG
14427098GABCDEF
15437199FGABCDE
164472 EDFEGFAGBACBDC
174573 CDEFGAB
184674 BCDEFGA
194775 ABCDEFG
204876 GFAGBACBDCEDFE
214977 EFGABCD
225078 DEFGABC
235179 CDEFGAB
245280 BACBDCEDFEGFAG
255381 GABCDEF
265482 FGABCDE
275583 EFGABCD
    1234567

[Page]

A Table Shewing by the help of ye Dominical Letter what day of the week any Month of the year begins for ever
 ABCDEFG
Ian 31Sund:Satur:Fryd:Thur:Wedn:Tuesd:Mun:
Feb 28Wedn:Tuesd:Mund.Sund.Satur.Fryd:Thur:
Mar 31Wedn:Tuesd:Mund:Sund:Satur:Fryd:Thur:
Apr 30Satur:Fryd:Thur:Wedn.Tuesd:Mund:Sund:
May 31Mund:Sund:Satur:Fryd:Thur:Wedn:Tuesd:
Iun 30Thur:Wedn:Tuesd:Mund.Sund:Satur.Fryd:
Iuly 31Satur:Fryd:Thur:Wedn:Tuesd:Mund.Sund:
Aug 31Tuesd:Mund.Sund:Satur.Fryd:Thur:Wedn:
Sep 30Fryd:Thur:Wedn:Tuesd.Mund.Sund:Satur.
Oct 31Sund:Satur.Fryd:Thur:Wedn:Tuesd:Mund.
Nou 30Wedn:Tuesd:Mund:Sund:Satur.Fryd:Thur:
Dec 31Fryd:Thur:Wedn:Tuesd:Mund.Sund.Satur.

[Page]

The 7 Varieties of the last Table Pag: 1 for finding ye day of the Month.
Wedn:18152229
Thursdays29162330
Frydays310172431
Saturdays4111825 
Sundays5121926 
Mundays6132027 
Tuesdays7142128 
Sund:18152229
Mundays29162330
Tuesdays310172431
Wednesdays4111825 
Thursdays5121926 
Frydays6132027 
Saturdays7142128 
Thursd:18152229
Frydays29162330
Saturdays310172431
Sundays4111825 
Mundays5121926 
Tuesdays6132027 
Wednesdays7142128 
Mund:18152229
Tuesdays29162330
Wednesdays310172431
Thursdays4111825 
Frydays5121926 
Saturdays6132027 
Sundays7142128 
Fryd:18152229
Saturdays29162330
Sundays310172431
Mundays4111825 
Tuesdays5121926 
Wednesdays6132027 
Thursdays7142128 
Tuesd:18152229
Wednesdays29162330
Thursdays310172431
Frydays4111825 
Saturdays5121926 
Sundays6132027 
Mundays7142128 
Saturd:18152229
Sundays29162330
Mundays310172431
Tuesdays4111825 
Wednesdays5121926 
Thursdays6132027 
Frydays7142128 

[Page 9] To find out the Prime or Golden Number for ever Divide the year of the Lord by (19) and to the Remaynder after the Division add (1) the sum: is the Prime for that year

Example Thus (1671) divided by (19) leaues (18) to which adding (1) makes it (19) for the Prime of yt year

[...]

A Table of Multiplication serving for the dividing of any year of the Lord by 19
1—196—114
2—387—133
3—578—152
4—769—171
5—95 

[Page 10]

A Table for the ready findiny of the Prime or Golden Number for euer.
 100200300400500600700800900
6111627121738
200021002200230024002500260027002800
120395877967121738131849
2214059789781318491419510
3224160799891419510151611
4234261809910151611162712
524436281 11162712173813
625446382 12173813184914
726456483 131849141951015
827466584 141951015161116
928476685 15161116271217
1029486786 16271217381318
1130496887 17381318491419
1231506988 18491419510151
1332517089 19510151611162
1433527190 1611162712173
1534537291 2712173813184
1635547392 3813184914195
1736557493 4914195101516
1837567594 5101516111627
1938577695 6111627121738

[Page 11]

The Table of Primes Continued.
 1000110012001300140015001600170018001900
1318491419510151
2900300031003200330034003500360037003800
120395877961419510151611162
22140597897151611162712173
32241607998162712173813184
42342618099173813184914195
524436281 184914195101516
625446382 195101516111627
726456483 16111627121738
827466584 27121738131849
928476685 381318491419510
1029486786 491419510151611
1130496887 510151611162712
1231506988 611162712173813
1332517089 712173813184914
1433527190 8131849141951015
1534537291 9141951015161116
1635547392 1015161116271217
1736557493 1116271217381318
1837567594 1217381318491419
1938577695 1318491419510151

[Page 12]

A Table to finde the moveable Feasts for ever. by the Dom [...] letter & Golden Number.
Domin [...] LetterGolden NumberFrom Christmas to Shrove sund:Shrove sundayEaster day
A2 5 13 166 weeksFebr 5Mar 26
7 10 15 187 weeksFebr 12Apr 2
1 4 9 128 weeksFebr 19April 9
3 6 11 14 179 weeksFebr 26Apr 16
8 1910 weeksMar 5Apr [...]3
B2 5 13 166 weeks 1 dayFebr 6Mar 27
4 7 10 15 187 weeks 1 dayFebr 13Apr 3
1 9 12 178 weeks 1 dayFebr 20Apr 10
3 6 11 149 weeks 1 dayFebr 27Apr 17
8 1910 weeks 1 dayMar 6Apr 2
C2 5 10 13 66 weeks 2 daysFebr 7Mar 28
4 7 15 187 weeks 2 daysFebr 14April 4
1 6 9 12 78 weeks 2 daysFebr 21Apr 11
3 11 14 199 weeks 2 daysFebr 28Apr 18
810 weeks 2 daysMar 7Apr 25
D165 weeks 3 daysFebr 1Mar 28
2 5 10 136 weeks 3 daysFebr 8Mar 29
4 7 12 15 187 weeks 3 daysFebr 15Apr 5
1 6 9 178 weeks 3 daysFebr 23Apr 12
3 8 11 14 199 weeks 3 daysMar 1Apr 19
E5 165 weeks 4 daysFebr 2Mar 23
2 10 13 186 weeks 4 daysFebr 9Mar 30
1 4 12 157 weeks 4 daysFebr 16april 6
6 9 14 178 weeks 4 daysFebr 23apr 13
3 8 11 199 weeks 4 daysMar 2apr 20
F5 165 weeks 5 daysFebr 3Mar 24
2 7 10 13 186 weeks 5 daysFebr 10Mar 31
1 4 12 157 weeks 5 daysFebr 18apr 7
3 6 9 14 178 weeks 5 daysFebr 24apr 14
8 11 199 weeks 5 daysMar 3apr 21
G5 13 165 weeks 6 daysFebr 4Mar 25
2 7 10 186 weeks 6 daysFebr 11apr 1
1 4 9 12 157 weeks 6 daysFebr 18apr 8
3 6 14 178 weeks 6 daysFebr 25apr 15
8 11 199 weeks 6 daysMar 4apr 22

