and alfo less than it, as being comprehended thereby, which is abfurd; therefore the Sphere A B C to a Sphere less than the Sphere DEF, hath not a triplicate Proportion of that which BC has to E F. After the fame manner it is demonftrated, that the Sphere DEF to a Sphere less than ABC, has not a triplicate Proportion of that which E F has to BC: I fay, moreover, that the Sphere A B C to a Sphere greater than D E F, hath not a triplicate Proportion of that which B C has to EF: For, if it be poffible, let it have to the Sphere L M N greater than D E F; then (by Inverfion) the Sphere L M N to the Sphere ABC, fhall have a triplicate Proportion of that which the Diameter E F has to the Diameter B C. But as the Sphere L M N is to the Sphere A B C, fo is the Sphere D E F to fome Sphere lefs than A B C, because the Sphere L M N is greater than DEF. Therefore the Sphere D E F to a Sphere lefs than ABC, hath a triplicate Proportion of that which EF has to BC, which is abfurd, as has been before proved. Therefore the Sphere A B C to a Sphere greater than DEF, has not a triplicate Proportion of that which BC has to E F. But it has also been demonftrated, that the Sphere A B C to a Sphere less than DE F, has not a triplicate Proportion of that which BC has to EF: Threfore, the Sphere ABC to the Sphere DE F, has a triplicate Proportion of that which BC has to EF; which was to be demonftrated.

THE

ELEMENTS

Of PLANE and SPHERICAL

TRIGONOMETRY.

DEFINITION S.

HE Bufinefs of Trigonometry is, to find the Angles when the Sides are given, and the Sides or the Ratios of the Sides, when the Angles are given, and to find Sides and Angles, when Sides and Angles are given: In order to which it is neceffary, that not only the Peripheries of Circles, but also certain Right Lines in and about Circles, be fuppofed divided into fome determined Number of Parts.

And fo the ancient Mathematicians thought fit to divide the Periphery of a Circle into 360 Parts, which they call Degrees; and every Degree into 60 Minutes; and every Minute into 60 Seconds; and, again, every Second into 60 Thirds; and fo on. And every Angle is faid to be of fuch a Number of Degrees and Minutes, as there are in the Are meafuring that Angle.

There are fome that would have a Degree divided into centefimal Parts, rather than fexagefimal ones; and perhaps it would be more useful to divide, not only a Degree, but even the whole Circle, into a duplicate Ratio ; which Divifion may fome Time or other gain Place. Now, if a Circle contains 360 Degrees, a Quadrant thereof, which is the Measure of a Right Angle, will

be 90 of thofe Parts: And if it contains 100 Parts, a Quadrant will be 25 of thefe Parts.

The Complements of an Arc is the Difference thereof from a Quadrant.

A Chord or Subtenfe, is a Right Line drawn from one End of the Arc to the other.

The Right Sine of any Arc, which is also commonly called only a Sine, is a Right Line drawn, from one End of an Arc, perpendicular to the Radius drawn through the other End of the faid Arc; and is, therefore, the Semifubtenfe of double the Arc; viz. DE=1 DO, and the Arc DO is double of the Arc D B. Hence, the Sine_of an Arc of 30 Degrees is equal to one half of the Radius. For (by Corol. 15. El. 4.) the Side of an Hexagon infcribed in a Circle, that is, the Subtenfe of 60 Degrees, is equal to the Radius. A Sine divides the Radius into two Segments CE, EB: one of which CE, which is intercepted between the Centre and the Right Sine, is the Sine of the Complement of the Arc D B to a Quadrant (for C E F D, which is the Sine of the Are DH), and is called the Cofine: The other Segment B E, which is intercepted between the Right Sine and the Periphery, is called a Verfed Sine, and fometimes a Sagitta.

And if the Right Line CG be produced from the Centre C, thro' one End D of the Arc, until it meets the Right Line BG, which is perpendicular to the Diameter drawn through the other End B of the Arc; then CG is called the Secant, and BG the Tangent, of the Arc DB.

The Cofecant and Cotangent of an Are are the Secant and Tangent of that Are which is the Complement of the former Arc to a Quadrant. Note, As the Chord of an Arc, and of its Complement to a Circle, is the fame; fo, likewife, are the Sine, Tangent, and Secant, of an Arca the fame as the Sine, Tangent, and Secant, of its Complement to a Semicircle.

The Sinus Totus is the greateft Sine, or the Sine of 90 Degrees, which is equal to the Radius of the Circle.

A Trigonometrical Canon is a Table, which, beginning frem one Minute, arderly expreffes the Lengths that every Sine, Tangent, and Secant have, in respect of the Radius, which is fuppofed Unity; and is conceived to be divided into 10,000,000 or more decimal Parts. And Jo the Sine, Tangent, or Secant, of an Arc, may be had

by

by Help of this Table; and, contra iwife, a Sine, Tangent or Secant, being given, we may find the Arc it expreffes. Take notice, That in the following Tract, R. fignifies the Radius, S. a Sine, Cof. a Cofine, V. a Tangent, and Cot. a Cotangent; alfo ACq fignifies the Square of the Right Line AC; and the Marks or Characters +, —, =,:::, and √, are, feverally, used to fignify Addition, Subtraction, Equality, Proportionality, and the Extraction of the Square Root: Again, when a Line is drawn over the Sum or Difference of two Quantities, then that Sum or Difference is to be confidered as one Quantity.

The two Sides of any Right-angled Triangle being given, the other Side is also given.

Fo

OR (by 47 of the first Element) A·C q=A Bq + BC q and A Cq-BC q-A B q and inter૧ changeably AC q-A Bq=BC q. Whence, by the Extraction of the Square Root, there is given A C= AB q+BCq; and A BACq-B Cq; and ACq-A Bq.

BC

PROPOSITION II.

PROBLEM.

The Sine DE of the Arc B D, and the Radius CD, being given, to find the Cofine D F.

THE Radius C D, and the Sine DE, being given in the Right-angled Triangle CD E, there will be given (by the laft Prop.)/CDq-DEq=(CE=)

DF.