Posthuma Fosteri the description of a ruler, upon which is inscribed divers scales: and the uses thereof: invented and written by Mr. Samuel Foster, late professor of astronomie in Gresham-Colledg. By which the most usual propositions in astronomy, navigation, and dialling, are facily performed. Also, a further use of the said scales in deliniating of far declining dials; and of those that decline and recline, three severall wayes. With the deliniating of all horizontall dials, between 30 and 60 gr. of latitude, without drawing any lines but the houres themselves.
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Posthuma Fosteri: THE DESCRIPTION OF A RULER, Upon which is inscribed divers SCALES: AND The Vses thereof:
Invented and written by Mr. SAMVEL FOSTER, Late Professor of ASTRONOMIE in GRESHAM-COLLEDG.
By which the most usual Propositions in Astronomy, Navigation, and Dialling, are facily performed. Also, a further use of the said Scales in delineating of far declining Dials; and of those that Decline and Recline three severall wayes.
With the delineating of all Horizontall Dials, between 30 and 60 gr. of Latitude, without drawing any lines but the Houres themselves.
LONDON: Printed by ROBERT & WILLIAM LEYBOURN, for NICHOLAS BOURN, at the South entrance into the Royall Exchange, 1654.
WE here present to thy view, this short TREATISE, (written by that learned Professor of ASTRONOMIE in Gresham Colledge, Mr. SAMUEL [Page] FOSTER deceased) containing in it the Description and Vse of certain Lines to be put upon a streight Ruler, in the ready solution of many necessary Questions, as well Geometricall, as belonging to Astronomie, Navigation, and Dialling.
We should not thus hastily have thrust this into the World without its fellows, had we not been assuredly informed that some people, greedy rather of unjust gain to themselves, then with honesty to sit still, had prepared one for the Presse, from a spurious and imperfect Copie, both to the abuse of thee, and discredit of the industrious Author: who had he thought [Page] such things as these worthy him or the Presse, could have daily cram'd thee with them, to his own losse of time, and thy satiety. However, such as it now is, we assure thee was his own, and doubt not, but thou wilt finde it pleasant in the use, profitable to thee, and portable in it selfe.
We thought fit farther to advertise thee, that there are abroad in particular hands, imperfect Copies of some other Treatises of the same Author: Namely, An easie Geometrical way of Dialling. Another most easie way to project houre-lines upon all kinde of Superficies, without respect had to their standing, either in respect of Declination[Page] or Inclination. A Quadrant fitted with lines for the solution of most Questions of the Sphere: with some other things of the like nature. We fear least sinister ends of some mean Artists, or ignorant Mechanicks, (for those of ingenuity in whose hands they may be, we no way distrust) may engage them to father these things as their own, or at least under the Authors name put out lame and imperfect Copies of otherwise good things: To prevent which we give thee this timely notice, assuring thee, that these, together with divers other pieces never yet seen, except by very few, and if we deceive not our selves, of much greater weight,[Page] are making ready for the Presse by the Authors approbation, and from his own Copie in our command, with his other papers, of which thou shalt be made partaker within few moneths. In the mean time, we desire thee not to lose thy time in reading, or money in buying any the forementioned Treatises put out by any other, either under their own, or our Authors name, except such as shall be attested by me, who am one of those intrusted for that purpose, and who shall be ever studious of thy good.
THe Scale of Inches is a Scale of equall parts, and will performe (by protraction upon paper) such conclusions as are usually wrought in Lines and Numbers, as in Master Gunters 10. Prop. 2. Chap. Sector, may be seen, and in others that have written in the same kinde.
An Example in Numbers like his 10th. Prop.
As 15 to 5, So 7 to what?
Here, because the second terme is lesse than the [Page 2] first, upon the line AB, I set AC the first terme 15, and AD the second terme 5, both taken out of the Scale of equall parts. Thence also the third number 7 being taken, with it (upon the center C) I describe the arke EF, and from A, draw the line AE, which may only touch that same arke. Then from D, I take DG, the least distance from the line AE, and the same measured in the same Scale of equall parts, gives 2 ½, the fourth terme required.
[figure]
But if your second terme shall be greater than the first, then the form of working must be changed; as in this Example.
As 5 to 15, So 20 to what?
Upon the line AB, I set the second terme 15, which here suppose to be AD: then with the first terme 5, upon the center D, I describe the arke GH, and draw AG that may just touch it. Again, having taken 20 the third term, out of the same Scale, I set one foot of that extent upon the line AB, removing it till it fall into such a place, as that the [Page 3] other foot being turned about will justly touch the line AG before drawn, and where (upon such conditions) it resteth, I make the point C. Then measuring AC upon your Scale, you shall finde it to reach 60 parts, which is the fourth number required.
The form of work (though not so Geometricall) is here given because it is more expedite than the other by drawing parallel lines. But in some practises, the other may be used.
I have been the more large upon this, because the solutions of proportions which follow must be referred hither, the forme of their operations being the same with this. In them therefore shall only be intimated what must be done in generall, the particular way of working being here explicated.
CHAP. II. Of the Scales of Chords.
THe Scales of Chords are to protract and measure angles. The manner how they must be used is well enough known.
Only note here, that you may make the line of Sines, the line of Versed Sines, or the Zodiacke, (beginning at the middle of these two last mentioned) to serve for Chords of severall extents, if you count each halfe degree for a whole degree, and so double all the numbers, accounting 10 to be 20; and 30 to be 60, 45 to be 90, &c. By this you [Page 4] are fitted with severall Scales of Chords which are of different lengths, and may be used, each of them, as occasion shall require. And (by the way) the Versed Scale being taken for Chords, it will be of the same Radius or length with the Sines, Tangents and Secants and so will protract Angles to a Circle of their Radius, which is usefull in Projections, and many other things: and so the little Scale of 60 Chords might be spared.
CHAP. III. The joynt use of equall parts and Chords.
BY these two together, may be resolved all Cases in Plaine Triangles without proportionall work, if the three quantities given be protracted by help of these two Scales. For their principall uses are to measure lines and angles.
Here must be remembered. First, that if the three angles alone be given, then will the proportions only of the sides be found, but not the sides themselves. Secondly, that if two angles be known, then is the third also known; because it is the residue of the sum of the other two to 180 gr.
One case which is of frequent use may here be given for an example. In mensurations of distances of places (as Towns or Forts) there are usually two stations taken as A and B, whose distance AB [Page 5] suppose known, 300 feet, and the two angles adjacent, at A 105 gr. at B 60 gr. known also by observation. To finde the other sides, draw AB, and upon it set 300 being taken from the Scale of equall parts.
[figure]
Then with a line of Chords protract the angles at A and B according to their known quantities, so shall the two legges of the same angles meet at C: And if the length AC be taken and measured upon your Scale of equall parts, the same will shew about 1004, intimating that C is from A 1004 feet. So again BC being measured in the same Scale will give 1120; shewing that from B to C are about 1120 feet.
In this manner may perpendicular altitudes (as of Towers or such like) be measured, though no accesse can be had to them; and that much better than by the Geometricall Square. For it is not here requisite that the ground whereon the mensuration is made, should be levell, as if you work by the Square it is most commonly required; neither are you tyed to right angled Triangles here, as there you are. As for example,
If one station be at A, and the other at B, you may, by the precedent work get the distance AD. Then (standing at A) observe the altitude of C, the complement of that altitude gives the angle ACD. [Page 7] And again from A, if you observe the altitude of D, the difference of these two altitudes observed, gives the angle CAD. Or if D had appeared lower then your station, then the summe of your two observed altitudes had made the angle CAD. However, you have now the three angles and side AD; you may therefore, by help of them, finde the length of CD.
