Mathematicall recreations. Or, A collection of many problemes, extracted out of the ancient and modern philosophers as secrets and experiments in arithmetick, geometry, cosmographie, horologiographie, astronomie, navigation, musick, opticks, architecture, statick, mechanicks, chemistry, water-works, fire-works, &c. Not vulgarly manifest till now. Written first in Greeke and Latin, lately compi'ld in French, by Henry Van Etten, and now in English, with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring: and the double horizontall diall. Invented and written by William Oughtred.

The description, and use of the double Horizontall Diall.

THere are upon the Plate two severall Dyals. That which is outermost, is an ordinary diall, divided in∣to houres and quarters, and every quarter into three parts which are five minutes a piece: so that the whole houre is understood to contein 60 minutes. And for this dyall the shadow of the upper oblique, or slanting edge of the style, or cocke, doth serve.

The other diall, which is within, is the projection of the upper Hemisphaere, upon the plain of the Horizon: the Ho∣rizon it self is understood to be the innermost circle of the limbe: and is divided on both sides from the points of East and West into degrees, noted with 10.20.30, &c. As far as need requireth: And the center of the In∣strument is the Zenith, or Verticall point.

Within the Horizon the middle straight line pointing North and South upon which the style standeth, is the Meridian or twelve a clock line: and the other short arching lines on both sides of it, are the houre lines, di∣stinguished accordingly by their figures: and are divi∣ded into quarters by the smaller lines drawn between them: every quarter conteining 15 minutes.

The two arches which crosse the houre lines, meeting on both sides in the points of intersection of the sixe a clocke lines with the Horizon, are the two semicircles of the Ecliptick or annuall circle of the sun: the upper of which arches serveth for the Summer halfe yeere; and the lower for the Winter half yeer: and therefore divi∣ded into 365 dayes: which are also distinguished into twelve moneths with longer lines, having their names set down: and into tenths and fifts with shorter lines:

and the rest of the dayes with pricks as may plainly be seene in the diall.

And this is for the ready finding out of the place of the Sun every day: and also for the shewing of the Suns yeerely motion, because by this motion the Sun goeth round about the heavens in the compasse of a yeer, ma∣king the four parts, or seasons thereof▪ namely, the Spring in that quarter of the Ecliptick which begins at the intersecti∣on on the East side of the diall▪ and is therefore called the Vernall intersection. Then the Summer in that quarter of the Ecliptick which begin at the intersection with the Meridian in the highest point next the Zenith. After that, Autumne in that quarter of the Ecliptick which beginneth at the intersection on the West side of the di∣all, and is therefore called the Au••umna••l intersection and lastly, the Winter in that quarter of the Ecliptic••, which beginneth at the intersection, with the Meridian i•• the lowest point next the Horizon.

But desides this yeerely motte••, the Sun hath a diurnall, or daily motion, whereby it maketh day and night, with all the diversities and inaequalities thereof: which is ex∣pressed by those other circles drawn crosse the houre lines; the middlemost whereof, being grosser then the rest, meeting with the Ec••iptick in the points of the Vernall, and Autumnall intersections▪ is the Equinoctiall: and the rest on both sides of it are called the parallels, or diurnall arch of the Sun, the two outermost whereof are the Tropicks, because in them the sun hath his furthe••t digres∣sion or Declination from the Aequinoctiall, which is de∣grees 23 1/••▪ and thence beginneth againe to return to∣wards the Equinoctiall. The upper of the two Tropicks in this nor Northerne Hemisphere is the Trop••ck of Can∣cer, and the sun being in it, is highest into the North, ma∣king

the longest day of Summer: And the lower next the Horizon is the Tropick of Capricorne; and the sun be∣ing in it, is lowest into the South, making the shortest day of winter.

Between the two Tropicks and the Aequinoctiall, infi∣nite such parallel circles are understood to be conteined: for the sun, in what point soever of the Ecliptick it is car∣ried▪ describeth by his Lation a circle parallel to the Aequi∣noctiall: yet those parallels which are in the instrument, though drawn but to every second degree of Declina∣tion, may be sufficient to direct the eye in imagining and tracing out through every day of the whole yeere in the Ecliptick, a proper circle which may be the diur∣nall arch of the sun for that day. For upon the right estimation of that imaginary parallel doth the manifold use of this instument especially rely: because the true place of the sun all that day is in some part or point of that circle. Wherefore for the bet••er con∣ceiving and bearing in minde thereof, every fift paral∣lel is herein made a little g••osser then the rest.

