The folution of the cases of oblique spherical tri


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Two sides ĄC The angle B (As fine BC : fine A : : fine AC : BC and an an opposite to fine B (by Cor. 1. 80 Theor. 1.) Notes gle A opposite the other This case is ambiguous when BC is to one of them

less than AC ;, since it cannot be determined from the data whether

B be acute or obtuse. Two fides AC, The included Upon AB produced (if need be) let BC and an an- angle ACB fall the perpendicular CD; then (by Igle A oppofite

Tbeor. 5.) rad. : co-fine AC :: Ito one of them

tang. A.: co-tang. ACD, but (by Cor. 2. to Tbeor. 1.) as tang. BC : tang. AC ; : co-fine ACD : co-fine BCD. Whence ACB ACD +

BCD is known. Two fides AC, The other As rad. : co-fine A: 1 tang. AC : BC and an an side AB

tang. AD (by i bor. 1.) and (by Cor. gle opposite to

10 Theor. 2.) as co-fine AC : Co-fine 3 one of them

BC: : co-fine AD: co-fine BD.
Note, This and the lạft case arc

both ambiguous when the first is fo. Two sides AC, The other As rad, : co-fin, A : : tang. AC : AB and the in fide BC

tang. AD (by Theor. 1.) whence BD 4 cluded angle A

is also known; then (by Cerol. 10 Theor. 2.) as co fine AD : co-fine

BD::C) fine 4C : co-fine BC. Two sides AC, Either of the As rad. : co-fine A : ;,tang. AC :

AB and the in- other angles, tang. AD (by Tbeor. 1.) whence BD 5 cluded angle A suppose B is known; then (by Cor. to Tbcor. 4.)

as fine BD : 'line AD : : tang. A : tang. B.

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angle B

(Two angles A,

The other As rad. ; Co-fine AC ::tang A : ACB and the

co-tang. ACD (by Tbeor. 5.) whence 6 fide AC be

BCD is also known; then (by Cor. twixt them.

to Theor. 3.) as fine ACD i fiue

BCD::co-fine A: co fine B. Two augles A, Either of the As rado: co-fue C : : tang. A :

ACB and the other fides, co-tang. ACD (by Theor. 5.) whence 7 lide AC be- suppose BC BCD is also known; then as co-fine twixt them.

of BCD: co-fine ACD : ; tan. AC

1: tang. BC (by Cor. 2. to Theor. I.) Swo angles A, Che fide BCAs fine B : fine AC : : sine A : fine Band a fide oppofite the BC (by Cor, 1. to Tbeor. 1.)

AC opposite to other one of them,

Two angles A, The fide AB: As rad. : co-line A : :tang. AC: B and a lide AC betwixt them.tang. AD (by Theor. 1.) and as tan, 9 opposite to one

B : tang. A : : sine AD : fine BD of them.

(by Cor. to Theor. 4.) whence AB is

also known Two angles A, The other

As rad. : co-fine AC :: tang. A ; B and a side Ac angle ACB co-tang. ACD (by Theor. 5.) and as iclopponte to one

co-fine A ; co-fine B: ; fine ACD of them.

1: fine BCD (by Cor. to Theor. 3.)

whence ACB is also knowil. All the thiee

An angle,
As tang: AB : tang.

AC + BC lides AB, AC fuppose A

AC-BC : ; tang

1 tang. DE, the Jdistance of the perpendicular from the middle of the base (by Cor. 10 Tbeor. 6.) whence AD is known;

then, as tang. AC : tang. AD ::

Irad. : co-fine A (hy Tbear. 1.)
All the thrce A fide, fup- As co-tari.


ABC . angles A, B pose AC and ACB

ACB | ; tang i tang. of the angle included by the perpendicular and a line bisecting the vertical angle: whence ACD is also known; then (by Theor. 5.) tang. A : co-tang. JACD:: rad. : co-fine Ac.

and BC





Note, In letting fall your perpendicular, let it always be from the end of a given side and opposite to a given angle.

