768

TRIGONOMETRY,

And tlie angle KNO = WQ Hie hour from midnight. Wliat wehave said with respect to the sun in cancer, is the same for itsplace in any other part of the ecliptic ; and die same triangles areto be drawn for the same purposes in the winter-signs, as we havehere done for the summer. And further, it is to be observed, thatsince any star may be supposed at ©, it is evident all the sametriangle-, serve for a star as for the sun. In the six triangles alreadyenumerated, no less than seventy-two problems may be stated inreference to the sun and stars; besides many other triangles that•might be added to these, to increase the number of aslrononiic.nlproblems. Ill the oblique triangle ©ZN , the side ©Z is the co-aititude of the sun; ©N, the co-declination; ZN , the co-lati-tude of the place. The angle ZN© is the hour from noon, andthe angle ©ZN or ©ZiE is the azimuth from the north or southpart of the horizon. This triangle, therefore, affords ten pro-blems more in respect of the sun, and if we consider the obliquetriangle ZHN, we shall find the sides and angles the same in re-gard to a star at R, and consequently oilier ten problems willthence be produced. Again, In the ' " _ue triangle RNP, theside RN is the star’s co-declination ; the side R 1’, its co-latitude;and the side NP = rEE the obliquity of liie Ecliptic. Also theangle NPR is the star’s longitude from the point E ; and the angleHNP its azimuth. From hence we have ten problems more; andtherefore this and the last triangle afford twenty principal problemsabout the stars only. In the Iasi place, let © be the moon ora planet, then if we draw a great circle through © and R, the placeof a star; we shall have an oblique triangle ©ZR, in which theside ©Z is the moon’s co-altitude ; ZR, that of the star, andQR the distance of the moon from the star. Also the angle ©ZII■the difference of the azimuths of the moon and star. This trian-gle affords four very useful problems relative lo the moon ands ars. Upon the whole, in these few triangles in this one projec-tion only, more than one hundred astronomical problems are con-tained. In regard to geography, if we suppose Z and R lo repre-sent two places on the surface of the terrestrial globe ; then in thetriangle ZRN, the side ZN is the co-latitude of the place Z ; theside RN in the co-latitude of the place R ; and the side. ZR is the.distance of the two places on the surface of the globe. The angleZN R = /EV, is the dilference of longitude of the two places, andthe angle RZN is the position of the place Rfrom Z. This one tri-angle therefore furnishes ten principal geographical problems,which depend on the doctrine of the sphere. In navigation,particularly that part of it called orlhodromics, which treats of theart of sailing on a great circle of the sphere, a spherical triangleis concerned. Thus let Z he the port from whence a ship is tosail to another port X.; then in the triangle ZN X ; the Side ZN Is the co-latitude of the port Z ; the side NX is the co-latitudeof the port X ; the side ZX is the nearest distance between themnr the ship’s way. Also the angle N ZX is the course lo be steeredat Z ; and the angle ZNX is the difference of longitude betweenthe ports. But in sailing from Z lo X, the course is perpetuallyaltering, or the angle made by tire ship’s way on the great circleZX with every meridian is different, or the latitudes alwayschanging, therefore it is necessary to calculate this for every fiveor six degrees of longitude, that the ship may be kept upon, ornear the circular arch ZX, every where ; and this is the whole artof great circle-sailing, which is entirely conversant in the solutionof spherical triangles. A great variety, therefore, of nauticalproblems here offer themselves from this one view of a sphericaltriangle. See Navigation. In dialing, the triangle NOK,right-angled at O, is used in finding all the angles NKO, whichthe several hour-circles NKS make with the horizon. For NObeing the latitude of the place, and flu: angle ONK being given,the arch of the horizon OK is easily determined. See Diaiing.