[Page 13]

The Table for the Moveable Feasts Continued
Dom: LetterGolden NumberRogat:Asens:Whit:TrinityAdvent
 Sund:daySund:Sund:Sund:
A2. 5. 13. 16.Apr. 30May. 4May. 14May. 21Dec. 3
7. 10. 15. 18.May. 7May. 11May. 21May. 28Dec. 3
1. 4. 9. 12.May. 14May. 18May. 28Iun. 4Dec. 3
3. 6. 11. 14. 17May. 21May. 25Iun. 4Iun. 11Dec. 3
8. 19.May. 28Iun. 1Iun. 11Iun. 18Dec. 3
B2. 5. 13. 16.May. 1May. 5May. 15May. 22Nov. 27
4. 7. 10. 15. 18.May. 8May. 12May. 22May. 29Nov. 27
1. 9. 12. 17.May. 15May. 19May. 29Iun. 5Nov. 27
3. 6. 11. 14.May. 22May. 26Iun. 5Iun. 12Nov. 27
8. 19.May. 29Iun. 2Iun. 12Iun. 19Nov. 27
C2. 5. 10. 13. 16.May. 2May. 6May. 16May. 23Nov. 28
4. 7. 15. 18.May. 9May. 13May. 23May. 30Nov. 28
1. 6. 7. 9. 12.May. 16May. 20May. 30Iun. 6Nov. 28
3. 11. 14. 19.May. 23May. 27Iun. 6Iun. 13Nov. 28
8.May. 30Iun. 3Iun. 13Iun. 20Nov. 28
D16.Apr. 26Apr. 30May. 10May. 17Nov. 29
2. 5. 10. 13.May. 3May. 7May. 17May. 24Nov. 29
4. 7. 12. 15. 18.May. 10May. 14May. 24May. 31Nov. 29
1. 6. 9. 1. 7.May. 17May. 21May. 31Iun. 7Nov. 29
3. 8. 11. 14. 19.May. 24May. 28Iun. 7Iun. 14Nov. 29
E5. 16.Apr. 27May. 1May. 11May. 18Nov. 30
2. 10. 13. 18.May. 4May. 8May. 18May. 25Nov. 30
1. 4. 7. 12. 15.May. 11May. 15May. 25Iun. 1Nov. 30
6. 9. 14. 17.May. 18May. 22Iun. 1Iun. 8Nov. 30
3. 8. 11. 19.May. 25May. 29Iun. 8Iun. 15Nov. 30
F5. 16.Apr. 28May. 2May. 12May. 19Dec. 1
2. 7. 10. 13. 18.May. 5May. 9May. 19May. 26Dec. 1
1. 4. 12. 15.May. 12May. 16May. 26Iun. 2Dec. 1
3. 6. 9. 14. 17.May. 10May. 23Iun. 2Iun. 9Dec. 1
8. 11. 19.May. 26May. 30Iun. 9Iun. 16Dec. 1
G5. 13. 16.Apr. 29May. 3May. 13May. 20Dec. 2
2. 7. 10. 18.May. 6May. 10May. 20May. 27Dec. 2
1. 4. 9. 12. 15.May. 13May. 17May. 27Iun. 3Dec. 2
3. 6. 14. 17.May. 20May. 24Iun. 3Iun. 10Dec. 2
8. 11. 19.May. 27May. 31Iun. 10Iun. 17Dec. 2

[Page 14]

Ye Table Continued for ye moveable Termes
Dom LetterGolden NumberEaster Terme beginsEaster Terme endsTrinity Terme beginsTrinity Terme ends
A2. 5. 13. 16Apr. 12May. 8May. 26Iun. 14
7. 10. 15. 18Apr. 19May. 15Iun. 2Iun. 21
1. 4. 9. 12Apr. 26May. 22Iun. 9Iun. 28
3. 6. 11. 14. 17May. 3May. 29Iun. 16Iuly. 5
B [...]. 5. 13. 16Apr. 13May. 9May. 27Iun. 15
4. 7. 10. 15. 18Apr. 20May. 16Iun. 3Iun. 22
1. 9. 12. 17Apr. 27May. 23Iun. 10Iun. 29
3. 6. 11. 14May. 4May. 3Iun. 17Iuly. 6
8 19May. 11Iun. 6Iun. 24Iuly. 13
C2. 5. 10. 13. 16Apr. 14May. 10May. 28Iun. 16
4. 7. 15. 18Apr. 21May. 17Iun. 4Iun. 23
1. 6. 9. 12. 7Apr. 28May. 24Iun. 11Iun. 30
3. 11. 14. 19May. 5May. 31Iun. 18Iuly. 7
8May. 12Iun. 7Iun. 25Iuly. 14
D16Apr. 8May. 4May. 22Iun. 10
2. 5. 10. 13Apr. 15May. 11May. 29Iun. 17
4. 7. 12. 15. 18Apr. 22May. 18Iun. 5Iun. 24
1. 6. 9. 17Apr. 29May. 25Iun. 12Iuly. 1
3. 8. 11. 14. 19May 6Iun. 1Iun. 19Iuly. 8
E5. 16.Apr. 9May. 5May. 23Iun. 11
2. 10. 13. 18Apr. 16May. 12May. 30Iun. 18
1. 4. 7. 12. 15.Apr. 23May. 19Iun. 6Iun. 25
6. 9. 14. 17.Apr. 30May. 26Iun. 13Iuly. 2
3. 8. 11. 19.May. 7Iun. 2Iun. 20Iuly. 9
F5. 16.Apr. 10May. 6May. 24Iun. 12
2. 7. 10. 13. 18Apr. 17May. 13May. 31Iun. 19
1. 4. 12. 15.Apr. 24May. 20Iun. 7Iun. 26
3. 6. 9. 14. 17.May. 1May. 27Iun. 14Iuly. 3
8. 11. 19.May. 8Iun. 3Iun. 21Iuly. 10
G5. 13. 16.Apr. 11May. 7May. 25Iun. 13
2. 7. 10. 18Apr. 18May. 14Iun. 1Iun. 20
1. 4. 9. 12. 15.Apr. 2 [...]May. 21Iun. 8Iun. 27
3. 6. 14. 17.May. 2May. 28Iun. 15Iuly. 4
8. 11. 19.May. 9Iun. 4Iun. 22Iuly. 11

Cambridg Commencment First Sunday in July. Oxon Act 2d. Sunday in July.

