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nutes in the laft Column to the right Hand, marked M, where the Numbers increafe as they afcend: Thus, to find the Sine of 51°. 20',

Find the 51° at the Bottom of the Page, and in the last Column to the right Hand, in the right-hand Page, find 20'; against which, in the Column having the Word Sine at the Bottom, you have 9.892536, for the Sine of 51°. 20'.

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To find the Sine of 48°. 39': Having found the 48° at the Bottom of the Table, as before, the Minutes, being more than 30, must be fought in the laft Column of the left-hand Page, where the Numbers, beginning at 30, increase as they afcend, to 60 at the Top; amongft which find the 39', and against it, in the Column of the fame Page marked Sine at the Bottom, you have 9.875459, for the Sine of 48°. 39'.

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In the fame Man-S51° 32'

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The Tangents may be found in the fame Manner, to any Degree and Minute of the Quadrant; always obferving, that, when the Degrees are found at the Top of the Page, the Minutes are to be fought in the first Column to the left Hand, where the Numbers increase as they defcend downward; and, when the Degrees are found at the Bottom of the Page, the Minutes must be fought in the laft Column, where the Numbers increase as they afcend upwards.

The Co-fines and Co-tangents are to be found in the fame Manner, being no more than the Sines and Tangents of the Complement of the given Number of Degrees and Minutes to 90°, or a whole Quadrant.

T

Plain TRIGONOMETRY.

RIGONOMETRY, or the Measuring of Triangles, is of two Kinds. The Measuring of right-lined Triangles, otherwife called plain Triangles, is called Plain Trigonometry: The Measuring of curve-lined Triangles, commonly called spherical Triangles, is called Spherical Trigonometry.

Definitions.

Definitions.

A plain Triangle is a Figure contained between three right Lines, which at their Meeting make three Angles.

If one of the three Angles of a Triangle be a right Angle, the Triangle is called a right-angled Triangle.

If neither of the Angles of a Triangle is a right Angle, the Triangle is called an oblique-angled Triangle.

If, in an oblique-angled Triangle, one of the Angles is obtufe, that is, greater than a right Angle, it is called an obtufe-angled Triangle.

If all the three Angles are acute, that is, lefs than right Angles, then is the Triangle called an acute angled Triangle. If all the three Sides of a Triangle are equal, it is called an equilateral Triangle.

If two of the Sides only are equal, it is called an Ifofceles. If the three sides are all unequal, it is called Scalenum.

In a right angled Triangle, as A B C, the longeft Side, A B, is called the Hypothenufe, and the other two Sides the Legs, of which, that on which the Triangle feems to stand, as A C, is called the Base, and the other Side, BC, the Per Apendicular.

B

In an oblique-angled Triangle, the longest Side is usually called the Bafe.

Angles are mea fured by the Arch of a Circle. The Periphery of every Circle, whether great or fmall, is divided into 360 Degrees, each Degree into 60 Minutes, every Minute into 60 Seconds, and fo on to Thirds, Fourths, &c.

Any Portion of the Periphery

of a Circle, as ECF, is called
an Arch; and a Line drawn from
the Ends of an Arch, as EIF, is
called the Chord of the Arch.
Half the Chord of any Arch, as A
EI, is called the Sine of the
Arch E C, and IC is called

the verfed Sine of the fame Arch

EC; fo alfo EG is the me of

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the Arch E D. A drawn perpendicular to the Diameter

of a Circle, fo as touch the Circle, and not cut it, is called a Tangent, as which is the Tangent of the Arch E C,

E e

because

because the Line B H, drawn from the Center B through E, called the Secant, meets it in the Point H. Any Radius, as BD, is the Sine of 90°, ufually called the whole Sine.

The Defcription and Ufe of a Scale of equal Parts.

The common plain Scales have on them a Line of equal Parts, marked L, divided into 10 equal Parts, marked 10, 20, 30, &c. to 100: At the Beginning of the Divifions there is one other Part not numbered, which is fubdivided into ten Parts by fmall Divifions; the middle one, being 5, is drawn longer than any of the reft to diftinguish it. The Ufe of this Line of Parts is, to lay any Thing down upon Paper, having the Dimenfions given: As fuppofe a Triangle was given, one Side being 21 Inches, another 28, and the third 35: To lay this down on Paper from the Line of Parts, you must draw a Line on your Paper; then, to fet off the longeft Side on the Line, fet one Foot of the Compaffes in the Divifion marked 30 of the larger Divifions, and extend the other backward on the Line to the 5 in the fmall Divifions; this Extent mark in your Line: Then, for the 28, fet one Foot of the Compafies on the Divifion marked 20, extending the other back to 8 on the fmall Divifions; with this Extent, fet one Foot in the Mark made at the End of the Line, and with the other draw a Portion of an Arch; which done, fet one Foot of the Compaffes on the fame Divifion, marked 20, and contract the Compaffes, till they reach to the first of the fmall Divifions; with this Extent, fet one Foot in the other Mark made in your Line, and draw another Arch, croffing the former: Laftly, from the Interfection of thefe Arches, draw Lines to the two Marks made in the firft Line; fo will you have a Triangle in the fame Proportion as was given. But, in Cafes where the Numbers are Hundreds, Tens, and odd Ones, there are on thofe Scales ufually two diagonal Scales, one divided into Inches, the other into Half-Inches; thefe Scales are numbered with 1, 2, 3, &c. the Divifions at the End are ufually numbered 2, 4, 6, 8 at the Side, there not being Room to put all the nine Digits, therefore 1, 3, 5, 7, 9, have no Numbers: There are likewife ten parallel Lines run the Length of the Scale, and at the End, among the fmall Divis fions, there are diagonal Lines, croffing from one of thofe Paral lels to another. To take off any Number from thefe Scales, fuppofe 768, look out the great Divifion marked 7, and then look on the fmaller Divifions at the End for that marked 6;