In such manner may all cases in plain Trigonometry be resolved.
CHAP. IV. Of the Scales of Sines, Secants, and Tangents.
THese being joyntly used with a Scale of equall parts, will resolve all things in plain Triangles, by working such proportions as are usually given for that purpose. The manner of the work may be gathered by the former delineation in the 1 Chapt. For if AC and AD had been taken out of the Scale of Sines, or Tangents, or Secants; and CE, DG, out of the Scale of equal parts; then had the work been resolved in Sines and equall parts, or Tangents and equall parts, &c. And so this kinde of work will produce the quantity required; although there be no delineation of the particular parts of the Triangle, as was before done by protraction.
[Page 8] By these same three Scales of Sines, Tangents, and Secants, may be wrought all things pertaining to Sphericall Triangles. That is to say.
1. Proportions in Sines alone.
2. Proportions in Tangents alone.
3. Proportions in Secants alone.
4. Proportions in Sines and Tangents together.
By naturall Sines and Tangents.
By ☉ and Versed-Sines.
5. Proportions in Sines and Secants together.
6. Proportions in Tangents and Secants together.
An Example in Sines alone What Declination shall the Sun have in the 10 gr. of Aries?
Upon the line AB (see Chap. 1.) set the Radius or Sine of 90 AC: and make AD equall to the Sine of 10 gr. (which is the Suns distance from the next Equinoctiall point.) Then with the Sine of 23½ (taken out of the same Scale of Sines) upon the center C, describe the arke EF; to which, from A, draw the Tangent line AE. Lastly, from D, to this line AE, take the least distance, the same measured [Page 9] in the line of Sines, gives about 4 gr. for the declination required.
The proportion that is here wrought stands thus.
As the Radius, to the Sine of 23 ½;
So the Sine of 10, to the Sine of 4 gr.
The like manner of work is to be used in Sines and Tangents (or any of the other two) joyned together; if it be remembred that the greater terms be kept upon the line AB; as was before prescribed in the first Chapter.
CHAP. V. OF NAVIGATION. Some things in this kinde will be performed very conveniently by these lines: As,
SECT. 1. To make a Sea-chart after Mercators projection.
A Sea-chart may be made either generall or particular; I call that a generall Sea-chart, whose [Page 10] line AE, in the following figure, represents the Equinoctiall, as the line AE there doth the parallel of 50 gr. and so containeth all the parallels successively from the Equinoctiall towards either Pole: but they can never be extended very neere the Pole because the distances of the parallels increase so much, as the Secants doe. But notwithstanding this, it may be termed generall, because that a more generall Chart cannot be contrived in plano, except a true Projection of the Sphere it selfe. And I call that a particular Chart which is made properly for one particular Navigation, as if a man were to sail between the Latitude of 50 and 55 gr. and his difference of Longitude were not to exceed 6 gr. then a Chart made (as the figure following is) for such a Voyage, may be called particular.
Now the making of such a Chart, is Master Gunters first proposition page 104 of the Sector, and this the line of Secants will sufficiently perform.
For it were required to project such a Chart: Having drawn the line AB, and having crossed it at right angles with another line AE, representing the parallel of 50 gr. you must then take the Secant of 51 from your Scale, and set it from 50 to 51 on both sides the Chart, and draw the parallel 51 51.
Again, take the Secant of 52 from your Scale, and set it upon your Chart from 51 to 52, and so draw the parallel 52 52. And so you are to draw the rest of the parallels.
Then for the Meridians, or divisions of the line BC, they are all equall to the Radius. [Page 11]
[figure]
[Page 12] If therefore you take the Radius, and run it above and below, you shall make the spaces or distances of the Meridians such as in the bottome of the Chart are figured with 1, 2, 3, 4, 5, 6.
These degrees thus set on the Chart, may be sub-divided into equall parts, which in the graduations above and below ought so to be. But in the graduations upon the sides of the Chart, they ought as they goe higher, still to grow greater. Yet the difference is so small that it cannot produce any considerable errour, though the sub-divisions be all equall. Divide them therefore either into 60 minutes, or English miles, or into 20 leagues, or into 100 parts of degrees, as shall best be liked of.
It a little more curiosity should be stood upon for the graduations of the Meridian, instead of the Secants of 51, 52, 53, &c. you may take 50½, 51½, 52½, &c. alwayes halfe a degree lesse than is the Latitude that should be put in.
Now if each of those divisions at the bottome of the Chart, as A 1, &c. be made equall to the common Radius of the Sines, Secants, and Tangents, and if a Chart be made to that extent upon a skin of smooth Velame; well pasted on a board; you may work upon it many conclusions very exactly.
The Vses of the Sea-Chart
Are set down in 12 Propositions by Master Gunter, beginning page 121. In each of which Propositions is shewed how to resolve the Question upon the Chart it selfe, which will be direction enough how [Page 13] the work must be performed, without any more words here used.
The working of these propositions also may be applyed to the Scales of Sines and Tangents, on the Ruler, and wrought by protraction, according to the rules given in the first Chapter, if the proportions, as he layes them down in the forecited pages, be so applyed.
If a Scale of Rumbs be thought more expedient for these operations then is a Scale of Chords, it may be put into some spare place of the Ruler.
His two Propositions, page 114. 116, may be done upon the Chart as is there shewed, but his second Proposition, which is,
SECT. 2. To finde how many Leagues doe answer to one degree of Longitude, in every severall Latitude.
THis (I say) may be done upon the Scales of Sines and equall parts: And for this purpose, the two last inches of the same Scale of equall parts, being equall in length to the Radius or Sine of 90, are divided into 20 at one end, and into 60 at the other end.
Take therefore upon the line of Sines, the complement of the parallels distance from the Equator, (or the complement of the given Latitude) and measuring it upon the Scale of 20 parts, it will shew [Page 14] you what number of Leagues make one degree of Longitude in that parallel of Latitude. And being measured upon the Scale of 60 parts, it gives so many of our miles, or so many minutes of the Equinoctiall, or any other great circle, as are answerable to one degree of Longitude in that Latitude.
Example,
Let it be required to finde how many Leagues doe answer to one degree of Longitude, in the Latitude of 18 gr. 12'.
Take out of the line of Sines, the complement of the given Latitude, namely. 71 gr. 48'. Then applying this distance to the Scale of 20 equall parts, you shall finde it to reach 19, and so many Leagues doe answer to one degree of Longitude, in the Latitude of 18 gr. 12'.
And the same distance being measured upon the Scale of 60 equall parts, will give you 57 parts, and so many minutes of the Equator are answerable to one degree of Longitude, in that parallel of Latitude.
So likewise, in the Latitude of 25 gr. 15', if you take the complement thereof 64 gr. 45', out of the Scale of Sines, and apply it to the former line of 20, you shall finde it to reach 18 parts, and so many Leagues doe answer to one degree of Longitude, in the Latitude of 25 gr. 15'.
¶In the Appendix to Master Norwoods Doctrine of Triangles, there is by him laid [Page 15] down 15 Questions of sailing by the plain Sea-chart, and others by Mercators Chart, all which the line of Chords and equall parts will sufficiently perform, if the work of the third Chapter of this Booke be rightly understood.
SECT. 3. How to set any place upon your Chart, according to its Longitude and Latitude.