For this inner diall serveth the shadow of the up∣right edge of the style; which I therefore call the upright shadow.

And thus by the eye and view onely to behold and com∣prehend the course of the sun▪ throughout the whole yeere both for his annuall and diurnall motion, may be the first use of this instrument.

II Use. To finde the declination of the sun every day.

Looke the day of the moneth proposed in the Ecliptick, and mark how many degrees the prick shewing that day, is distant from the Equinoctiall, either on the Summer or Winter side, viz. North or South.

Example 1. What will the Declination of the sun be upon the eleven••h day of August? look the eleventh day of August and you shall finde it in the sixth circle above the ••quinoctiall: Now because each parallel standeth (as hath been said before) for two degrees, the sun shall that day decline Northwards 12. degrees.

Example 2. What declination hath the sun upon the 24 day of March? look the 24 day of March, and you shall finde it betweene the second and third northern parallels, as it were an half and one fift part of that di••tance from the second: Reckon therefore four de∣grees for the two circles, and one de••ree for the halfe space: So shall the Suns declination be five degrees, and about one fift part of a degree Northward that same day.

Example 3. What declination hath the sun upon the 13 day of November? look the 13 day of November, and you shall finde it below the Equinoctiall ten parallels, and about one quarter which is 20 degrees and an halfe southward. So much is the declination. And according to these examples judge of all the rest.

III. Use. To finde the diurnall arch, or circle of the suns course every day.

The sun every day by his motion (as hath been said) describeth a circle parallel to the ••quinoctiall, which is either one of the circles in the diall, or some-where ••e∣tween two of them. First, theref••re se••k the day of the moneth; and if it fall upon one of those parallels; that is the circle of the suns course that same day: But if it fall be∣tweene any two of the parallels, imagine in your mind••, and estimate with your eye, another parallel th••ough that point betweene those two parallels keeping still the same distance from each of them.

As in the first of the three former examples, The circle of the Suns course upon 11 of August▪ shal be the very sixt circle above the Equinoctiall toward the cente••.

In ••xample 2. The circle of the suns cou••se upon the 24 of March shall be an imaginary circle between the se∣cond and third parallels still keeping an half of that space, and one fifth part more of the rest, from the second.

In example 3. The circle of the suns course upon the 13 of November: shall be an imaginary circle between the tenth and eleventh parallels below the Equinoctiall, still keeping one quarter of that space from the tenth.

IIII Use. To finde the r••sing and setting of the sun eve••yday.

〈…〉〈…〉 (as was last shewed) the imaginary circle or parallel of the suns course for that day, and marke the point where it meeteth with the horizon, both on the East and W••st sides, for that is the very point of the suns r••sing, and setting that same day, and the houre lines which are on both sides of it, by proportioning the di∣stance reasonably, according to 15 minutes for the quarter of the houre, will shew the houre of the suns rising on the East side, and the suns setting on the West side.

V Use. To know the reason and manner of the In∣creasing and decreasing of the dayes and nights ••hroughout the whole yeere.

When the Sun is in the Equinoctiall, it riseth and set∣teth at 6 a clock, for in the instrument the intersection of the Equinoctiall, and the Ecliptick with the Horizon is in the six a clocke circle on both sides. But if the sun be out of the Equinoctial, declining toward the North, the in∣tersections of the parallel of the sun with the Horizon is

before 6 in the morning, and after 6 in the evening: and the Diurnall arch greater then 12 houres; and so much more great, the greater the Northerne Declination is. Againe, if the sun be declining toward the South, the in∣tersections of the parallel of the sun, with the Horizon is after 6 in the morning, and before 6 in the evening: and the Diurnall arch lesser then 12 houres; and by so much lesser, the greater the Southerne Declination is.