The table of Briggs’s logarithms for every number from 1 fo100, and for every prime number from 100 to 200, calculated toupwards of 60 places of decimals, is inserted in the two followingpages for the gratification of those curious readers who would hepleased to find the logarithms calculated to so many places: asvery many important problems in the mathematics may he per-formed by their means. And the logarithm of almost any othernumber not contained in the table may be found, (provided it benot a prime,) by the addition, subtraction, multiplication, or di-vision, of two or more numbers ; as is sufficiently manifest from (henature of logarithms. See Logarithms . Thus the logarithm of

343 is easily found by multiplying the logarithm of 7, the cuberoot of .343, by 3, the product 2.53529412004277049213664877-577790S58045071718897189(12196109060 being the logarithmrequired. But when the logarithm of a prime number above 200is required, we must make use of one of the two following rulesof Dr. Halley:

I. Let the number, whose logarithm is sought, and either ofthe other two numbers next it, greater or less by an unit, ((helogarithms of one or both of which are given) be made an impro-per fraction; to the denominator doubled, add 1.0, that shall bethe first divisor, and the square of that must divide the first quo-tient, and all the rest: the dividend must always be the reciprocalof the natural logarithm of 10 = R6S5S.8963S.06503.65530.2257-8.37833.21010.45887.94011.60733.31322.28907.5663, &c. All therest must he as is directed for making natural logarithms; onlythe sum of the series, or logarithm of the fraction, when the givennumber is less than that sought, must be added to its logarithm ;when greater, subtracted from it. Ex. Gr.

The series to make Briggs’s log. of 251, 250, the next less,to make the fraction m, whose logarithm is first sought 2 x 250-f- 1 = 501 is the first divisor, the square thereof = 251001 dividesall the rest, as in the following operation.

501 .S6858.89638.06503.65530.225784

(.00173.37105.06599.80769.521409251001)1733&c. (69.07185.65503.64647825 1001) 690 718 5 & c. ( 27.518 5 5.831266251001) 2751855831266( 10.963525

251001) 10963525( 44

The odd powers divided by 1, 3, 5, &c.

.00173.37105.06599.S0769.5214093) 23.02395.21834.548826

5) 5.50371.166253

7) 1.566218

9) 5

= 00173.37128.09000,52976.802711250 = 2.39794.00086.72037.60957.25222!

__25l_ = 2.3 9967 .37214.81038.13934.054932

II. 1 he second method may be this: if two numbers, nextthat whose log. is sought (either one greater, and the other less;or both greater, or both less) have known logarithms, the squareof the middle number shall be the numerator of the improperfraction ; the product of the other two, on each side, the deno-minator. For example, Take 239, its square 57121, and the pro-duct of 238 and 240 = 57120; whence the fraction and

their sum 114241, the first divisor, and its square 13051006081divisor for all the rest, as in the following series.

114241)86858.89638'.06503.65530.225784( 00000.76031.28157.19841.0862151305100608 1)7603128 &e. (.58 257.027159The Odd Powers divided by 1, 3.

.00000.76031.28157.19841.0862153) 19419.009053

Tne sum is Bi iggs's fLog, of = (

0.00000.76031.2S157.39260.095268

If the log. of the middle number be sought, the log. of thefraction, added to tile log. of the greatest and least, will be thelog. of the square of the middle number; the half of which is itslogarithm, as in the following work;

24x10 = 240 1 = 240 =2.38021.12417.11606.02293.624459

34X 7 = 238 2= 238 =2.37657.69570T65113J5446.612505

240 x 238=57120 3 =57120=4,75678.81987.681 1 7T97740~.23(>9()4[*i=0.00000.76031.28157.39260.095268239 X 239=57121 5=57121 =4.75679.58018 96275.37000.332232Athelog.of 57121 = log.of 339=2.37839.79009.48137.62500.166116

If (he log of the greatest or least number be sought, subtractthe log. of life traction Irom the log. of the square of the middlenumber, the remainder shall be the log. of the product of l!' eother two; from which subtract the log. of the known number,the remainder will be the log. of the other.

Suppose the log. of 239, 240 given, and 23S sought, then 5—-4 = 3—1 = 2; if the log. of 239, 238 given, and 240 sought)then 5 — 4 = 3 — 2 = 1.

Mr.