IF the two places lie under one parallel, and so differ only in Longitude, then the course leading from one to the other is East or West: As A and E being two places under the parallel of 50 gr. and differing 5½ gr. in Longitude.
But if the two places differ only in Latitude, and lie under one Meridian, as A and B, then the course is North or South.
But if the places differ both in Longitude and Latitude as AC, then the course is upon some other point so much distant from the Meridian, as is the quantity of the angle BAC. [Page 16]
FOr this purpose chiefely, is the lesser line of Chords added, being made to the same Radius that belongs to the Sines, Secants, and Tangents. For when any Projection is to be made, the fundamentall Circle must be of that common Radius, and then the angles to be inserted upon it may be taken out of this line of Chords which is fitted to it. See the second Book of the Sector, Chap. 3. For these Tangents and Secants will performe the same things in those Stereographicke projections that there are done; and in all other irregular projections likewise.
By this kinde of work may any Sphericall conclusions be performed by protraction in plano. Also true Schemes of the Sphere may be drawn, sutable to any question, which will not a little direct in Sphericall calculations.
As suppose it were required to project the Sphere sutable to this Question.
Having the Latitude of the place, the declination of the Sun, and the Altitude of the Sun, to finde either the Azimuth or the houre of the day.
First, With the Radius of the line of Chords, upon the center C describe the fundamentall Circle ZHNO representing the Meridian, and draw the [Page 18] diameter HO for the Horizon, and ZCN at right angles thereto, ZN being the Zenith and Nadir points. Then by your line of Chords set the Latitude of your place (which let be 51 gr. 32') from Z to E, and from N to Q, drawing the line ECQ for the Equinoctiall, and at right angles thereto, the line MP for the axis of the World, P representing the North, and M the South Pole
Secondly, Supposing the Sun to have 20 gr. of North Declination, take 10 gr. (the Semitangent of the Declination) out of the line of Tangents, and set it from C to G. Likewise, take 20 gr. (the Declination) from your line of Chords, and set that distance upon the Meridian from E unto D, and from Q unto K: then describe the arke of a Circle which shall passe directly through the points DGK, the center whereof will alwayes fall in the line CP if it were extended, and this arke DGK shall be the line of the Suns course when his Declination is 20 gr. from the Equinoctiall Northward.
Thirdly, Supposing the Altitude of the Sun to be 50 gr. take 25 gr. (the Semitangent of the Altitude) out of the line of Tangents, and set that distance from C to F. Also take 50 gr. (the Altitude) from your line of Chords, and set them upon the Meridian from H unto A, and from O unto B, drawing the arke AFB, the center whereof will fall in the line CZ being extended, and this arke shall represent the Almicanter of 50 gr. And where this parallel of Altitude crosseth the parallel of Declination, which is at S, that is the place of the Sun at the time of the Question: Therefore, if you draw [Page 19]
[figure]
the arke of a circle which shall passe through the points MSP, it shall represent the houre of the day; and another arke through ZSN shall represent the Azimuth of the Sun at the same time. And the distance CT being measured on the Tangent line will fall upon 20 gr. 4', the double whereof is [Page 20] 40 gr. 8', which is the Azimuth of the Sun from the East or West, and the complement thereof to 90 gr. is the Suns Azimuth from the South.
¶1. The centers of the parallels of declination, and of the parallels of altitude, may readily be fouud by the Scale it selfe; as in this projection, having found the point F upon the Line ZC, extend the line ZC without the circle, and because the Suns altitude is 50 gr. take therefore out of your Scale the Secant of 40 gr. (the complement of the Altitude) and set that distance from C to I, so shall the point I be the center of the parallel of Altitude.
Or take the Tangent of 40 gr. out of your Scale and place it from F to I, either of which will fall in the point I, the center of the parallel of 50 gr.
In the same manner may the center of the parallel of Declination be found, by taking out of your Scale the Secant complement of the Declination, and setting it from C, upon the line CP, (being extended) and where that distance ends, that is the center of the parallel.
Or the Tangent complement of the Declination being set from G on the line GP shall give the center also. [Page 21]
[figure]
¶2. For the finding of the centers of the Hours and Azimuths, the Scales of Secants and Tangents will much help you; So the Azimuth from the South being 49 gr. 52', if you take the Tangent thereof out of the Scale of Tangents, & set it upon the Horizon [Page 22] from C to L: the point L shall be the center of the circle NTZ.
Or the Secant of 49 gr. 52' being set from T, that also shall give you the center Las before.
The center of the Houre-circle is found in the same manner, for the houre from the Meridian being 31 gr. 28', if you take the Tangent thereof our of your Scale, it shall reach from C to R, the point R being the center of the Houre-circle MVSP.
Or the Secant of 31 gr. 28', being set from V, shall give the point R for the center of the same houre. And in this manner may any Houre or Azimuth whatsoever be drawn.
Many other propositions in Astronomie, may be wrought upon this projection, and indeed any of the 28 cales of Sphericall Trigonometry, may by this kinde of projection be easily illustrated and resolved, which will cleerely informe the fancie in the resolving of Sphericall Triangles. An Example or two for practise shall be,
1. To finde the Suns Amplitude.
In this projection, the Amplitude from the East or West is represented by the line CX, take therefore the distance CX in your compasses, and apply it to the line of Tangents, (counting every degree [Page 23] of the Tangents to be two degrees) and where it resteth, that shall be the Amplitude from the East or West, which will be found to be 33 gr. 22x.
Or if you lay a Ruler upon Z and X, it will cut the Circle in Y, and the distance NY being measured on the line of Chords, shall give the Amplitude also.
2. To finde the distance of the Sun from the Zenith.
The distance of the Sun from the Zenith is the arke Z S, therefore to finde the quantity thereof, you must first finde the pole of the circle NS Z, which is done after this manner.
Lay a Ruler from Z to T, and it will cut the circle in a, then take in your compasses a quadrant of the outward circle, and set it from a to b, then lay a ruler from Z to b, and it shall cut the Horizon in e, which point e is the pole of the circle ZTN.
Now to measure the arke Z S, you must lay a ruler upon e and S; which will cut the outward circle in the point A, so shall A Z, being measured upon the line of Chords, give you the quantity of degrees contained in the arke Z S, which will be 40, equall to the complement of the Suns Altitude.
¶This latter proposition was inserted rather to shew how the arke of any great Circle of the Sphere (the sides of all Sphericall Triangles being such) may be measured, then for any need there was to finde the distance of the Sun from the Zenith, for that mighe have [Page 24] been more easily effected, it being only the complement of the Suns Altitude; but according to this operation, may the side of any Sphericall Triangle whatsoever be measured.
The line of Sines also will project the Analemma, as Master Gunter sheweth, if this proposition be added.
How to divide any line given, into such parts as the Scale of Sines is divided.
Which proposition may be done by that which is set down in the 1 Chapt. For if AD mn C were parts or divisions made equall to those upon the Scale of Sines, and CE were a line in the same manner to be divided: After you have prepared your work as is there prescribed, you need only to take the least distances between the points C nm D and the line AE, and insert the same into your given line, so shall the divisions thereof be proportionall to the line of Sines.
CHAP. VII. Of the line of Versed-Sines.
THe generall use of this Scale is principally to resolve these two Sphericall Cases. First, By having three sides of a Sphericall Triangle, to finde an angle. Secondly, By having two [Page 25] sides and the angle comprehended, to finde the third side. According to which two generall cases you shall finde particular examples; namely, the first and third Sections of this Chapter sutable to the first Case: and the 5 Section answerable to the second.