And in those places of the Ecliptick in which the sun most speedily changeth his declination, the length also of the day is most a••tered: and where the Ecliptick goeth most parallel to the Equinoctiall changing the declina∣tion, but little altered. As for example, when the sun is neer unto the Equinoctiall on both sides, the dayes in∣crease and also decrease suddenly and apace; because in those places the Ecliptick inclineth to the Equinoctiall in a manner like a streight line, making sensible declination. Again, when the sun is neere his greatest declination, as in the height of Summer, and the depth of Winter, the dayes keep for a good time, as it were, at one stay, because in these places the Ecliptick is in a manner parallel to the Equinoctiall, the length o•• the day also is but little, scarce altering the declination: And because in those two times of the yeer, the sun standeth as it were still at one declinati∣on, they are called the summer solstice, and winter solstice. And in the mean space the neerer every place is to the Equinoctiall, the greater is the diversity of dayes.

Wherefore, we may hereby plainly see that the com∣mon received opinion, that in every moneth the dayes doe equally increase, is erroneous.

Also we may see that in parallels equally distant from the Equinoctiall, the day on the one side is equall to the night on the other side.

VI. Vse. To finde how far the sun riseth, and setteth from the true east and west points, which is called the suns Amp••itude ortive, and occasive.

Seek out (as was shewed in III Vse) the imagina∣ry circle, or parallel of the suns course, and the points of that circle in the horizon, on the East and West sides cutteth the degree of the Amplitude ortive, and occasive.

VII Use. To finde the length of every day and night.

Double the houre of the sunnes setting, and you shal have the length of the day; & double the hour of the sunnes rising, and you shal have the length of the right.

VIII Vse. To finde the true place of the sun upon the dyall, that is, the point of the instrument which answereth to the place of the sun in the heavens at any time, which is the very ground of all the questions following.

If the dyall be fixed upon a post: Look what a clock it is by the outward dyall, that is, look what houre and part of houre the shadow of the slanting edge of the style sheweth in the outward limbe. Then behold the shadow of the upright edge, and marke what point thereof is upon that very houre and part in the inner dyall among the parallels, that point is the true place of the Sunne at the same instant.

If the dyal be not fixed, and you have a Meridian line no••ed in any window where the Sunne shineth: place the Meridian of your dyal upon the Meridian line gi∣ven, so that the top of the stile may point into the north: and so the dyal is as it were fixed, wherefore by the former rule you may finde the place of the Sunne upon it.

If the dyal be not fixed, neither you have a Meridian line, but you know the true houre of the day exactly: hold the dyal even and parallel to the Horizon, moving

it till the slanting edge of the stile cast his shadow justly upon the time or houre given; for then the dyal is truly placed, as upon a post. Seek therefore what point of the upright shadow falleth upon that very houre, and there is the place of the Sun.

But if your dyal be loose, and you know neither the Meridian nor the time of the day. First, by the day of the moneth in the Ecliptique, finde the su••s parallel, or d••urnall arch for that day▪ then holding the dyal level to the horizon, move it every way untill the slanting sha∣dow of the style in the outward limbe, and the upright shadow in the Sunnes diurnal arch, both shew the very same houre and minute, for that very point of the Sunnes parallel, which the upright shadow cutteth, is the true place of the Sun on the dyal at that present.

But note that by reason of the thicknes of the style, and the bluntnesse of the angle of the upright edge, the Sun cannot come unto that edge for some space before and after noone. And so during the time that the Sunne shineth not on that upright edge, the place of the Sunne in the dyal cannot be found. Wherefore they that make this kinde of double dyal, are to be careful to file the up∣right edge of the style as thinne and sharpe as possible may be.

That which hath here bin taught concerning the finding out the Suns true place in the dyal, ought perfect∣ly to be understood, that it may be readily, and dexterious∣ly practised, for upon the true performance thereof de∣pendeth all that followeth.

IX Vse. To finde the houre of the day.

If the dyal be fastned upon a post, the houre by the outward dyal, or limbe, is known of every one, and the

upri••ht shadow in the Suns parallel, or diurnal arch will also shew the very same houre.

But if the dyall be loose, either hold it or set it paral∣lel to the Horizon, with the style pointing into the north and move it gently every way untill the houre shewed in both dialls exactly agreeth, or which is all one, finde out the true place of the Sun upon the dyall, as was taught in the former question, for that point among the houre lines sheweth the houre of the day.

X Vse. To finde out the Meridian, and other points of the Compasse.

First, you must seek the tru•• houre of the day (by the last question) for in that situation the Meridian of the dyall standeth direct••y north and south: and the east pointeth into the east, and the west into the west, and the rest of the points may be given by allowing degrees 11. 1/•• unto every point of the compasse.