D 3

Of the nature and construction of Logarithms

with their application to the doćirine of Triangles.


S the business of trigonometry is wonder, the

lo. garithms; which are a set of artificial numbers, fa proportioned among themfelves and adapted to the natural numbers 2, 3, 4, 5, &c. as to perform the same things by addition and subtraction, only, as these do by multiplication and division : I shall here, for the sake of the young beginner (for whom this finall tract is chiefly inrended) add à few pages upon this subject. But, first of ail, it will be neceffáry to premise something, in general, with regard to the indices of a geometrical progression, whereof logarithms are a particular species.

Let, therefore, i, a, a, a, a, a, a, a, &c. be a geomecrical progression whole firft term is unity, and common ratio any given quantity' a. Then it is manifest,

1. That, the sum of the indices of any two ter11s of the progression is equal to the index of the produst of those terms. . Thus 2 + 3 (5) is = the index of a'xa', or a'; and 3+ 4(37) is = the index of a' X at, or a'. This is universally demonstrated in p. 19. of my book of Algebra?

. That, the difference of the indices of any two terms of the progrenzo is equal ti the index of the quotient of one of them divided by ihe other. Thus 53 is=the index of or a. Which is only

a. the converse of the preceding article,

3: That,

3. That, the product of the index of any term by a given number (n) is equal to the index of the power phose exponent is the faid number (n). Thus 2x3 (6) is the index of a raised to the 3d power (or a). This is provęd in p. 38, and also follows from article 1,

4. That, the quotient of the index of any term of the progression by a given number (n) is equal to tbe index of the root of that term defined by tbe fame number

' (n). Thus (2) is the index of (24) the cube root of do. Which is only the converse of the last article.

These are the properties of the indices of a geometrical progression; which being universally true, let the common ratio be now supposed indefinitely near to that of equality, or the excess of a above unity, indefinitely little; so that some term, or other, of the progression 1, l, a', a', as, a', &c. may be equal to, or coincide with, each term of the series of natural numbers, 2, 3, 4, 5, 6, 7, &c. Then are the indices of those terms called low garithms of the numbers to which ebe terms them. felves are equal. Thus, if an = 2, and a" = 3, then will m and n be logarithms of the numbers 2 and 3 respectively. Hence it is evident, that what bas been above

per cified, in relation to the properties of the indices of powers, is equally true in the logarithms of numbers; since logarithms are nothing more than the indices of such powers as agree in value with those num . bers. Thus, for instance, if the logarithms of 2 and 3 be denoted by m and n; that is, if "=3, and a"=3, then will the logarithm of 6, (the product of 2 and 3) be equal to m+ (agreeable fo article 1); because 2x3 (6)=am xaaaa D4


But we must now obferve, that there are various forms or species of logarithms ; because it is evident that what has been hitherto said, in respect to the properties of indices, holds equally true in relation to any equimultiples, or like parts, of them; which have, manifestly, the same properties and proportions, with regard to each other, as the indiçes themselves. But the most simple kind of all, is Neiper's, otherwise called the hyperbolical.

The hyperbolical logarithm of any number is the index, of that term of the logarithmic progression agreeing with the proposed number, multiplied by the excess of the common ratio above unity.

Thus, if e be an indefinite small quantity, the hyperbolic logarithm of the natural number agreeing with any term ite]" of the logarithmic progression i, 5+e, 1 + el, i + el, it et, &c. will be expressed by ne.


The hyperbolic logarithm (L) of a number being gi

ven: to find the number itself, answering thereto.

Lët 1-fuel". He that term of the logarithmic progreffion', T-rel'; i+el', itel', it e1*, &c. which is equal to the required number (N). Then, because it el" is, universally, = 1 + ne + n. - tn.

e}.&c. we shall, also,

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have 1 +ne+ n.

3 &c, = N. But, because n (from the nature of logarithros) is here supposed indefinitely great, it is vident, fi it, that the numbers. connected to it by the fign, may be rejected, as far as any assigned $


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