SECT. 1. To finde the Suns Azimuth.
FIrst, Finde the summe and difference of the complement of your Latitude, and complement of the Suns altitude. Then having made AB equall to the length of the whole Scale, count upon the same Scale the summe and difference before found.
[figure]
After this, take with your Compasses the distance from the Suns place to the summe, and setting one foot of that extent upon B, with the other describe the arke CD. So again, take the distance upon the [Page 26] Scale from the Suns place to the difference, and with that extent upon the center A, describe the arke EF: Which done, draw the streight line DE, so as it may justly touch those two arks, cutting the line AB in G: so shall BG (being measured upon the Scale, from the beginning of it) shew the Azimuth from the South. And AG measured upon the same Scale will give the Azimuth from the North.
SECT. 2. To finde the Amplitude of the Suns Rising or Setting.
IF you suppose the Sun to be in the Horizon, or 00 gr. high, and so the complement of the Altitude to be 90, and if (upon these suppositions) you work as in the last Section is shewed, then shall BG give the graduall distance of the Suns rising or setting from the South; AG from the North, and from the midst of the line to G, is the Amplitude from East or West.
SECT. 3. To finde the houre of the Day.
MAke AB equall to the whole Scale, as before: and count from the beginning of the Scale to the Suns place what number of degrees there are; the same number shews the graduall distance of the Sun from the North Pole. Of this distance and the complement of your Latitude, finde the sum and [Page 27] difference, and count them both upon the Scale, as was done before. Then again, count thereon also the complement of the Suns altitude: Upon which point, setting one foot of your Compasses, extend the other to the forenamed summe; and with that
[figure]
extent upon the center B describe the arke CD. Again, setting one foot of your Compasses upon the complement of the Suns altitude, extend the other to the forenamed difference, and with that extent upon the center A, describe the ark EF. Lastly, draw the streight line DE, which only touching the two former arks, may cut the line AB in G: so shall AG (measured on the Scale, from the beginning of it) give the degrees of the Suns distance from the South. These may be turned into houres, counting 15 gr. for one houre, and 1 gr. for 4 minutes of an houre.
SECT. 4. To finde the Semidiurnall and Seminocturnall arks.
IF you suppose the Suns altitude to be 00 gr. and so the complement of it to be 90, and then work as is directed in the 3. Sect. of this Chap. then shall AG give the Semidiurnall arke, and BG the Seminocturnall arke: Each of these turned into Houres and minutes, and doubled, will give the length of the Day and Night.
SECT. 5. The Suns place being assigned in any point of the Ecliptick, to finde his Altitude at all houres.
BY this, may Tables of the Suns Altitude be made to all houres, the Sun being in any Signe of the Zodiacke, whereby many particular Instruments for finding the houre of the day, may be made, as Rings, Quadrants, Cylinders▪ and such like.
Draw the line AB, and upon it, with CA or C B, equall to halfe your Scale, describe a Semicircle. Then count upon your Scale the Suns distance from the North Pole, as was done in the 3. Sect. of this Chapt. and in the same manner also finde the summe and difference of this distance and the complememt [Page 29]
[figure]
of your Latitude. Then take the distance of the said sum and difference in your Compasses, and set it upon the Semicircle from A to D, and draw BD. Now because AB is equall to your whole Scale, you may divide the same into houres, by transferring each 15' gr. from your Scale to the line. This done, take the least distance from the point 1 to the line BD, and set one foot of this distance upon the forenamed difference counted upon the Scale, and let the other foot stand further onwards upon the Scale, and where it falls, it sheweth how many degrees that houre of 1 is distant from the Zenith. Or if you count the degrees from the middle of the Scale, it shews the Altitude of the Sun in that Houre. Thus doe for the points of 2, 3, 4, &c. and you shall in the same manner finde their Altitude: And if you go on to the end, you shall (most commonly) finde your Compasses at last to reach beyond the middle of the Scale.
[Page 30] [This alwayes, and then only, happens, when the sum (found at first) is greater than 90 gr.] Look then how much it is beyond, for so many degrees is the Sun below the Horizon at that houre of the night: Or (which is all one) so many degrees is the Sun elevated above the Horizon in that Signe or point of the Eclipticke which is so much on the other part of the Equinoctiall. That is, If the Suns place given were the beginning of Taurus or Virgo, and your Compasses (suppose at the 9th. houre) goe beyond the 90thgr. of the Scale, you shall there see how low the Sun is under the Horizon at 9 a clock at night, or at 3 in the morning. And the same also sheweth how high the Sun is at 9 in the morning, or at 3 afternoon, if his place were in the entrance of Scorpio or Pisces, which two Signes are so much beyond the Equinoctiall on the other part, as Taurus and Virgo are on this side.
SECT. 6. All Proportions in Sines alone, where the Radius stands first, may be wrought upon this Scale, without any protraction at all.
THe manner of the work will best appear by an Example. Let the proportion set down before in Sines alone be here repeated. The terms stand thus:
As the Radius, to the Sine of 23½;
So the Sine of 10, to the Sine of what?
[Page 31] Take the sum and difference of the second and third arks, the sum is 33½ the difference is 13½: count these both upon the Scale, and there take their distance: apply the same to the middle of the Scale; so as that the same number of degrees may be above 90, that is below; so shall the degrees either above or below, be about 4; and this is the Sine required for a fourth proportionall to the former.
CHAP. VIII. How to work proportions in Sines and Tangents, by the lines of Versed-Sines and ⊙.
DEscribe a Circle, as ABD, of the same Radius with the line ⊙. The Versed Scale is in length four times the same Radius.
[figure]
Let the Sines (given or required) be measured out of ⊙, and let them be set upon the Radius from A, to AC or AE.
[Page 32] Let the Tangents (given or required) be measured out of the Versed Scale, from 90 to 00, or to 180, which are 90 Chords belonging to 90 equall parts of the Semicircle ABDP, and the same Tangents must ever be set upon the Circle from A, as AB, AD:
Then draw a right line through the first and third of the given terms, as from B and C to O; and another right line from O to D or to E.
So the fourth terme required shall be either the Sine AE, or the Tangent AD, each to be measured in its proper Scale.
[figure]
If the Radius be ingredient in the proportion, then this of Sines and Tangents may be wrought by the draught of one line: see this third Scheme. The Sine is to be taken or set on the Radius from A, as AF, the Tangents are to be set and taken upon the Circle (in this case alwayes) the lesser of them from A, the greater from P; as AG and PH. So that whatsoever is given or required will here be found.
The further use of this line is shewn afterwards in the making of declining reclining Dials.
TO effect this, there are required two observations: the first is of the Horizontall distance of the Sun from the pole of the plain, the second is of the Suns Altitude, thereby to get the Azimuth. And these two observations must be made at one instant of time, as neer as may be, that the parts of the work may agree together the better.
1 For the horizontall distance of the Sun from the pole of the plain: Apply one edge of a Quadrant to the plain, so that the other may be perpendicular to it, and the limbe may be towards the Sun, and hold the whole Quadrant horizontall as neer as you can conjecture: Then holding a threed and plummet at full liberty, so that the shadow of the threed may passe through the center and limbe of the Quadrant, observe then the degrees cut off by the shadow of the threed, and number them from that side of the Quadrant that standeth square to the plain, for those degrees are the distance required.
2 At the same instant observe the Altitude of the Sun, these two will help you to the plains declination by the rules following.
First, By having the Altitude, you may finde the Azimuth by the 1. Sect. of the 8. Chap. then by comparing the Azimuth and distance together, you may finde the plains declination in this manner.