XI Vse. To finde out the Azumith of the sun, that is, the distance of the Verticall circle, in which the sun is at that present, from the Meridian.

Set your diall upon any plain or flat which is paral∣lel to the horizon, with the Meridian pointing directly north or south, as was last shewed: then follow with your eye the upright shadow in a streight line, till it cutteth the horizon: for the degree in which the point of intersection is, shal shew how far the suns Azumith is distant from the east and west points, and the comple∣ment thereof unto 90; shal give the distance thereof from the meridian.

XII Vse. To finde out the Declination of any Wall upon which the sun shineth, that is, how far that wall swer∣veth from the north or south, either eastward or west∣ward.

Take aboard having one streight edg▪ & a line strick∣en perpendicular upon it; apply the streight edg unto the wall at what time the sun shineth upon it, holding the board parallel to the horizon: Set the dyal thereon, and move it gently every way, untill the same hour and mi∣nute be shewed in both dyals: and so let it stand: then if the dyal have one of the sides parallel to the Meridian strike a line along that side upon the board, crossing the perpendicular, or else with a bodkin make a point upon the board, at each end of the meridian, and taking away the instrument from the board, and the board from the wall, lay a ruler to those two points, and draw a line crossing the perpendicular: for the angle which that line maketh with the perpendicular, is the angle of the decli••nation of the wall. And if it be a right angle, the wall is exactly east or west: but if that line be parallel to the perpendicular, the wall is direct north or south without any declination at all.

You may also finde out the declination of a wall, if the dial be fixed on a post not very far from that wall; in this manner. Your board being applyed to the wall, as was shewed, hang up a thred with a plummet, so that the shadow of the thred may upon the board crosse the perpendicular line: make two pricks in the shadow and run instantly to the dyal and look the horizontal distance of the suns Azumith, or upright shadow from the meri∣dian. Then through the two pricks draw a line crossing the perpendicular: and upon the point of the intersection, make a circle equal to the horizon of your Instrument, in which Circle you shal from the line through the two pricks measure the Horizontal distance of the upright shadow, or Azumith from the me∣ridian, that way toward which the Meridian is: draw

a line out of the center, to the end of that arch measured: and the angle which this last line maketh with the per∣pendicular, shall be equall to the declination of the wall.

XIII Vse. How to place the dyall upon a post without any other direction but it selfe.

Set the diall upon the post, with the stile into the North, as neere as you can guesse: then move it this way and that way, till the same houre and minute be shewed, both in the outward and inward dials by the severall sha∣dowes, as hath been already taught, for then the diall standeth in its truest situation; wherefore let it be nailed down in that very place.

XIIII Vse. To finde the height of the sun at high noon everyday.

Seeke out the diurnall Arch or parallel of the suns course for that day, (by Vse III.) and with a paire of Compasses, setting one foot in the center, and the other in the point of intersection of that parallel with the Meridi∣an, apply that same distance unto the Semidiameter di∣vided: for that measure shal therein shew the degree of of the Suns altitude above the the Horizon that day at high noon.

XV Vse. To finde the height of the sun at any houre or time of the day.

Seeke out the diurnal Arch, or parallel of the suns course for that day: and marke what point of it is in the very houre and minute proposed. And with a paire of Compasses, setting one foot in the Center, and the other in that point of the parallel, apply the same distance upon the Semidiameter divided: for that measure shall shew the degree of the suns altitude above the Horizon at that time.

And by this meanes you may finde the height of the Sun above the Horizon at every houre throughout the whole yeere, for the making of rings and cylinders and other instruments which are used to shew the houre of the day.

XVI Vse. The height of the sun being given, to finde out the houre, or what it is a clocke.

This is the converse of the former: Seeke therefore in the Semidiameter divided, the height of the sun given. And with a paire of Compasses, setting one foot in the center, and the other at that height, apply the same di∣stance unto the diurnall arch, or parallel of the Sun for that day: for that point of the diurnall arch, upon which that same distance lights, is the true place of the sun upon the dial; and sheweth among the houre lines, the true time of the day.

XVII Use. Considerations for the use of the instrument in the night.