the

the Compaffes being extended from 7 in the greater Divifions to 6 in the small, is 760; but, for the 8, you must count eight of the Parallels, and fet one Foot of the Compaffes where the eighth Parallel croffes the Line marked 7, before found, and extend the other Foot to where the Diagonal 6, before found, croffes the fame Parallel; that Extent is 768, the given Number.

The Defcription and Ufe of a Line of Chords.

I have before obferved, that all Circles are fuppofed to be divided into 360 Parts, called Degrees; therefore each Quadrant contains 90 Degrees, as appears by the Diameters cutting each other at right Angles: If then a Quadrant of a Circle be divided into 90 Degrees, and the Distance from one End of the Arch to each of thefe Degrees be transferred to a straight Line, this is what we call a Line of Chords. The Radius of every Circle, being taken in the Compaffes, will, if the Compaffes be fet in the Periphery and turned about, at fix Times go round the Circumference; therefore the Radius is the Chord of 60 Degrees. On the plain Scale there are ufually two Lines of Chords to different Radii, marked at the End Cho. the Ufe of which is to measure Angles; for, if you take in your Compaffes 60 Degrees from the Line of Chords, and, fetting one Foot of the Compaffes in the angular Point, draw an Arch from one Leg of the Angle to the other, and then take in the Compaffes the Length (or rather the Chord) of the Arch, that, measured on the fame Line of Chords, will fhew the Degrees of the Angle, as has been already fhewn, Prob. 8.

There are usually on the Scale divers other Lines; as a ́ Line of Sines, a Line of Tangents, a Line of Secants, a Line of Half-Tangents, &c. whofe Use we shall have Occafion to explain hereafter.

The great Ufe of Logarithms is in Trigonometrical Calculations. All the Cafes in Trigonometry being done by the Rule of Proportion, to avoid the Trouble of Multiplication and Divifion, Mr. Gunter, as was obferved above, calculated the Logarithms of the Sines and Tangents, by which all the Problems in Trigonometry are performed: So that, having allo Tables of the Logarithms of Numbers, nothing more is neceffary, but to add the Logarithms of the fecond and third Numbers in the Proportion together, and from their Sum fubtract the Logarithm of the first; the Remainder is the LogaE e 2

rithm

rithm of the fourth Number fought; that is, it is the Logarithm of the Length of the Side, when a Side is required, or of the Sine of the Angle, when an Angle is required: And, when the first Number in the Proportion happens to be the Radius whofe Logarithmic Sine is 10.000000, that is, 10. there is then no Occafion for any other Subtraction, but only to reject 10 out of the Sum of the second and third Numbers.

Of right-angled plain Trigonometry.

There are eight Cafes in right-angled plain Trigonometry, feven of which are folved by the following general Rule: The Method of folving the other will be explained when we come to it. The general Rule is this:

As the Sine of any Angle

Is to its oppofite Side,

So is the Sine of any other Angle
To its oppofite Side:

Or,

As the Length of any Side

Is to the Sine of the Angle oppofite,

So is the Length of any other Side

To the Sine of the Angle oppofite to it.

Prob. 18. In the right-angled plain Triangle A B C, rightangled at B, given the Bafe A B 47.5, the Hypothenufe AC 59.4 to find the Perpendicular BC, and the Angles at A and C.

Conftruction. Draw the Bafe-Line A B at Pleasure; fet one, Foot of the Compaffes on the Divifion of the diagonal Scale at 4, and extend the other Foot to feven of the Divifions at the End of the Scale; this is 47: Now, to take off the .5, take your Compaffes from the Scale,

C

B

and go down five of the Lines which are drawn parallel to the Sides of the Scale; place one Foot of the Compaffes there, and you will obferve the other will fall a little fhort of the Line they before touched at the End of the Scale; extend the Compaffes till they reach that Line; this Extent is 47.5 i which place on the Bafe-Line from A to B, and at B erect BC perpendicular to A B, by Prob. 3, and continue it at Pleafure. Now, for the Hypothenufe, fet one Foot of the Compaffes on the diagonal Scale at 5, and extend the other Foot

to

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