When you make your observation of the Suns [Page 34] horizontall distance, marke whether the shadow of the threed fall between the South and that side of the Quadrant which is perpendicular to the plain. For,
1. If the shadow fall between them, then the distance and Azimuth added together, do make the declination of the plain, and in this case, the declination is upon the same coast whereon the Suns Azimuth is.
2. If the shadow fall not between them, then the difference of the distance and Azimuth is the declination of the plain, and if the Azimuth be the greater of the two, then the plain declineth to the same Coast whereon the Azimuth is: Otherwise, if the distance be the greater then the plain declineth to the contrary Coast to that whereon the Suns Azimuth is.
¶Note here further, that the Declination so found is alwayes accounted from the South, and that all Declinations are numbred from either South or North towards either East or West, and must not exceed 90 gr.
1. If therefore the number of declination exceed 90, you must take the residue of that number to 180 gr. and the same shall be the declination of the plain from the North.
2. If the number of declination doe exceed 180 gr. then the excesse above 180 gives the plains declination from the North, towards that Coast which is contrary to the Coast whereon the Sun is.
[Page 35] ¶And here note, that wheresoever in this Chapter the use of a Quadrant is required, the Scale of Chords will effect the same; if upon a piece of plain board you describe a Quadrant, whose sides may be parallel to the edges of the board, upon which you may set off the Horizontall distance and Altitude, which will performe the work thereof when a Quadrant is not at hand.
CHAP. X. OF DIALS. To draw upright declining Dials, by the lines of Sines and Tangents.
THe declination of the plain being found by the last Chap. Upon your plain describe a rectangled parallelogram, in which let the sides AB and CD be perpendicular to the Horizon, and each of them equall to the Tangent of your Latitude: and let AC and BD be equall each of them to the co-tangent of your Latitude, and let BD be prolonged if need be.
Then taking that side of the parallelogram (for the houre of 12) which looketh towards that coast unto which the plain declineth, as here namely, the side AB; and on that line having assumed the superiour [Page 36] angle A in South-declining-plains, or the inferiour in North-decliners, for the center of your Diall: Let BE and CG be made equall to the Sine of the plains declination, so AE being drawn, shall be the substilar, and AG shall be the houre of 6. Then from E, raise EF perpendicular to AE, and make A 12 and EF equall to the co-sine of the declination: and if you draw AF, the same shall represent the Axis, and the angle FAE sheweth how much the same is to be elevated above the substylar. Again, make AH equall to the co-sine of your Latitude, and draw H6 parallel to AB; which will cut AG in the point noted with 6. To this A6, let A6 also beyond the center be made equall, and then draw the lines 12 6 and 12 6, which lines must have the houre points set upon them; and to performe that worke doe thus.
Draw upon paper, or some other plaine, the line LM, upon which set LR and RM, each of them equall to your Tangent of 45 gr. Then make RN equall to the Tangent of 30, and RO equall to the Tangent of 15, so shall you have points to finde all the houres, and if you desire halves and quarters, you must also put their Tangents into the same line RM. Being thus prepared, if you would divide the lesser line 12 6 into its requisite parts, take the same line in your Compasses, and with it, upon L as a center describe the arke PQ, and from M draw MP, which may only touch the same arke. Then from N take the least distance to the line MP, and the same will reach from 12 to 11, and from 6 to 7; so the least distance from O to the line MP, will [Page 37] give from 12 to 10, and from 6 to 8. And the least distance from R will reach from 12 or 6 to 9.
In the same manner you must divide the larger line 12 6. Take it out of your Diall, and with it describe the arke ST from the same center L, and draw MS touching only the same arke. Then the least distances from N, O, R, to the line MS, will give the points or distances 12 1; 6 5; and 12 2; 6 4; and 12 3; or 6 3. These upon the South-decliner; the like may be done upon the North-decliner. Lastly, from the center A, through these points you must draw the Houre-lines.
CHAP. XI. Of the Horizontall and full South Dials.
THese are done more easily, for having made ZX for the line of 12, and WV perpendicular thereto for the two sixes, in them both make ZX equall to the Radius, and in the Horizontall let ZV and ZW be equall (each of them) to
[figure]
the Sine of your Latitude: in the South plain let ZV and ZW be (each of them) equall to the cosine of your Latitude. Then draw the lines XV and XW, and divide them as was now shewed in the declining plains; so may you from the center Z, [Page 41] and these points, draw all the houres, as you see in these figures.
[figure]
The Styles are to stand over the line of 12: that in the Horizontall must be elevated so much as your Latitude comes to; the other according to the complement of your Latitude.
The upright North plain is the same with the South, only turned upside down, and the course of the figures altered.
The East and West upright plains may be made by the Tangent line, in such manner as others have prescribed.
CHAP. XII. Of the Scale of Horizontall Spaces.
FOr the Horizontall plains in speciall, there is a peculiar Scale by which the houres may sodainly be described, to any Latitude between 30 and 60 degrees.
The manner of which work is easie. For you [Page 42] have the numbers from 30 to 60 five times repeated, serving for the five houres in so many Latitudes. Suppose then a Horizontall Diall were to be described for the Latitude of 51½ gr. First, by the Radius (which is from the beginning of the line to R) describe a Circle, and draw the line of 12 from the center. Then take from the beginning of the line to VI, and set it in the Circle both wayes from 12, these two are the points of the two sixes. Again out of the same Scale take the length from the beginning to 51½ in the remotest numbers, and set that upon the Circle on both sides 12, these are the points of 5 and 7. So from the beginning of the Scale to 51½ in the next remotest numbers, being set as the other were, will give the points of 4 and 8. The third 51½ will give the points of 3 and 9. And the fourth gives 2 and [...]0. The last gives 1 and 11.
¶The Chord line that is fitted to this Horizontall Scale, is of good use in other delineations: But the further use of these two joyntly, must be referred to another place.
CHAP. XIII. How to draw upright declining Dials when the Latitude of the place is very little or very great.
IN the work of the 10 Chap. it may fall out that either the Tangent or co-tangent of the Latitude may be too great, such as the Scale wil not afford. [Page 43] This will frequently fall out in the new Latitude of re-in-cliners: to remedy that inconvenience, I have added these helps.
Where the Latitude is but small
[figure]
A rectangle parallelo.
1. AB=CD=tang. of Latit.
2. BD=AC=Radius.
3. BE=CG=Sine declination.
4. A12=EF=cosine of declination.
5. AH=consine of Latitude.
6. H m♒AB.
7. Draw EG, it will cut CD in K. AK is the line of six: it cuts H m at 6, make A6=A6, on both sides, and draw 12 6; 12 6; and divide them as the other are in the 10 Chap.
[Page 44] Or you may draw BC the Diagonall, and EK ♒ thereto, and so omit CG.
Or you may make the ∠DEK=to your Latitude, and so omit the two former.
Or thus.
After the 1, 2, 3, 4, 5, you may omit the 6. Then 7thly. Draw EG it will cut CD in K, and AK is the line of six.
Then lay a Ruler from 12 to H, cutting DC in L.
Make 12M=CL, and AN=AK.
So shall KM, MN, be ♒ to the two former lines 6 12; 6 12; and may supply their Offices somwhat better, because they are larger.
Where the Latitude is great.
a rectangle parallelogr.