The Returnes of The Act of

  • Trin. Terme are j. Crast: Trin 2 Oct: Trin: 3 Quind Trin: 4 Tivs. Trin
  • Easter Terme. are. j. Quind. Pasch: 2 Tres Pa. 3. Mens Pa. 4 Quind: Pa. 5: Crō. Asc:
  • Batchelors in Cambr: & Oxf: ye 1st. day of Lent and of Masters in
    • Cambr. Iuly. 3
    • Oxford. Iuly. 8

[Page 15]

A Table of the Fixed Feasts, and other Solemn Days to be observed in the Church of England.
  • All Sundays.
    • NEw-years-day, or Circumcision Jan. 1
    • Twelf day, or Epiphany Jan. 5
    • Martyrdom of K. Charles I. Jan. 30
    • Purification of the Virgin Mary Febru. 2
    • LADY-DAY, or the Annuntiation of the Virgin Mary Mar. 25
    • Mark Evang. April 25
    • May day, or Phillip and Jacob May 1
    • Birth and return of Charles II. May 29
    • MIDSUMMER or John Baptist. June 24
    • James Apostle July 25
    • Bartholomew Apostle Aug. 24
    • Matthew Apostle Sept. 21
    • MICHAELMAS, or Mich. Archangel Sept. 29
    • Luke the Evangelist Osto. 18
    • Simon and Jude Octo. 28
    • All Saints Nov. 1
    • Powder Treason Nov. 5
    • Andrew Apostle Nov. 30
    • Thomas Apostle Dec. 21
    • CHRISTMAS, or Birth of our Lord Dec. 25
    • St. Stephen Dec. 26
    • St. John Evang. Dec. 27
    • Innocents Dec. 28
  • [Page 16]
    Other Remarkable Days.
    • VAlentine Febr. 14
    • Equal day and night Mar. 10
    • St. George April 23
    • Longest day, or Barneby June 11
    • Swithin July 15
    • Lammas Augu. 2
    • Equal day and night Sept. 12
    • Shortest day Dece. 11
  • HILLARY TERM.
    • BEgins Jan. 23
    • Ends and hath Four Returns. Febr. 12
    • 1. Octab. Hil. Jan. 20
    • 2. Quind. Hil. Jan. 29
    • 3. Crast Pur. Febr. 3
    • 4. Octab. Pur. Febr. 10
  • MICHAELMAS TERM
    • Begins Octob. 23
    • Ends and hath Six Returns. Nov. 28
    • 1. Tres Mich. Octo. 21
    • 2. Mens. Mich. Octo. 29
    • 3. Crast. An. Nov. 4
    • 4. Crast. Mar. Nov. 11
    • 5. Oct. Mar. Nov. 18
    • 6. Quin. Mar. Nov. 27

[Page] [Page]

 Gol: NumD: of ye Mon.A Table for the ready finding what Sign the Moon is in or shall be for euer And what part of Mans body every Sign doth govern
Febr: Nov:3127262524232221201918171615141312111098765432Head & Face
March 212726252423222120191817161514131211109876543
 14321272625242322212019181716151413121110987654Neck & Throat
December6432127262524232221201918171615141312111098765
  543212726252423222120191817161514131211109876Arms shoulders and Hands
Aprill17654321272625242322212019181716151413121110987
 9765432127262524232221201918171615141312111098Breast and stomack
May1876543212726252423222120191817161514131211109
  987654321272625242322212019181716151413121110
 12109876543212726252423222120191817161514131211Heart and Back
 4111098765432127262524232221201918171615141312
Iune 121110987654321272625242322212019181716151413Bowels & Belly
 15131211109876543212726252423222120191817161514
Iuly7141312111098765432127262524232221201918171615Reyns & Loyns
  151413121110987654321272625242322212019181716
 18161514131211109876543212726252423222120191817Secrets
 10171615141312111098765432127262524232221201918
August2181716151413121110987654321272625242322212019
  191817161514131211109876543212726252423222120Thighs
 13201918171615141312111098765432127262524232221
 5212019181716151413121110987654321272625242322Knees
September 222120191817161514131211109876543212726252423
 16232221201918171615141312111098765432127262524Leggs
Ian: Oct:8242322212019181716151413121110987654321272625
  252423222120191817161514131211109876543212726Feet
 19262524232221201918171615141312111098765432127
 11272625242322212019181716151413121110987654321

The Ʋse of this Table.

1. SEek the name of the Month in the left-hand Margent, and guiding your Eye to (1) in the Table, find out the day of the Month, which you shall find either above, or beneath in that Column.

2. From that day of the Month, guide your eye back to the Number in the left-hand Margent that stands against it in the Column under the Title (Day of the Month.)

3. From that Number guide your Eye to (1) in the Table, and in that Column find the Number that is the Number of the Prime for that Year, and from thence guide your Eye to the right-hand Margent, so have you your desire.

Example.

The Tenth day of May, 1665. I desire to know what Sign the Moon is, &c.

1. The Prime for that Year is 13.

2. I find May in the left-hand Margent, and guiding my Eye to (1) in the Table, and in that Column to 10 the Day of the Month, I bring my Eye back to (17) in the left-hand Margent.

3. I look for 13 among the Primes, and from that guide my Eye to (1) in the Table, and finding the aforesaid Number (17 in that Column, I do from that, guide my Eye to the right-hand Mar­gent, and find that the Moon upon the Tenth of May, 1665. is entring into Leo Ω, and governs the Bowels and Belly.

A Table shewing: the time of the Moons com­ing to the South, and quantity of her shining.

The Moons age.Moons southing & shin.Moons age for her shi.
116048129
217136228
318224327
419312426
520400515
621448624
722536723
823624822
924712921
10258001020
11268481119
12279361218
132810241317
142911121416
1530120015

The use of this Table.

FInd the Moons age in the first Column, and next against the same towards the right hand, is the time of her coming to the South; which from the New Moon to the Full Moon, is always in the Afternoon, but from the Full to the New, it is in the Morning.

Example.

May 12. 1671. the Moon is fourteen days old, which I find in the first Column, against which, towards the right hand in the second Column is 11. 12, which being before the Full of the Moon, I conclude that the Moon comes to south May 12. 1671. at 11 a clock at Night, and 12 Minutes past.

[Page] To know how long the Moon Shineth.

Enter the third Column with the Moons age, and against it, on the left hand, you have the time of her shining, which all the time of her Encrease, being added to the hour of Sun rising, gives the time of her rising: But if added to the time of Sun setting, gives the time of her setting.

But after the Full.

Take the time of her shining from the Suns ri­sing, and it gives her rising; and then take the same from the Sun setting, it gives the time of her setting.