In such questions as concerne the night▪ or the time before sun rising, and after sun setting, the instrument re∣presenteth the lower Hemisphaere wherein the Southerne pole is elevated. And therefore the parallels which are above the Aequinoctiall toward the center shall be for the Southerne, or winter parallels: and those beneath the Aequinoctiall, for the Northerne or Summer pa∣ral••els; and the East shall be accounted for West, and the West for East; altogether contrary to that which was before, when the Instrument represented the upper Hemisphaere.

XVIII Use. To finde how many degrees the sun is un∣der the Horizon at any time of the night.

Seeke the Declination of the sun for the day propo∣sed

(by Vse II.) And at the same declination the con∣trary side imagine a parallel for the sun that night▪ and mark what point of it is in the very houre and mi∣nute proposed: And with a pair of compasses, setting one foot in the center, and the other in that point of the pa∣rallel, apply that same distance unto the semidiameter di∣vided: for that measure shall shew the degree of the suns depression below the Horizon at that time.

XIX Use. To finde out the length of the C••epusculum, or twylight, every day.

Seek the declination of the sun for the day proposed (by Vse II.) And at the same declination on the contrary side imagine a parallel for the sun that night. And with a paire of compasses setting one foot in the center, and the other at 72 degrees upon the semidiameter divided, apply that same distance, unto the suns nocturnall parallel: for that point of the parallel, upon which that same di∣stance shall light, sheweth among the houre lines, the be∣ginning of the twilight in the morning, or the end of the twilight in the evening.

XX Use. If the day of the moneth be not known, to finde it out by the dyall.

For the working of this question, either the diall must be fixed rightly on a post, or else you must have a true Meridian line drawn in some window where the sun shineth, wherefore supposing the diall to be justly set ei∣ther upon the post, or upon the Meridian. Look what a clock it is by the outward diall, and observe what point of the upright shadow falleth upon the very same minute in the inner diall, and through that same point imagine a parallel circle for the suns course; that imagi∣nary circle in the Ecliptick shall cut the day of the moneth.

I The description of it.

THis Instrument serveth as a Diall to finde the houre of the day, not in one place onely (as the most part of Dials do) but generally in all Countreys lying North of the Aequinoctiall: and therefore I call it the generall H••rologicall ••ing.

It consisteth of two br••zen circles: a Diameter, and a little Ring to hang it by.

The two circles are so made, that though they are to be set at right angles, when you use the Instrument: yet for more convenient carrying, they may be one folded in∣to the other.

The lesser of the two circles is for the Aequinoctiall, having in the midst of the inner side or thicknesse, a line round it, which is the true Aequinoctiall circle, divided into twice twelue hours, from the two opposite points in which it is fastened within the greater.

The greater and outer of the two circles is the Meridi∣an: One quarter whereof, beginning at one of the points in which the Aequin••cti••ll is hung, is divided into nine∣ty degrees.

The Diameter is fastened to the Meridian in two op∣posite points or poles, o••e of them being the very end of the Quadrant, and is the North Pole. Wherefore it is perpendicular to the ••quinoctiall, having his due position. The diameter is broad, and slit in the middle: and about the slit on both sides are the moneths and dayes of the yeer. And within this slit is a litt••e sliding plate pierced through with a small hole: which hole in the motion of it, while it is applied to the dayes of the yeer, representeth the Axis of the world.

The little Ring whereby the Instrument hangeth, is made to slip up and down along the Quadrant: that so by help of a little tooth annexed, the Instrument may be rectified to any elevation of the Pole.

II. The use of it.

IN using this Instrument, First, the tooth of the little Ring must carefully be set to the height of the Pole in the Quadrant, for the place wherein you are.

Secondly, the hole of the sliding plate within the slit, must be brought exactly unto the day of the moneth.

Thirdly, the Aeqinoctiall is to be drawn out, and by means of the two studs in the Meridian staying it, it is to be set perpendicular thereto.

Fourthly, Guesse as neer as you can at the houre, and turn the hole of the little plate toward it.

Lastly, Hold the Instrument up by the little Ring, that it may hang freely with the North Pole thereof toward the North: and move it gently this way and that way, till the beams of the Sun-shining thorow that hole, fall upon that middle line within the Aequinoctiall: for there shall be the houre of the day: And the Meridan of the Instrument shall hang directly North and South.

These Instrument all Dials are made in brasse by Eli∣as Allen dwelling over against St. Clements Church without Temple Barre, at the signe of the Horse∣shooe neere Essex Gate.

FINIS