1. AB=CD=Radius,
2. BD=AC=co-tang. lat.
3. BR=CG=Sine declination. GR a right line cutting DB in E. AE Substilar. AG houre of 6.
4. GP=RT=A12=co-sine of declination. TP a right line, cutting BD in O. AEF a right angle.
5. EF=EO. AF the Style.
6. AH=co-sine Latitude. [Page 45]Hm♒AB, cuts AG in 6. A6=A6, on both sides. Draw 6 12; 6 12, &c. [figure] Or after the 1, 2, 3, 4, 5.
6. Draw 12 H, it cuts DC in L. Make 12 M=LC: and AN=AG. Then GM, MN shall be ♒ to 6 12; 6 12: and may therefore supply their uses.
CHAP. XIIII. Concerning Reclining and Inclining Plains, how to draw houres upon them.
THey may be referred to a new Latitude, in which they shall stand as upright plaines: and then the delineation will be the same with those in the 10 Chap.
The Meridian line is not here to be taken for the line of 12 at mid-day (for it often represents the mid-night) but for that part which helps to describe the Diall.
1. The first thing to be done upon these plains, is (by some levell) to draw the Horizontall, and then the Verticall line perpendicular thereto.
2. Next is the placing of the Meridian upon the plain, in a true position. In direct plains that re / in-cline, and in upright decliners, the Meridian is the same with the plains Verticall line. In East and West re / in-cliners, it is the same with the horizontall line. In the rest, it ascendeth or descendeth from the horizontall line, and must be placed according to the rules hereafter given.
3. For which purpose, In the Circle FZFN (made of the same Radius with that of Sines and Tangents, &c. upon the Ruler) set FP, ZAE, FP, NAE, [Page 47] each equall to your Poles altitude. Then count the Reclination or Inclination from Z downwards, towards P if it be North, towards AE if it be South; and there set B. Then proceed by one of these two wayes.
¶How the Meridian line is to be placed, whether ascending above, or descending below the horizontall line: and from which end of that line, whether that which looks the same way with the declination of the plain, or that which looks the contrary way.
In North re / in-cliners.
If D fall below P, the
Recliners are North plains, and the Meridian ascends above the horizontall line, from that end of it which looks to the same Coast of declination.
Incliners are South plains, and the Meridian descends below that end of the horizontall line, which looks to the contrary Coast of declination.
If D fall above P, the
Recliners are South plains, and the Meridian goes below: contrary,
Incliners are North plains, and the Meridian goes above the end looking the same way with declination.
In South re / in-cliners
If D fall above AE, the
Recliners are North plains, and the Meridian goes above the horizontall line, from the same end with the Coast of declination.
Incliners are South plains, and the Meridian goes below the horizontall line, from that end which is contrary to the Coast of declination.
If D fall below AE, the
Recliners are North plains, and the Meridian goes below the horizontall line: contrary,
Incliners are South plains, and the Meridian goes above the horizontall line, from that end which looks to the Coast of declination.
If D fall into P, both re / in-cliners, are called Polar plains, and the Meridian, in both, ascends from the
Same end in Recliners.
contrary end in Incliners.
If D fall into E, the
Recliners are North plains, and the Meridian ascends from the same; descends from the contrary end to that which looks upon the Coast of declination.
Incliners are South plains, and the Meridian ascends from the contrary; descends from the same end that looks upon the Coast of declination.
¶East & West
Recliners are North plains, declining from North,
So much as the complement of their re / in-clination comes to. This is their new declination, & their new Latitude is the complement of the Latitude of your place.
Incliners are South plains, declining from South,
[Page 52] 4. For that which follows, take notice of these four things. First, That from D to the neerest AE (measured by the line of Chords) gives the new Latitude, in which the re / in-clining plain, is an upright declining plain. Secondly, That OR (measured upon the line of Sines) gives the complement of the plains new declination in that new Latitude: this New declination is to the same Coast with the Old, but alwayes lesse in quantity than it. Thirdly, That DS (measured upon the Sines) gives the quantity of the Meridians ascension or descension. This gives the quantity, the former rules gave the Coast. Fourthly, That in the description of the Diall, you must only make use of the new Latitude, and new Declination: having nothing to doe with the other.
5. Having the former things known, you must (by the Tangent and co-tangent of the new Latitude) describe your Rectangled Parallelogram (as in the 10 Chap.) and according as the plain was discovered to be a decliner from the North or South, you must make choice of your center, place the substylar, style, and six a clock line, by help of the Sine and co-sine of the new declination, and new Latitude, and then prick down and draw the houres, all in the same form that was before shewed in the 10 Chap. for upright decliners.
This for the Dials description.
6. Lastly, for placing your Diall. First, Consider which way, and how much, your Meridian [Page 53] ascended or descended from the horizontall line. Then goe to your plain, and there draw the same Meridian line answerably, setting off so many degrees by your Scale of Chords. When this is done, take your paper description, and lay the Meridian of it, either upon, or else parallel to, the Meridian drawn upon the plain, and take care to place it the right way; namely so, as that the imaginary style of your paper (or a reall pattern of the style cut fit and set upon the paper Diall) may point into the North or South Pole, according as the plain is esteemed to be a North or South plain. After this is performed, you may transfer each houre from the paper to the plain, and so finish all the work.
CHAP. XV. Concerning full East and West re-inclining plains.
HEre in this sort of plain, you are only to take notice, that the new Latitude (wherein they stand as erect plains) is ever the complement of your own Latitude. And the new declination (in that Latitude) is the complement of their re / in-clination. By knowing these, you may describe the diall according to the 10 Chap. The Meridian line (in all these) lyeth in (or parallel to) the horizontall line. All which things will appear also [Page 54] out of the former figures, if according to them you should make a draught, and suppose your plain to decline 90 degrees, as all these East and West plains do. All other things will follow of themselves, agreeable to other plains.
CHAP. XVI. Concerning re-in-cliners, that are direct, or have no declination.
IF the line CB be placed (as is prescribed in the former figure) and drawn quite through, it will represent your plain that is re / in-clining towards the North, and without any declination. So also BL, if it be drawn quite through, will represent such plains as re / in-cline towards the South, and have no declination.
For which lines so drawn (or imagined only) you may gather (according to the former rules) which of the Poles (A or X) is elevated, and how much it is elevated (which is shewed by the arke CA or LX.) You may also see which end of the Meridian is to be taken for the substilar line, over which (in these direct plains) the stile is ever to be erected, and must stand.
Then for drawing the houres, you have no more to doe, but to describe an Horizontall Diall to that [Page 55] elevation, which is due to the plain. The manner whereof is shewed before in the 11 Chap.
CHAP. XVII. How to deal with those plains, where the Pole is but of small elevation.
SUch plains whose styles lie low, cannot have the houre-lines distinctly severed, unlesse the center of the Diall be cast out of the plain. In such cases therefore the Diall is to be made without a center, in this manner.
1. Place AB the Meridian, A [...] the substilar, AF the style, by the rules before given in the 10 and 13 Chapt. omitting what is done for the line of six, being here of no consequence.
2. Finde the plains difference of Longitude by the 18 Chap. following.
3. Assume any two points in the substilar AE, as at R and S, and through them draw two infinite right lines, at right angles to AE.
4. To the style AF, draw the parallel GH, at any convenient distance, such as you shall think fit, for your new style to stand from your plain.
5. Take the least distance from R to GH, and set it upon the substilar from R to K. So from S to GH, set from S to L.
[Page 56] 6. Upon the two centers K and L, describe two Circles: And in them both, make the two angles RKM, SLM, equall to the plains difference of Longitude; and set it on that side the substilar RS, upon which the Meridian AB standeth.