Example.

May 12. 1671. the Moon is 14 days old, and I find 11 hours, 12 minutes, for the time of her shining, (which being added to the Suns rising) upon the twelf of May, 1671. (viz. four hours) makes 3 of the clock, 12 minutes, for the time of the Moons rising the next Morning.

Again, to the said 11 hours, 12 minutes, add 8 hours from the Sun setting, it gives 7 hours, 12 minutes for the time of his setting.

Though these Rules are not altogether exact, yet they come near enough the truth, for ordinary use.

A Tide-Table of certain Havens in and about Eng­land, whereby may be known what Moon makes a Full-Sea in any of the said places; and how many ho. and min. are to be added to the time of the Moons coming to the South for the time of High-water.

South and NorthQueenborough Southamton, Ports­mouth, Isle of Wight, Spits, Ken­tish Knock, half-Tide at Dunkirk0. H.0 M.
S by W N by ERochester, Maldon, Aberdeen, Redband, West-end of the Nowr-Blacktail.0 H.45 M.
S S W N N EGravesend, Downs, Rumney, Ten net, Silly halfe tide, Blackness, Ramkines, Senebead.1 H.30 M.
S W by S N E by NDundee, St. Andrews, Lisborn St. Lucas, Bell Isle, Holy Isle.2 H.15 M.
S W N ELondon, Tinmouth, Hartlepool Whitebay, Amsterdam, Gascoign, Britain, Galicia.3 H.0 M.
S W by W N E by EBarwick, Hambrough-head, Brid­lington bay, Burdeux, Ostend, Flushing, Fountness.3 H.45 M.
W S W E N EScarborough quarter-tide, Lawre nas, Severn, Horkhave, Dungarum, Mounts-bay, Kingsale, Calice-Creek4 H.30 M.
W by S E by NNewcastle, Humber, Falmouth Sal ly, Dartmouth, To bay, St. Mal­lows, Foy, Garsy. Liz.5 H.15 M.
East and WestPlimouth, Weymouth, Hull, Lyn, Davids head, Antwerp, Lundy, Holms of Bristol.6 H.0 M.
E by S W b NBristol, Foulness at the Start.6 H.45 M.
[Page]E S E W N WMilford, Bridgewater, Lands-end, Waterford, Abermorick, Cape-Cleer, Texel.7 H.30 M.
S E by E N W by WPortland, Peterport, Harflew, the Hague, S. Magnes. south, Dublin, Lambay. Macknels Cape.8 H.15 M.
S E N WPool, S. Hellen, Catnes, Orkney, Fair-Isles, Kilden, Man-Isle, Bass-Islands.9 H.0 M.
S E by S N W by NNeedles, Laisto, North & South Foreland.9 H.45 M.
S S E N N WTarmouth, Dover, Harwich, S. John de Luce, Calice Road, Bullein.10 H.30 M.
S by E N by WRye, Winchelsey, Goree, Thames, Rhodes.11 H.15 M.

The Use of the Tide-Table.

Example.

May 12. 1671. I would know the Full Sea at London.

1. By the foregoing Rules I find the Moon comes to South at 11 of the clock, and 12 Minutes past, at Night.

There I seek for London in this Table, where I find that a S. W. or N. E. Moon makes a Full Sea, and on the right-hand I find 3 hours 0 minutes, which must be added to the Moons Southing. That is 3 hours, 0 minutes, added to 11 ho. 12 min. makes 2 a clock and 12 minutes the next Morning, for High Water at London-Bridge.

So that for any place and day, the hours and min. in the Table, are to be added to the Moons Southing, which gives the true time of High­water for that place and day.

The time of the Suns Rising and Setting throughout the whole Year.

Days of the Month.JanuaryFebruaryMarch
Sun rises.Sun sets.Sun rises.Sun sets.Sun rises.Sun sets.
H.MHMH.MH.MH.M.H.M.
189351718442620540
288351717443618542
387353715445616544
486354713447614546
584355711449611549
6823567945168551
7813587745866554
880407545564556
9758417345762558
1075643714596060
11755456595155862
12754466575355664
13752486545655466
14751496525855268
15750410650510550610
16748412648512548612
17746414646514546614
18744416644516544616
19743417642518542618
20741419640520540620
21739421638522538622
22737423636524536624
23735424634526534626
24734426632528532628
25732428630530530630
26730430629531528632
27728431627533526634
28726434624536524636
29724436622  522638
30722438    520640
31720440    518642

[Page]

Days of the Month.April.May.June.
Sun rises.Sun sets.Sun rises.Sun sets.Sun rises.Sun sets.
H.M.HM.H.M.H.M.H.MH.M.
1516644419742342818
2514646417743342818
3512648415745342818
4510650414746341819
558652412748341819
656654411749341819
755655410750341819
85365748752341819
95265847753341819
10507046754341819
114577344756341819
124557543757341819
134537742758341819
14451794080341819
1544971135981341819
1644771335882341819
1744571535783341819
1844371735684341819
1944171935585342818
2043972135387342818
2143872235288342818
2243672435189343817
23434726350810344816
24432728349811344816
25430730348812345815
26428732347813346814
27427733346814347813
28425735345815348812
29423737345815349811
30421739344816350810
31    343817    

[Page]

Days of the Month.July.August.September
Sun rises.Sun sets.Sun rises.Sun sets.Sun rises.Sun sets.
H.M.H.M.H.M.H.M.H.M.H.M.
135189435726535625
235188436724537623
335288438722539621
435387440720541619
535486441719543617
635684443717545615
735783445715547613
835882447713549611
93598144871255169
10408045071055367
11427584527855664
12437574547655862
1344756456746060
14467544587262558
1547753507064556
16487525265866554
174107505465669551
1841174956654111549
1941274858652613547
20414746510650615545
21415745512648617543
22417743514646619541
23418742516644621539
24420740518642623537
25422738520640625535
26423737522638627533
27425735524636629531
28427733526634631529
29428732528632633527
30430730530630635525
31432728532628    

[Page]

Days of the Month.October.Novemb.December.
Sun rises.Sun sets.Sun rises.Sun sets.Sun rises.Sun sets.
H.M.H.M.H.M.H.M.H.M.H.M.
1638522738422816344
2640520739421816344
3642518741419817343
4644516742418817343
5646514744416818342
6648512746414818342
7650510747413818342
865258749411819341
96545675149819341
106565475248819341
116585275446819341
12705075545819341
137245875743819341
147445675842819341
15764548040819341
167845281359818342
1771045083357818342
1871244884356818342
1971444685355817343
2071644487353817343
2171844288352816344
2272044089351816344
23722438810350815345
24723437811349814346
25725435812348814346
26727433812348813347
27729431813347812348
28731429814346812348
29733427814346811349
30734426815345810350
31736424    89351

A Table shewing the length of the longest Artificial Day, in all places from the Equinoctial, to the Poles of the World.

