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4 AB, BC, CA, A, B, C ihe 2 AC CB: ACq+CBq the three sides, three angles. - ABq::R : CUS, C. If

ACq+CBq be greater
than ABq. Fig. 16.
2 AC x CB: ABq-ACq
-CBq::R: Cos, c. If
AB be greater than ACI
+ CBq. Fig. 17. (4.)

Otherwise,
Let AB + BC + AC=2P.
PxP-AB:P-AC X
P-BC :: Rq : Tq, į C,
and hence Cis known. (5.)

Otherwise,
Let AD be perpendicu-
lar to BC. 1. If ABq be
less than ACq+CBq. Fır.
16. BC: BA +AC:: BA
- AC : BD - DC, and
BC the sum of BD, DC
is given; therefore each of
them is given. (7.)

2. If A Bq be greater than ACq+CBq. Fig. 17.BC: BA+AC: BA-AC: BD+DC; and BC the difference of BD, DC, is given, therefore each of them is given. (7.)

And ČA: CD :: R: COS, C. (1.) and C being found, A and B are found by case 2 or 3.

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CONSTRUCTIONS

OF THE

TRIGONOMETRICAL CANON.

A Trigonometrical Canon is a Table, which, beginning

from one second or one minute, orderly expresses the lengths that every sine, tangent, and secant have, in respect of the radius, which is supposed unity; and is conceived to be divided into 10000000 or more decimal parts.

And so the sine, tangent, or secant of an arc, may be had by the help of this table; and, contrariwise, a sine, tangent, or secant being given, we may find the arc it

expresses.

Take notice that in the following tract, R signifies the radius, S a sine, Cos. a cosine, T a tangent, and Cot. a cotangent; also ACq signifies the square of the right line AC; and the marks or characters, +,-,=, :,::, and v, are severally used to signify addition, subtraction, equality, proportionality, and the extraction of the square root. Again, when a line is drawn over the şum or difference of two quantities, then that sum or difference is to be considered as one quantity.

Constructions of the Trigonometrical Canon.

PROP. I. THEOR.

The two sides of any right angled triangle being given, Fig. 28:

the other side is also given. For (by 47. 1.) ACq=ABq + BCq, and ACq-BCq. = ABğ, and interchangeably. AČI-ABq = BCq. Whence, by the extraction of the square root, there is given AC=> ABq+ BCq; and AB=w ACq-BCq; and BC=ACq-ABq.

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