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Note, The 10th, 11th and 12th cafes are ambiguous; fince it cannot be determined, by the data, whether AB, C, and AC; greater or less than 90 degrees each.

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3

4

5

Given

Two fides AC,
BC and an an-
gle A oppofite
to one of them

Two fides AC,
BC and an an-
gle A oppofite
to one of them

Two fides AC,
BC and an an-
gle oppofite to
one of them

Two fides AC,
AB and the in-
cluded angle A

Two fides AC,
AB and the in-
cluded angle A

Sought

angles.

The angle B
oppofite to
the other

The included
angle ACB

The other
fide AB

The other
fide BC

Either of the
other angles,
fuppofe B

Solution.

As fine BC: fine A:: fine AC: fine B(by Cor. 1. to Theor. 1.) Note, This cafe is ambiguous when BC is lefs than AC; fince it cannot be determined from the data whether B be acute or obtufe.

Upon AB produced (if need be) let fall the perpendicular CD; then (by Theor. 5.) rad. : co-fine AC :: tang. A co-tang. ACD; but (by Cor. 2. to Theor. 2.) as tang. BC: tang. AC: co-fine ACD: co-fine BCD. Whence ACBACD+ BCD is known.

As rad. co-fine A:: tang. AC: tang. AD (by Theor. 1.) and (by Cor. to Theor.2.) as co-fine AC: co-fine BC co-fine AD: co-fine. BD. Note, This and the last case are both ambiguous when the first is fo.

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

D. 3

of

Of the nature and conftruction of logarithms, with their application to the doctrine of triangles.

A

S the business of trigonometry is wonderfully facilitated by the application of logarithms; which are a fett of artificial numbers fo proportioned among themselves and adapted to the natural numbers 2, 3, 4, 5, &c. as to perform the fame things by addition and subtraction, only, as thefe do by multiplication and divifion : I fhall here, for the fake of the young beginner (for whom this small tract is chiefly intended) add a few pages upon this fubject. But, firft of all, it will be neceffary to premise something, in general, with regard to the indices of a geometrical progreffion, whereof logarithms are a particular fpe

cies.

Let, therefore, 1, a, a2, a3, aa, a3, ao, a2, &c. be a geometrical progreffion whofe firft term is unity, and common ratio any given quantity a. Then it is manifeft,

1. That, the fum of the indices of any two terms of the progreffion is equal to the index of the product of those terms. Thus 2+3 (5) is the index of a2x a3, or a3; and 3 + 4 (7) is the index This is univerfally demonftrated book of Algebra.

of a3 x a*, or a". in p. 19. of my

2. That, the difference of the indices of any two terms of the progreffion is equal to the index of the quotient of one of them divided by the other. Thus

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53 is the index of or a2. Which is only

a3

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 whofe exponent is the faid number (n). Thus 2 x 3 (6) is the index of a' raised to the 3d power (or a). This is proved in p. 38, and alfo follows from article 1.

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4. That, the quotient of the index of any term of the progreffion by a given number (n) is equal to the index of the root of that term defined by the fame number (n). Thus Thus (2) is the index of (a2) the cube root of a°. Which is only the converfe of the laft article.

6

Thefe are the properties of the indices of a geometrical progreffion; which being univerfally true, let the common ratio be now fuppofed indefinitely near to that of equality, or the excefs of a above unity, indefinitely little; fo that fome term, or other, of the progreffion 1, a, a, a3, aa, a3, &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 thofe terms called logarithms of the numbers to which the terms themJelves are equal. Thus, if am = 2, and a" = 3, then will m and n be logarithms of the numbers 2 and 3 refpectively.

Hence it is evident, that what has been above fpecified, in relation to the properties of the indices of powers, is equally true in the logarithms of numbers; fince logarithms are nothing more than the indices. of fuch powers as agree in value with thofe numbers. Thus, for inftance, if the logarithms of 2 and 3 be denoted by m and n; that is, if am 2, and a" = 3, then will the logarithm of 6, (the product of 2 and 3) be equal to m+n (agreeable to article 1); because 2 × 3 (6) am × a" = am+".

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