Heig.Long.dayHeig.Long.dayHeig.Longest day.
Pole.H.M.Pole.H.M.Pole.D.H.M.
00120047154268420116
06122048155269541625
12124249160070641346
16125850161071740000
20131251162072820636
24133052163073890458
27134253164274961700
301356541654751040104
321406551708761100727
341416561722771161422
351422571736781221706
361428581752791270955
371434591810801340458
381438601830811393136
391444611854821450643
401452621920831520206
411458631950841560303
421504642024851610523
431512652110861661123
441518662218871712147
45152666½2400881760529
4615346724 Days891812158
      901870639

[Page] The following TABLES Are of excellent use, and do readily discover the exact time of the New Moon, Full Moon, As likewise the First and Second Quadrats; And consequently her true Age.

And this from the year of our Lord 1673, to the year 1700.

EXAMPLE.

In the Moneth of April 1673, and the 14th day of the Moneth, the Table for that year, will discover, over against the said Moneth, April

First That the New Moon happens to be the fixth day of that Moneth, and the 13th hour of that day; That is, 10 Minutes past 1 of the Clock at night (remembring always that the dayes are to be accompted from Noon.)

Secondly. That the first Quadrat is the 13th day, 10 min. past 9 at night.

Thirdly. That the Full Moon is the 20th day, 11 min. past 12 at night.

Fourthly. The Second Quadrat is the 28th day, 1 Min. past 12 at night.

Fifthly, And Lastly, because the Moon changes on the sixth day, and 8 added to 6 makes 14, therefore the Moon is 8 dayes old, upon the said 14th day of April.

But if you will be more exact, you must Accompt

For theFirst Quarter of the Moon7 d.09h.11 m
Full Moon14 d.18 h.22 m
Last Quart.22 d.03 h.33 m
Time from Moon to Moon29 d.12 h.44 m

An Explanation of the double Numbers in the Table.

EXAMPLE.

In the first Quarter of the Moon, in the Moneth of May, in the year 1674. I find the 2 Numbers, viz.

[...]

The meaning whereof is, that in the said moneth of May, the first quarter of the Moon happeneth to be both upon the first day, 12th hour, 45th minute, And likewise upon the 3 [...]th day, 16 houres, 30 m. of the same Moneth. The which is to be so read, and so understood in any other year or month.

[Page]

1673.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January71407151537220110290520
Februa.60629132058201206272311
March80012150406212338291812
April6131013091420121128121
May52311121412200135281026
June40710102105181557261820
July31409100638180659260342
August1 3120 544 1781924162221241501
Septem.29151571117151342222222
October2935875511542522441
Novem.271937615913186201147
Decem.279175223313619192059

1674.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January26084141617111722180909
Februa.15031530951100315162347
March26185242244111245181612
April2509003071992216170925
May2421011 3112 1645 30 90627170254
June23062429210771942151944
July2214222904247084051132
August2021582714475231214156
Septem.1905265041549121425
October18154025224148181219
Novem.173724193311810945
Decem.1617142415542171691714

[Page]

1675.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January1593623122617228623
Februa.143422638No Full6957
March1520502321271 305 18 17 5472053
April1413282281528234161034
May14423211421286186116
June12171619194926153441721
July124161823312624741011
August101546174392416253317
Septem.822541512442383011957
October874715111232221 3111 139 47
Novem.617241317621218291327
Decem.641013121321152282245

1676.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January4263812857206582768
Februa.363911449182156251258
March3227120431979252050
April214111017417145924640
May2 316 218 161067162217231829
June131118817361552622844
July300377221314132322033
August28932625912232201810
Septem.262316474811132191234
October26939314151123919649
Novem.241952206920491802
Decem.246141 3114 64 57916417155

[Page]

1677.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M
January22173030318111616239
Februa.2152728215171417141111
March221453017598198151719
April217532911107630132230
May20223429196152513520
June19131527131342245111411
July9335162256453011146
August17195254122 319 018 2591622
Septem.16918231157308448943
October152322217512922498437
Novem.14115212122813467023
Decem.132282013232810566161

1678.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January12521193362761451343
Februa.101852171958260204422
March1252419123427155151433
April10162818882643032132
May1025018129251441322
June815251617442322401 306 1222 35
July8952168182362292125
August6203114205721132628102
Septem.512411373019215227146
October557121641988262052
Novem.3212102320172053251730
Decem.3112410639171225251433

[Page]

1679.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January201781520166124958
Februa.No New72261502823226
March1 3120 548 328154716168241515
April297267656142345222357
May282325623415023221233
June27959515581315542098
July2622315851122243191312
August2513144161161517203
Septem.24555216529143516617
October243312 316 1839 3882316152039
Novem.221543304137910141420
Decem.2210262912206219141035

1680.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January2105927191951061373
Februa.19125726336421120358
March19222026112641839122155
April186172515231125111254
May17135024145933911256
June15213123617118139720
July1516392223121 3039 1922 1781215
August13732211647295436152
Septem.12657201031278842128
October1325820130271194523
Novem.10181318185025112021619
Decem.101333187342420222934

[Page]

1681.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January9752161725231061 314 018 27
Februa.723481503721236No2.Qu.
March91243166452313241 3120 15 45 15
April72331413242242930734
May7711132147211958292041
June51371285320104528624
July4214111224020118271425
August34571014318142025207
Septem.1144398917452413
October1 303 189 439320161642381
Novem.291256722381558211750
Decem.29826716361413522188

1682.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January28332682113012192320
Februa.26204842058111125181737
March2810566511122145201233
April26229410511193419645
May267231529102217182349
June2414521204991157171450
July2321311 305 144 13923217357
August22444266317176151430
Septem.2013592802693413230
October2011827193961413558
Novem.1815222616174153111453
Decem.18832261220457102037

[Page]

1683.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January173102562221828975
Februa.15215723216123972017
March171539257362 317 2158 4591132
April1664623142830758415
May15192622183729174072132
June145402022312851261434
July1314152054027193674
August1122918124326103142231
Septem.105551717253233129
October91455161717242052308
Novem.81261512252313421 3010 172 42
Decem.7141615921235230134

1684.New2. Quar.Full1. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January6534146252120328942
Februa.422161315120449261930
March515251318132014027729
April4911278182155252110
May40511116318534251232
June214429215016142524516
July2 312 1426 7913716031232152
August292220773514125222154
Septem.2873351220138021740
October271646421201221520230
Novem.2625831411111537191118
Decem.251432362711101818230

