TO fhew the use of Logarithms in Trigonometry.

Let ABC be a plane triangle, right angled at A, of which the hypothenufe BC is 368 feet, and the angle at B 42° 35', and let the fides AC and AB be fought.

B

C

A

b

By the 2d Cor. to 2d Prop. of Plane Trig. R, is to the fin. B, as BC to CA: confequently, from the nature of numbers, CA is found by multiplying BC by the fin. B, and dividing the product y R; and this is done, by adding the logarithm of BC, and the artificial fine of B together, and fubtracting the artificial radius from the fum, for the remainder will be the logarithm of CA. And in the fame manner may BA be found.

To CA 249.01 f. 2.3962195 ||To BA. 270.96.

10.000000

2.4328990

When the first term is not radius, inftead of fubtracting its logarithm, we often add what it wants of the radius to the other two, and take 10 from the index of the fum; this want is eafily got, by fubtracting the right-hand figure of the logarithm from 10, and all the reft from 9; and it is called the Arithmetical Complement of the Logarithm.

And because the fquare of a number is obtained by adding its logarithm to itself, or by doubling it; the fquare root of number will be got by dividing its logarithm by 2.

2

To illuftrate this, let the three fides of a plane triangle ABC be, AB 228, AC 136, and BC 318 feet; and let the angle at A be required.

By the 8th Prop. of Plane Trig. the rectangle contained by AB, AC is to the rectangle contained by the fum of the three fides and its excefs above BC, as the fquare of R to the square of the cofine of A. And the rectangles are got by adding their logarithms, and the fquares by doubling them. Therefore, add the three fides together, and from the fum fubtract BC. Then add into one fum the Arithmetical complements of AB

Rr

and

LOGAR.

LOGAR. and AC, and the logarithms of the fum and remainder, and the fum of these four will be the cofine of the angle A.

AB 228 Arith. comp. 7.6420652

AC 136 Arith. comp.

BC 318

7.8664611

Half the angle A 59° 43′ 22′′ cofine 9.7015042

The angle A 118 36 44

B

PRAC

PRACTICAL

GEOMETRY.

N order to determine the magnitudes of any kind of quantities, PART I.

quantity

ciently known, and the others are said to be known, when their relation to it is known.

To avoid fractions, this affumed quantity is commonly among the least that are in common ufe. But after it is affumed, any multiple or part of it may be affumed for the fame purpose. Quantities thus affumed, are called Measures of other Quantities.

As measures are of general utility, it is often neceffary to determine their form, or other circumstances, by means of which their magnitude may be known.

This Treatife is divided into three parts. The first treats of Lines and Angles; the fecond of Superficies; and the third of Solids.

PART I.

OF LINES AND ANGLES.

THE lcaft measure of lines with us, is an inch, 12 of which make a foot; and 3 feet make a yard. But distances are often measured by a chain of 22 yards, of which 80 make a mile. If the length of a pendulum, vibrating feconds at London, be divided into 313 equal parts, eight of these parts will make an inch.

PART I.

The Scots inch is a little longer than the English inch, for 185 Scots inches make 186 English inches. There are 37 Scots inches in an ell, and 24 ells in a Scots chain, 80 of which make a Scots mile; fo that 55 Scots miles are equal to 62 English.

T

PROB. I. FIG. 1.

Plate I. O defcribe the conftruction and ufe of the inftruments used for taking angles.

Fig. 1.

Fig. 2.

Pl. I. Fig. 3.

The Quadrant is the fourth part of a circle divided into ninety degrees, and, if the limb be large enough, each degree fubdivided into quarters or minutes. To the radius which paffes through the ninetieth degree, two fights are adapted; and a thread with a plummet is hung from the centre. The use of it is to take angles in a vertical plane.

The Theodolite is a circle divided into 360 degrees, with one or more indices fixed on its centre; thefe indices are fitted with Nonuis's divifions, for finding the parts of a degree. The use of it is to take angles in an horizontal plane. It has a great deal of apparatus, in order to adjuft it, fuch as three staffs, with joints and fcrews, and a level, for fupporting it and placing it horizontally; it has alfo a telescope for obferving objects; and befides thefe, it has a vertical arch for fhewing the inclination of lines to the horizon, and a compass for fhewing their inclination to the meridian.

PROB. II. FIG. 3.

O defcribe the conftruction of the Geometrical
Square.

This is a fquare with a line and plummet hanging from one of its angles A, and each of the fides BE and ED is divided into Ico equal parts and there are two fights C and F fixed on the fide AD.

There is also an index GH, with fights, which, when there is occafion, can be joined to the inftrument, and made to move about the centre A.

Pl. I.

T

O defcribe the conftruction and ufe of a line of chords, and of a line of equal parts.

It is often neceffary to lay down a figure on paper, like to PART I. another figure, for which purpose, thefe lines are required, in w order to make their angles equal, and their fides proportionals.

The line of chords is made thus: Let BAC be a quarter of a Fig. 4. circle, and join BC, and divide the arch BC into 90 degrees: Then from C transfer the distances of the divifions to the straight line CB, and mark them with the degrees in the corresponding arch. The chord of 60 degrees being the fide of an inscribed hexagon, is equal to the radius of the circle, by cor. to the 15th of 4th B. of El.

If now the angle EDF is to be measured, take the chord of Fig. 5. 60°, and from the centre D defcribe with that diftance the arch FG. Then, if the distance FG applied on the line of chords, from C towards B, gives 25°, this fhall be the measure of the angle propofed.

When an obtufe angle KDE is to be measured, produce KD to F, and measure its fupplement FDE, and then KDE will be

known.

But if an angle of 50° is to be made at a given point M, in Fig. 6. the line KL, from the centre M, with the distance MN, equal to the chord of 60°, defcribe the arch NR. Take sc° from the line of chords, and make NR equal to it, and join MR; and it is plain, that NMR is an angle of 50°.

The line of equal parts is made by taking any convenient Fig. 7. diftance for one of the parts, and laying it on the line, from one end to the other; after which, one of the parts is ufually fubdivided into 10 equal parts. From fuch a fcale, any number lefs than 100, may be taken, by calling each of the greater divifions 10, and each of the lefs divifions one.

If upon one fide of the line, thus divided, there be drawn ten others, at equal distances from one another, perpendiculars to the divided line, through the larger divifions, will divide them all; and if one of the fpaces between the perpendiculars. of the two outermoft lines, be each divided into ten equal parts, and from the end of one of them a line be drawn to the first divifion of the other, and the rest of the divifions be joined in their order, a diagonal fcale will be made, from which any number lefs than a thoufand may be taken.

These lines, with feveral others, as lines of fines, tangents, and fecants, are ufually marked on the fcales used by artists. But they are moft convenient for ufe, when placed on a fector, which is a jointed fcale, like a carpenter's rule; for by means of it, an arch may be made of a given number of degrees to any radius, and distances may be taken greater or lefs, without altering their ratios; this is done, by opening the sector, and extending the compaffes from the number on one of the legs, to the fame number on the other.

PROB.