THE Refraction of a Star is equal to the Difference between the obferved Altitude, and the true Altitude, which is known by Calculation, and thus Refractions are eafily known. Then to our purpose:

IF it were to be folved Geometrically, it would be brought to this Problem :

LET the Star be in S (Fig. 29.) fending out the Ray Sf, and the Refraction nfL.

AND in the Altitude fg it's Refraction m r L. THEREFORE in the Circle drf, whose Center is T, there is given T L the Semidiameter of the Earth, and drawing Tr, Tf, Lf, Lr, the Angles Tlf and T Lr may be had; the latter being made of the Star's Altitude, and a right Angle, and the Angles n fL and mr L are given; and we know that the Proportion of the Sine of the Angle nfT to LfT is the fame as the Sine of the Angle mr T to the Sine of Lr T. From these to find the Semidiameter Tf or Tr, and the Proportion of the Sine of nfT to the Sine LfT, or to find the Angle TfL. Which will give the Proportion of the Sines.

Lem

THE Algebraic Solution is fomething difficult, but the common fynthetic way requires many mata to be premifed, which the former Solution doth not. Let us therefore produce the analytic Solution, to fhew that it will confirm the preceding Propofition. Let the Sine of the right Angle TLF, or

FIRST, because there is given the Sine of both the Angles TfL and Lfn, the Sine of the whole Angle nf T is given, viz. if the Sine of each Angle be multiplied into the Co-Sine of the other, and the Sum of their Products divided by the Radius. Thus the Sine of the Angle nfT will be az + d√ bb — aa

MOREOVER, feeing the Sine T Lf is to the Sine TfL (So is Tfto TL or Tr to TL) fo is the Sine TLr to the Sine TL; that Sine

ca

TrL will be. And seeing there is given also

the Sine Mr L, let there be found, according to the former Rule, the Sine of the whole mr T, kca+b√b4-cca a which is

bb

Thus we have the

Sine of four Angles LfT, nfT, LrT, mrT, for we know they are proportional fince as a: са kca

a g + d √ bb — a a

b

:: :

b

b b

abba

-ссаа

And therefore cgated √ b b — a a = k ca + b

√64-ccaa; or if

b4

сс

then, after due Reduction, it will be nad √ bb — aa = b √ min-aa. And both Sides fquared bbm m-bbaa-nnaa-ddbb + d2 a2 = 2 nad√bb-aa. For p write bbmm-ddbb, and 99 q q for d d-bb-nn, and fquare again p++qqa a = 2 nad√bb-aa, and it will be p2 + q a+ + 2 p + q qaa=4nnbbddaa—4 nnd da*. And dividing by 4nd d-qt, and fubftituting other

8

Sines

Sines arraa-s 4.
$ 4. And a a =

FROM this Equation it appears that the Problem is determined, and that a, which is the Sine of the Angle TfL, may be found by extracting the fquare Root. And from thence 'tis found, that two Refractions are fufficient to find the Altitude of the Air TF, and the Rule of Proportion between them; which I take Notice of because I fee Kepler, in his Epitome of Aftronomy p. 65. takes three Refractions, tho' he did not try this Method himself.

THE Refolution of this Problem may be alfo had by the Rule of Pofition, by affuming Tf in a certain Proportion to TL, and trying if, by that Affumption, the Sines of the four Angles TfL, Tfn, TrL, Trm will be proportional.

THEREFORE, in the Triangle ƒLT, let there be found the Angle TfL from having ƒT, TL, and TLf. And likewife in the Triangle T Lr, find the Angle Tr L from having Tr, TL, and TLr.

LET there be then taken the Sine of the Angles TfL, Tfn, TrL, Trm; and let there be a fourth Proportional taken to the Sines TfL, Tfn, TrL. And if T rm be equal to this fourth Proportional, then the affumed Height of the Air Tf will be juft; but if the Sine Trm be greater than the fourth Proportional, then Tf must be taken lefs; but if lefs, then it must be taken more; and fo always 'till they become equal,

EXAMPLE.

SUPPOSE the Virgin's Spike, or any other Star, or the Sun, to be feen in the Horizon Lf

when

when 32 Minutes under it, as in S; thus the Refraction nfL is 32.

THEN when the Sun hath the apparent Altitude gx 1 degr. 22 min. or the true Altitude 1 degr. the Refraction Lrm is 22 min.

THE Semidiameter T L is 860 German Miles. But fuppofe it 10000, and the Altitude of to be 5

meter TL; that is, abont of a Mile.

THEREFORE in the Triangle T Lf, the

Radius being 10,000,000.

AS ƒT to TL, fo is the Sine TLf to the Sine TfL.

2001: 2000 :: 10,000,000: 9,995,992, the Sine of 88 degr. 22 min. 40 fec.

AND thus Tfn will be 88 deg. 54 min. 40 fec. whofe Sine is 9,998,200.

AGAIN, in the Triangle Tr L.

AS Tr: TL, fo is the Sine of the Angle T Lr to the Sine TrL.

2001 2000 :: 9,997,155: 9,992,159, the Sine of 87 degr. 43 min. 40 fec.

THEREFORE Trm is 88 degr. 5 min. 40 fec. whofe Sine is 9,994,500.

THEN let there be found a fourth Proporional to the Sines of TfL, Tfn, TrL. AS TfL: Tfn:: Tr L.

AS 9,995,992 : 9,998,200 :: 9,992,159 : 9,994,366.

AND with that fourth Number compare the Sine of the Angle Trm, which is 9,994,500.

AND we find that this Sine is very near to that fourth Number; and therefore the affumed Altitude of the Air, viz. & of a Mile, is not far from the Truth. And if any one defire it more accurately, he may affume another Altitude, and work the fame way, 'till the Sine of Trm be

nearer

nearer to the fourth Proportional; or, by the Rule of Falfe, having it twice too little, you may find the true Altitude as near as poffible, for it cannot be found perfectly true; because a small Difference in the Sines changes it very much if it be but half a Minute: and befides this the Canon of Sines must be very exact.

WE conclude therefore, that the Height of the Air is about the 2000 part of the Semidiameter of the Earth, which is 1,633,190 Perches ; and the Altitude of the Air 816 Perches, one Perch being twelve Rhinlandish Feet: but 'tis better allowed to be half a German Mile, for the Refraction Lfn was found, by Tycho, to be greater, and may be thirty fix or forty eight Minutes; and then the Height of the Air will be one Mile.

THE Height of the Air being known, there is alfo known the Proportion of the Denfity of the Air to that of the Ethereal Matter, or the Law of Refraction, in that Air making fuch Refractions in fuch Altitudes, i. e. the Proportion of the Sine TfL to the Sine Tfn, before found, is the Proportion fought.

AS 9,995,992 to 9,998,200. And the Reason why thefe Refractions are so small is, because we fuppofed a clear Air, not much differing from the Ethereal Matter in Density; as fome have imagined.

MOREOVER, whether the Altitude of the Air be the fame in all Places and Times may be known; if we ufe the fame way two Refractions at two Altitudes in a different Air and Time. And that Students may understand thefe Secrets of Nature, I have, that they may try a Calculation, fet down Examples from Tycho's Obfervations, who obferved the Refractions of the Sun and Moon for every Degree of their Height; and because they differ from the Obfervations of Lanfberg,

made