[Page]

1685.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January243382/312/2328/4103341775
Februa.221819No1.Qu.81752151359
March24945220410549162058
April2313519448145815525
  1329   
May221813 [...]145772248141559
June2174928231065013440
July202122284505134121941
August1910102691932141111237
Septem.172153241457283910955
October178452322501/3122/1536/2610131
Novem.15201622105230102481942
Decem.155272222030556885

1686.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January1311592020502822127112
Februa.121262015372716205757
March13142211112965661839
April124322051327151507
May11169192118262304541
June109161811422555521250
July923591721572412401 3123 145 12
August8157164152220503042
Septem.75581413152164628232
October6201131918201922281817
Novem.591612213191124271430
Decem.4214111351951527921

[Page]

1687.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January3755100241802726128
Februa.1181381238161926222247
March34310849181217252247
April11489225172182442
May1 300 1240 24819551614423812
June2912971256142230211252
July28164745114624202012
August277595183812151119256
Septem.26029462102130172116
October2510503141010649171453
Novem.24838202281826161056
Decem.2322301 307 1516 1688311685

1688.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January22101229057711215450
Februa.2020242712395194132230
March215152822961358141314
April1915272618145641304
May1816102610414213612656
June178525334395610546
July1619472420342203891510
August159102312341 305 1449 26 72045
Septem.1404922342282306428
October13184021165928

[Page]

1689.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January102210172026242212312
Februa.911212634123141211156
March102157171217251643178
April9632152242232302931
May813341511212314191 3122 726 46
June62281415122418291356
July652817135421165828187
August4152112111320417262238
Septem.33391151718174625813
October218551023018040241421
Novem.112359153516103023522
Decem.1 311 34 29601522222224 [...]

1690.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January2920177165314153821188
Februa.281146149122039201424
March2922878614101422946
April2865951421321342137
May2714314210121519201753
June25212136301141319543
July2544221827102128181445
August2313211 319 218 2391145162132
Septem.2203229211281421538
October21131029151971426141035
Novem.20734289516025121652
Decemb.2024228555131112425

[Page]

1691.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January1822826191032340101855
Februa.1716282553929519129
March197522613363202511630
April1721402419427221015
May176292521501 3120 712 5491737
June1514382242429222681046
July142149211343291329812
August1345020222285166136
Septem.111313181813262184230
October1114181392612294642
Novem.91215179442523021342
     12054
Decem.9151176332415931429

1692.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January72152160462323291734
Februa.6163114182621114828737
March7103015142212053282336
April6361315332054927166
May516481220481915402799
June442811037182222624
July216251051417159251751
August1 3121 548 4981144165552495
Septem.291424622442226222113
October290366123014155222932
Novem.271148564613946201843
Decem.2713552401315020046

[Page]

1693.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January2517384011116018955
Februa.2410502195310352162013
March26424141711133218519
April242020345292144161756
May241172 315 347 6953716759
June2223563034471336142337
July221172973362240141622
August202126271239595013958
Septem.1965725116323341210
October1816172591431541111925
Novem.172924042291610102
Decem.161117232032 315 2316 2492228

1694.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January151142219393015348753
Februa.13145021131No Full6157
March15541237521 304 1413 3172140
April1321622052822136450
May131241211311285565142
June123221923222611514128
July1118616612520503159
August1072717111224630274
Septem.8201015171221981 300 1955 44
October8732142248221034301436
Novem.61820138462144229814
Decem.64431222202108282331

[Page]

1695.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January4155111459191741271323
Februa.31521094018121325201
March4132112532025027146
April314110234618135725711
May2145910165417238241253
June1 304 1 [...]56 41975616617222126
July3010528202015131022842
August292137620132033202310
Septem.271655514512530191630
October276574926111645191156
Novem.2516733101070181037
Decem.256112 3111 2112 439235918425

1696.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January23173630114981836162159
Februa.213202834371410151221
March2212592921228751152225
April202302814416231014542
May209522881261145131033
June182232270842148111442
July1811522614564656101954
August17310254152 3113 2135 219236
Septem.1519462315293055271717
October151136230522916307923
Novem.1451221816285226431
Decem.132042015492720496042

[Page]

1697.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January12859190126135342228
Februa.1021341710302581031743
March125141823242712351020
April1013321713522517273230
May9213917537257133746
June86361522162318591 3011 17 10 2
July71658151522351292110
August6527148121135428348
Septem.4201712235619221026146
October413712144019744261428
Novem.374611313171751242214
Decem.3201013251753124184

1698.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January1/3120/1348/168212415197231526
Februa.No New742214102221133
March1 3119 834 1381241641824520
April2914146228141819221955
May28211068241495522711
June274584173213040201459
July26135141255121420192036
August2505235401122518025
Septem.23142912336913301651
October237241 3171 11 53 26 90215143
Novem.222103031679591423
Decem.2121372916061954131748

[Page]

1699.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January201558282205659121210
Februa.19752269423144911758
March202054271513571413345
April19646251432050112144
May181443255731113111410
June162132231145201210316
July16438234531 3117 722 5291427
August14123221204929222272210
Septem.122229201542814106415
October121041201043280265947
Novem.1124319637261233825
Decem.1020511905625225822011

1700.New1. Quar.Full2. Quar.
D.H.M.D.H.M.D.H.M.D.H.M.
January916261714324941 3116 727 14
Februa.811311659221911No 2. Qu
March10418171334245392050
     311933
April818181519322170301322
May853214232221563066
June6142613434201825282110
July52142121213209228110
August4571023101904226222
Septem.212549133317164825420
October1 3121 956 36971217854241545
Novem.29235083101601222138
Decem.2918072344151362290

[Page]

A Catalogue of all the Eclipses of the Sun and Moon which will be visible in England, from the year 1672, to the year 1700.
Year.Moneth.D.H.M.Lun.Dig.Min
1674January1117221121
1674July718391217
1675January17291853
1675June261530152
1676May312120450
1677May61525815
1678October198172218
1681August1814221035
1682August717561850
1682February1111281948
1683January174271030
1684June161417135
1684July231080
1685November3010262145
1686November191122735
1687May1125140
1688April564649
1689March256412119
1689September1814462039
1690March141014553
1692July171591228
1693February1116241932
1693June2306234
1694June261250647
1695November1070655
1696May611452148
1696October2916302145
1697October19744854
1699March571495
1699September122130100

A Table shewing the beginning of every Kings Reign, from the Conquest; Together with the year of Christ, answering to every year of each King or Queens Reign, from Henry 8. to Charles 2. inclusive. The year beginning on the 25th of March.

WILLIAM the Conquerer, BEgan his Reign the Fifteenth of October, 1066. Ended it the Ninth of September, 1087. Reigned 20 Years, 10 Months, 21 Days.