7. The rest of the work will be easie to finish, if you begin (in each circle) from the points at M, to divide them into 24 equall houres; and from the centers to those equall divisions, draw out lines to cut their respective contingent lines in 12, 11, 10, &c. And from each correspondent houre, you must draw the lines 12 12, 11 11, 10 10, &c.
CHAP. XVIII. Having the Latitude of the place, and the plains declination, to finde the plains difference of Longitude.
IT must be understood, that the plain is supposed (in this work) to be alwayes erect; and that therefore for re / in-clining plains, the Latitude and declination here mentioned is meant of the new Latitude and new declination.
Two wayes to doe it.
Make ABC a right angle.
I.
AB Sine of new Latitude.
BC Tangent of new declination.
BAC is the difference of the plains Longitude from your Meridian.
HDE is the complement of the difference of Longitude.
Or DHE is the difference it felfe.
If this work be done for upright plains in your own Latitude, which will be needfull in far decliners, then instead of the new Latitude and new Declination here mentioned, you are to use your own Latitude, and the upright plains Declination. The new Latitude and Declination are for re-in-clining plains.
CHAP. XIX. Of Polar Plains, on which the Pole is not elevated at all.
THose are called Polar plains, upon which neither of the two Poles is elevated at all, but the plaine lies parallel to the Axis, such are the upright East and West: and in every declination [Page 60] from the South some one recliner: in every declination from the North some one Incliner.
The new declination of all Polar plains is their difference of Longitude, in these you must work by the 10 and 14 Chap. to place AB the Meridian, AE the substilar; & for the style AF, it hath no elevation from the substilar, but is the same with it. So that the work will be much like that in the 17 Chap.
Make GH for the style, parallel to the substyle AE, at some convenient distance. Then assigning any point in the line AE, as S, through it draw an infinite right line perpendicular to AE. And take the least distance from S to GH, make SL equall thereto. Upon L describe a circle, and make SLM equall to the difference of Longitude, on the same Coast from SL unto which the plain declineth, or to the same Coast upon which the first line of 12 namely AB standeth. Then having found the houre points upon the line which passeth through S, namely, 6, 7, 8, 9, 10, &c. draw lines through them, all parallel to thē substilar AESL.
CHAP. XX. Another way to prick down the hourepoints, by the Tangent line on the Scale.
LEt the first four Sections of the 17 Chap. be performed according to the directions there given. After them, you must gather the [Page 61] angles at the Pole, by help of the plains difference of Longitude in this manner. Let the former example serve. The difference of that plains Longitude will be 82 gr. 08'. Out of this, take the greatest number of some just houre; viz. 75 gr. The remainder is 7 gr. 8'. Having then set down the substile 00 00, as in the Margin, write this 7 gr. 8'. next under it, to which adde 15 gr. continually, and you shall produce all the following numbers as you there see them. And note, that in this work 82 gr. 8'. the difference of Longitude will ever stand against the houre of 12, if you work right. Then take the first number 7 gr. 8'. out of 15 gr. the remainder is 7 gr. 52'; set this above the substyle, and to this number adde continually 15 gr. (or one houre) the numbers will be produced such as you here see.
gr.
'
3
52
52
4
37
52
5
22
52
6
7
52
Substyle
00
00
7
7
08
8
22
08
9
37
08
10
52
08
11
67
08
12
82
08
When this is done, draw a right line, therein assuming the point S or R. Then upon your Scale of Tangents, count the numbers 7 08, 22 08, &c. in the Table, and take them off from the same Scale, setting them severally from S to a, b, c, d, e. So again, upon the same Scale of Tangents count the other numbers, 7 52, 22 52, 37 52, &c. and take them off thence severally, and place them from Sat f, g, h, i.
[Page 62] After this, take 45 gr. out of your Tangent Scale, and place it upon this line from S to K both wayes. Then (as in the 19 Chap.) take the least distance
[figure]
from the point S (in your Diall) to GH the fiduciall edge of your style, and setting one foot of that extent upon K, with the other describe the arke MP, and from S draw the lines SP only touching the said arks. Being thus fitted, you must from a, b, c, d, e, take the least distances to the line LP, and set them (respectively) upon the contingent line of your Diall, from S to 7, 8, 9, 10, 11, And again, the least distances from f, g, h, i, to the line PS, will give the poins 6, 5, 4, 3, upon the same contingent line of your Diall. Thus is one of the contingent lines divided [Page 63] into his requisite houres. The like work must be done with the other. For you must take the least distance from the point R (in your Diall) to GH the fiduciall edge of your style, and setting one foot of that extent upon K, with the other describe the arke TV, and draw the touch-line SV. Then from a, b, c, d, e, take the least distance to SV, and set them on your Diall from R to 7, 8, 9, 10, 11. So from f, g, h, i, take the least distances, and set them from R to 6, 5, 4, 3, by which means you have the other contingent line divided into its requisite hours. The rest of the work for finishing the Diall will be the same with that in the 17 Chap.
But because the tangents upon the Scale goe but to 63 gr. 26', it must therefore here be shewed how those that exceed that quantity may be supplyed. Namely thus, Double the number of degrees and minutes, and from the sum take 90 gr. so shall the Tangent and Secant of the remaining arke (both of them put together) give the Tangent required. As if in the former example, it were required to finde the Tangent of 67 gr. 8' noted upon the line by the length S e, we must doe thus. The double of 67 gr. 8' is 134 gr. 16', from which takeing 90 gr. the remainder will be 44 gr. 16'. Accordingly we must first take the Secant of 44 gr. 16', and set it from S to y; then take the Tangent of the same 44 gr. 16'. and set it also forward from y to e, so shall you have S e the whole Tangent of 67 gr. 8'. as is required. Thus doe for any other which shall goe beyond the Scale.
TAke notice of these terms. 1. Verticall distance, is the distance of the plains pole from the Vertex or Zenith of the place.
2. Polar distance, is the distance of the plains pole from the North pole.
Preparatory works.
1. Draw the horizontall line upon the plain,The wayes how to effect these are shewed other-where, and are here taken as known. and crosse it at right angles with a Verticall line.
2. Get the plains re / in-clination, and consequently the distance of the plains Pole from the Zenith of the place: which is here called the Verticall distance.
3. Get the plains declination, and alwayes account how much it is from the North. For that is here called the angle of Declination.
SECT. 1. By the Scale of Versed Sines, how to finde the elevation of the Pole above the plain: and which Pole it is, whether North or South, that is elevated.
First, finde the summe and difference of
The complement of your Latitude,
The plains Verticall distance.
[figure]
Then take halfe the length of your Versed Scale; and with that Radius (upon the center C) describe the Semicircle ADB. Afterwards, upon the same Scale, count the former summe and difference, and take the length betwixt them, and set it from A [Page 66] to D, and draw BD. Also count (upon the same Scale) the angle of the plains declination, and set that length from B to E, and take the least distance from E to BD. This least distance being rightly applyed to the Scale, namely, from the fore-named difference forward upon the Scale, will give the distance of the plains Pole from the North Pole, which is to be made use of hereafter, and is called, The Polar distance.
[figure]
And observe likewise, that
If the point of your Compasses (applyed to the Scale) doe fall just upon 90, then is your plain a Meridionall or Polar plain, and hath no pole elevated above it.
If it fall short of 90 then is the North Pole elevated; and the elevation is so much as the point fals short of 90.
If it fall beyond 90, so much as it falls beyond, so much is the South Pole elevated.