WILLIAM RƲEƲS. Began his Reign September the Ninth, 1087. Reigned 12 Years, 11 Months, 18 Days.

HENRY I. Began his Reign the First of August, 1100. Reigned 35 Years, 4 Months, 12 Days.

STEPHEN, Began his Reign December the Second, 1135. Reigned 18 Years, 11 Months, 20 Days.

HENRY II. Began his Reign October the Twenty Fifth, 1154. Reigned 34 Years, 9 Months, 5 Days.

RICHARD I. Began his Reign July the Ninth, 1189. Reigned 9 Years, 9 Months, 19 Days.

JOHN. Began his Reign April the Sixth, 1199. Reigned 17 Years, 7 Months, 0 Days.

[Page 2] HENRY III. Began his Reign October the Nineteenth, 1216. Reigned 56 Years, 1 Month, 0 Days.

EDWARD I. Began his Reign November the Sixteenth, 1272. Reigned 34 Years, 8 Months, 9 Days.

EDWARD II. Began his Reign July the Seventh, 1307. Reigned 19 Years, 7 Months, 9 Days.

EDWARD III. Began his Reign January the Twenty Fifth, 1326. Reigned 50 Years, 5 Months, 7 Days.

RICHARD II. Began his Reign June the Twenty First, 1377. Reigned 22 Years, 3 Months, 14 Days.

HENRY IV. Began his Reign September the Twenty Ninth, 1399. Reigned 13 Years, 6 Months, 3 Days.

HENRY V. Began his Reign March the Twentieth, 1412. Reigned 9 Years, 5 Months, 24 Days.

HENRY VI. Began his Reign August the Thirty First, 1422. Reigned 38 Years, 6 Months, 16 Days.

EDWARD IV. Began his Reign March the Fourth, 1460. Reigned 22 Years, 1 Month, 8 Days.

RICHARD III. Began his Reign June the Twenty Second, 1483. Reigned 2 Years, 2 Months, 5 Days.

HENRY VII. Began his Reign August the Twenty Second, 1485. Reigned 23 Years, 8 Months, 19 Days.

[Page 3] HENRY VIII. Began his Reign April the Twenty Second, 1509. Reigned 37 Years, 10 Months, 1 Day.

Years of his Reign.Anno Dom.Years expired, Mar. 25. 1672.
11510162
21511161
31512160
41513159
51514158
61515157
71516156
81517155
91518154
101519153
111520152
121521151
131522150
141523149
151524148
161525147
171526146
181527145
191528144
201529143
211530142
221531141
231532140
241533139
251534138
261535137
271536136
281537135
291538134
[Page 4]301539133
311540132
321541131
331542130
341543129
351544128
361545127
371546126

EDWARD VI. Began his Reign January the Twenty Eighth, 1546. Reigned 6 Years, 5 Months, 19 Days.

Years of his Reign.Anno Dom.Years expired, Mar. 25. 1672.
11547125
21548124
31549123
41550122
51551121
61552120

MARY, Began her Reign July the Sixth, 1553. Reigned 5 Years, 4 Months, 22 Days.

Years of her Reign.Anno Dom.Years expired, Mar. 25. 1672.
11554118
21555117
31556116
41557115
51558114

ELIZABETH, Began her Reign November the Seventeenth, 1558. Reigned 44 Years, 4 Months, 15 Days.

[Page 5]

Years of her Reign.Anno Dom.Years expired, Mar. 25. 1672.
11559113
21560112
31561111
41562110
51563109
61564108
71565107
81566106
91567105
101568104
111569103
121570102
131571101
141572100
15157399
16157498
17157597
18157696
19157795
20157894
21157993
22158092
23158191
24158290
25158389
26158488
27158587
28158686
29158785
30158884
31158983
32159082
[Page 6]33159181
34159280
35159379
36159478
37159577
38159676
39159775
40159874
41159973
42160072
43160171
44160270

JAMES, Began his Reign March the Twenty Fourth, 1602. Reigned 22 Years, 0 Months, 3 Days.

Years of his Reign.Anno Dom.Years expired, Mar. 25. 1672.
1160369
2160468
3160567
4160666
5160765
616864
7160963
8161062
9161161
10161260
11161359
12161458
13161557
14161656
15161755
[Page 7]16161854
17161953
18162052
19162151
20162250
21162349
22162448

CHARLES I. Began his Reign March the Twenty Seventh, 1625. Reigned 23 Years, 11 Months, 0 Days.

Years of his Reign.Anno Dom.Years expired, Mar. 25. 1672.
1162547
2162646
3162745
4162844
5162943
6163042
7163141
8163240
9163339
10163438
11163537
12163636
13163735
14163834
15163933
16164032
17164131
18164230
19164329
20164428
[Page 8]21164527
22164626
23164725
24164824

CHARLES II. Began his Reign January the Thirtieth, 1648. and is now Reigning.

Years of His Reign.Anno Dom.Years expired, Mar. 25. 1672.
1164923
2165022
3165121
4165220
5165319
6165418
7165517
8165616
9165715
10165814
11165913
12166012
13166111
14166210
1516639
1616648
1716657
1816666
1916675
2016684
2116693
2216702
2316711
2416720

Advice touching the POSTS, and ROADS, more exactly than hath hitherto been published.

I. Concerning Letters, which may be sent from LONDON.

On Mondays, TO France, Spain, Italy, Germany, Flanders and Sweedland, Denmark, Kent and the Downs.

On Tuesdays, To Holland, Germany, Sweedland, Denmark, Ireland, Scotland, all parts of England and Wales.

On Wednesdays, To all parts of Kent and the Downs.

On Thursdays, To France, Spain, Italy, all parts of England and Scotland.

On Fridays, To Flanders, Germany, Italy, Sweedland, Den­mark, Holland, Kent and the Downs.

On Saturdays, All parts of England, Wales, Scotland, Ireland.

Letters are returned from all parts of England and Scotland, certainly every Monday, Wednesday, and Friday; from Wales every Monday and Friday; and from Kent and the Downs every day; But from other parts more uncertainly, in regard of the Sea.

POST STAGES.

NORTHERN-ROAD.Miles.Total.
FRom London to Waltham-cross1212
From Waltham-crose to Ware820
From Ware to Royston1333
From Royston to Caxton841
From Caxton to Huntington950
From Huntington to Stilton959
From Stilton to Stamford1271
From Stamford to Southwitham879
From Southwitham to Grantham887
From Grantham to Newark1097
From Newark to Tuxford10107
From Tuxford to Bautry12119
From Bautry to Doncaster6125
From Doncaster to Ferrybrigg10135
From Ferrybrigg to Tadcaster9144
From Tadcaster to York8152
From York to Burrowbrigg12164
From Burrowbrigg to North-allerton12176
From North-allerton to Darlington10186
From Darlington to Durham14200
From Durham to Newcastle12212
From Newcastle to Morpeth12224
From Morpeth to Alnwik12236
From Alnwik to Belford12248
From Belford to Berwick12260
From Berwick to Cockburnspeth14274
From Cockburnspeth to Haddington14288
From Haddington to Edenbrough12300

[Page 11]NORWICH-ROAD.Miles.Total.
From Royston to Cambridge1010
From Cambridge to Newmarket1020
From Newmarket to Bury1030
From Bury to Thetford1040
From Thetford to Attlebrough1050
From Attlebrough to Norwich1262

CHESTER-ROAD.Miles.Total.
From London to Barnet1010
From Barnet to St. Albans1020
From St. Albans to Dunstable1030
From Dunstable to Fenistratford838
From Fenistratford to Tosseter1250
From Tosseter to Daintry1060
From Daintry to Coventry1474
From Coventry to Coshall882
From Coshall to Lichfield1294
From Lichfield to Stone16110
From Stone to Namptwich16126
From Namptwich to Chester14140
From Chester to Northope18158
From Northope to Denbeigh12170
From Denbeigh to Conway14184
From Conway to Blewmorris10194
From Blewmorris to Hollyhead24218

DARBY-ROAD.Miles.Total.
From Tosseter to Northampton66
From Northampton to Harbrough1218
From Harbrough to Leicester1230
From Leicester to Loubrough838
From Loubrough to Darby1250

[Page 12]WESTERN-ROAD.Miles.Total.
From London to Stanes1616
From Stanes to Hartford bridge1632
From Hartford bridge to Basingstoke941
From Basingstoke to Andover1859
From Andover to Salisbury1675
From Salisbury to Shaftsbury1994
From Shaftsbury to Sherborn16110
From Sherborn to Crookhorn13123
From Crookhorn to Huniton19142
From Huniton to Exeter15157
From Exeter to Ashburton20177
From Ashburton to Plymouth24201
From Plymouth to Foye  
From Foye to Trowro  
From Trowro to Merkejew  

PORTSMOUTH-ROAD.Miles.Total.
From Stanes to Hartford bridge1616
From Hartford bridge to Petersfield1026
From Petersfield to Portsmouth2450

YARMOUTH-ROAD.Miles.Total.
From London to Burntwood1616
From Burntwood to Witham1834
From Witham to Colchester1246
From Colchester to Ipswich1662
From Ipswich to Saxmundum1678
From Saxmundum to Beckles1694
From Beckles to Yarmouth10104

[Page 13]BRISTOL-ROAD.Miles.Total.
From London to Huntslo1010
From Huntslo to Maidenhead1626
From Maidenhead to Reading1238
From Reading to Newberey1654
From Newberey to Malbrough1569
From Malbrough to Chippenham1584
From Chippenham to Bristol20104

GLOCESTER-ROAD.Miles.Total.
From Maidenhead to Abbington1616
From Abbington to Farrington1531
From Farrington to Cirencister1546
From Cirencister to Glocester1864

KENTISH-ROAD.Miles.Total.
From London to Dartford1414
From Dartford to Rochester1428
From Rochester to Sittingburn1240
From Sittingburn to Canterbury1555
From Canterbury to Deal Dover157 [...]

Forreign Weights and Measures, carefully compared with the English, by the great pains and industry of the famous, and my worthy Friend, Sir Jonas Moor, Knight.

 English Foot, into 1000 e­qual parts.English foot, in­to inches, and tenth parts of an inch.The pound A­verdupois into 100 parts.
London Foot10000.12.0100
France.   
Paris, the Royal Foot1.0681.00.80.93
Lyon Ell3.9763.11.71.09
Boloyne Ell2.0762.00.80.89
The 17 Provinces.   
Amsterdam Foot0 9420.11.30.93
Ell2.2692.03.2 
Antwerp Foot.9460.11.30.98
Ell2.2732.03.3 
Brill Foot1.1031.01.2 
Dort Foot1.1841.02.2 
Rynland or Leyden foot1.0331.00.40.96
Ell2.2602.03.1 
Lorain Foot.9580.11.40.98
Mecalin Foot.9190.11.00.98
Middlebourg Foot.9910.11.90.98
[Page 15] Germany.   
Strasbourgh Foot.9200.11.00.93
Bremen Foot.9640.11.60.94
Cologn Foot.9540.11.40.97
Frankford and Me­nain Foot.948.11.40.93
Ell1.8261.09.9 
Hambrough Ell1.9051.10.80.95
Leipsig Ell2.2602.03.11.17
Lubick Ell1.9031.09.8 
Noremburgh1.0061.00.10.94
Ell2.2272.03.3 
Bavaria.9540.11.4 
Vienna1.0531.00.60.83
Spain and Portugal.   
Spanish Palm, or the Palm of Castile..7510.09.00.99
The Spanish Vare or Rod, (four Palms)3.0043.00.0 
Their Foot is ⅓ of the Vare1.0011.00.0 
Lisbon Vare2.7502.09.01.06
Gibralter Vare2.7602.09.11.03
Toledo Foot.8990.10.71.00
Vare2.6852.08.2 
Italy.   
Roman Foot, on the Monum. of Cossutius.9670.11.61.23
Of Statelius.9720.11.7 
Roman Palm, for building, where­of ten make the Cauna.7320.08.8 
[Page 16] Bononia Foot1.2041.02.41.27
Ell2.1472.01.7 
Perch, whereof 500 to a Mile.12.04012.00.5 
Florence Brace or Ell1.9131.11.01.23
Naples Palm.8610.09.61.43
Brace2.1002.01.2 
Cauna6.8806.10.5 
Genua Palm.830.09.61.42
Mantoua Foot1.5691.06.81.43
Milan Calamus6.5446.06.51.40
Parma Cubit1.8661.10.41.43
Venice Foot1.1621.01.91.53
Other Places.   
Dantzick Foot.9440.11.31.19
Ell1.9031.10.8 
Copenhagen Foot.965.11.60.94
Prague (in Bohemia) Foot1.0261.00.31.06
Riga Foot1.8311.09.9 
China Cubit1.0161.00.2 
Turin Foot1.0621.00.7 
Cairo Cubit1.8241.09.91.61
Persian Arash3.1973.02.3 
Turkish Pike, at Con­stantinop. the greater2.2002.02.40.86
The Greek Foot1.0071.00.1 
Montons Universal foot0.675.08.11 
A Pendulum of the just length whereof will Vi­brate 132 times in a Minute.   
Ex. by me, Jonas Moore,

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