SECT. 2. To finde the plains difference of Longitude from the South part of your Meridian, and which way the said difference of Longitude is to be taken.
First, finde the sum and difference of
The complement of your Latitude,
The fore-mentioned polar distance.
Then make AB equall to your whole line of Versed Sines. And upon your Scale count your difference now found, and the fore-mentioned verticall distance, taking the distance of these two as they are numbered upon the Scale. With which length upon A, describe the arch CD. Take also upon the Scale, from the verticall distance to the fore-mentioned sum, and with that length upon B, describe the arke EF. Then draw the line FC, so as to touch both these arks, cutting the line AB in G: so shall AG (being measured upon the Scale) give the plains difference of Longitude from the South, which is here required.
¶This difference of Longitude is to be taken to the same Coast in the heavens unto which the plain declineth, and may afterwards, in the description of the Diall, be easily accounted either from the South or North part of the Meridian, viz. so as that the said difference may be alwayes lesso than 90 gr.
SECT. 3. To finde how much the Substilar (or plains proper Meridian) must lie from the Verticall line of the plain, and which way.
Frist, Finde the summe and difference of
The Polar distance,
The Verticall distance.
[figure]
Then make AB equall to your whole Versed Scale. And on the same Scale, take the extent from the complement of your Latitude to the difference now before found, with which length, upon A as the center, describe the arke CD. Also upon the Scale, take from the complement of your Latitude [Page 69] to the summe here before found, and with that length, upon the center B, describe the arke EF. then draw the line FC, justly touching both these arks, and cutting the line AB in G, so shall AG (being applyed to the Scale) give the quantity of the angle here required. According to this angle the substylar line must alwayes stand off from the verticall line of the plain.
Which way must the Substilar lie from the Verticall line.
If the plain hath the North Pole elevated upon it, then must the substilar alwayes lie from the upper end of the Verticall line towards the North Pole, so much as the angle was (in the last Section) found to be.
If the South Pole be elevated, then the substilar lyeth alwayes from the lower end of the verticall line towards the same South Pole, according to the forenamed angle.
If the plain be Meridionall (upon which neither of the Poles is elevated) then the substilar must doe either, or both; these two: according to the angle before found.
According to these Rules you may place the substilar line upon the plain in its true position requisite.
FIrst, consider by the first of these Sections, whether it is the North or South pole that is elevated upon your plain. If it be the North pole, then must the center of your Diall stand downward, and the style must point upward to the said North pole. But if the South pole be elevated, then the center of the Diall is to be set upward, and the style comming from thence must point downwards into the South Pole.
This being remembred, you may upon paper draw the right line ACB, and upon C as the center, with the extent of the line ⊙ taken from your Scale, describe a circle. If then the
and the poles elevation (taken from the Scale ☉) must be set betwixt A and C, B and C, namely, at the mark ☉. Then (counting halfe your Versed Scale as a line of 90 Chords, beginning at 90, and ending at 00, or 180) from that end of the diameter A or B which is the center of your Diall, set off your difference of Longitude (taken out of the Versed Scale as it was now taken for Chords) that way to which the plain declineth, set it off at M, this point M is the point of 12. Then from M, divide your circle into 12 equall parts, at a, b, c, d, e, f, g, h, i, k, l, and from each of those points through the point ⊙ make touches of right lines, cutting again the same circle on the opposite part into 12 unequall parts.
[figure]
[Page 72] Lastly, From the center of the Diall A or B, and through the said unequall parts, draw right lines. These last lines shall give you 12 of your houres required: And if you draw each of them quite through the center, you shall have the whole number of 24, of which, you may take such as are sutable and necessary for your plain.
When your paper Diall is thus finished, you may transfer it to your plain, by laying the substilar upon (or parallel to) the substilar before placed upon the plain, and so insert all the houres from the paper to the plain.
After all this, you may make the style to the angle of the Poles elevation, and fit it in according to its requisite place and position.
¶Note, that because some of the houre points found in the Circle will happen so neere to the center of the Diall that you cannot well draw the houre-lines true; you may therefore help your selfe by that direction which I have given in my Geometricall way.
[‘This Geometricall way shall shortly be published by the Authors own copie, with his own Demonstrations of the whole work.’]
For drawing houres upon plains that have small elevations, and upon Polar plains, use the former directions.
CHAP. XXII. A third way for all re-in-clining Dials.
SECT. 1. To finde a re-in-clining plains difference of Longitude from the South part of your Meridian: and how much the plains Meridian or (substyle) must lie from the Verticall line of the plain.
I.
II.
III.
Complement of your Latit.
38 30 K
38 30 K
38 30 K
Plains verticall distance.
100 00 Z
60 00 Z
30 00 Z
Their Summe.
138 30
98 30
68 30
Their Difference.
61 30
21 30
8 30
Their halfe Summe.
69. 15 R compl. 20 45 V
49. 15 R comp. 40. 45 V
34. 15 R compl. 55. 45 V
Their halfe difference.
30. 45 S. compl. 59. 15 X
10. 45 S compl. 79. 15 X
4. 15 S compl. 85. 45. X
Plains declinat. from Sou.
50. 00 Y
140 00 Y
160 00 Y
Describe a Circle with your common (or lesser) Scale of Chords.
[Page 74] And out of the same Scale make A Y = plains declination from South.
Out of the line ⊙ make A R = R, and A S = S, & draw Y R M and M S B and make A D = A B.
Out of the same line ⊙ make A V = V, and A X = X, and draw Y V N, and N X C.
¶Then if K be lesse than Z
C A D is the differ. of Longitude required.
and
C B is the angle between the substile & the verticall line.
¶But if K be greater than Z
C A D is the forementioned angle.
and
C B is the difference of Longitude.
These two arks C D and C B, must be measured from 90 in the line of Versed Sines, and looke what number of degrees they there cut, the same must be accounted for their quantities.
SECT. 2. To finde the elevation of the Pole above the plain: and which of the Poles it is, whether North or South, that is elevated.
MEasure A B upon the Versed Sines (from 90) as before: the complement of that is E B. Measure also E C upon the same Scale, in the same manner. Count these quantities E B and E C (so found) upon the line ⊙, and set them from E, to F and G, and make Er = R (taking E r out of the Scale of Versed Sines from 90) Draw r F O, and O G P. Measure E P upon your Scale of Chords, it will there give you the polar distance.
If E P fall to be 90, it is a Meridionall plain, and hath no Pole elevated.
If it be lesse than 90, the complement of it is the elevation of the North Pole.
If it be greater than 90. the excesse is the elevation of the South Pole.
¶Note, that the three figures following have relation to the three Columns of the foregoing Table; and to these rules last delivered.
SECT. 3. Which way must the Substilar lie from the Verticall line?
THe Rules are the same with those before in the second way of Dialling, where the same Question is propounded. You may therefore have recourse to them. Or thus.
Upon all plains whereon the
North pole
South pole
is elevated; the substilar must lie from the
upper end
lower end
of the Verticall line towards the full
North.
South.
For drawing the houres, and finishing the Diall, you must doe as is prescribed in the 4th. Sect. of the former second way. For, having placed the Substilar, and knowing the plains difference of Longitude, you are to use the same course here that was there given.
It will be easie to doe these things in plains that are upright, and have no re-in-clination.
All directions here given suppose you to be in the Northern Hemisphere of the world. If therefore you should be in the Southern Hemisphere, you may easily make these precepts serve there too, by only altering the name of North, Northen, &c. and South Southern, &c. one into the other.