VIII. On a new Property of the Tangents of the three Angles of a Plane Triangle. By Mr. William Garrard, Quarter Master of Instruction at the Royal Naval Asylum at Greenwich. Communicated by the Astronomer Royal.

Read February 11, 1808.

PROPOSITION I. In every acute angled plane triangle, the sum of the three tangents of the three angles multiplied by square of the radius, is equal to the continued product of the tangents.

the

Demonstration.-Let AH, HI, and IB

be the arches to represent the given angles; and AG, HK, and BT be their G tangents, put r the radius, AG= a, and BT=b,

Then and will be the tangents of

a

HD and DI.

Now by Prop. VIII. Sect. I. Book I. EMERSON'S Trigonometry,

F

K

As radius square-product of two tangents
Is to radius square,

So is the sum of the tangents

To the tangent of their sum.

PROPOSITION II. In every obtuse angled plane triangle, the sum of the three tangents of the three angles multiplied by the square of radius, is equal to their continued product. Demonstration.-Let AH be an obtuse

arc, and HE, ED the other two.

Then BF, ED, and AG are the three tangents.

Put BFt and DE u radius=r,

then per trigonometry, as before,

H

IX. On a new Property of the Tangents of three Arches trisecting the Circumference of a Circle, by Nevil Maskelyne, D. D. F.R. S. and Astronomer Royal.

Read February 18, 1808.

MR. WILLIAM GARRARD having shewn me a curious property of the tangents of the three angles of a plane triangle, or in other words, of the tangents of three arches trisecting a semicircle, in a paper which I have communicated to this Society, I was led to consider whether a similar property might not belong to the tangents of three arches trisecting the whole circumference; and, on examination, found it be so.

Let the circumference of a circle be divided any how into three arches A, B, C; that is, let A+B+C be equal to the whole circumference. I say, the square of the radius multiplied into the sum of the tangents of the three arches A, B, C, is equal to the product of the tangents multiplied together. I shall demonstrate this by symbolical calculation, now commonly called (especially by foreign mathematicians) analytic calculation.

It may be proper to premise, that the signification of the symbolical expressions of the tangents of an arc, whether with respect to geometry or numbers, are to be understood according to their position as lying on one side, or the other side of the radius, passing through the point of commencement of the arc of the circle; those tangents which belong to

the first or third quadrant of the circle being considered as positive, and those belonging to the second and fourth quadrant, being of a contrary direction, as negative; in like manner as the sines in the first semi-circle are considered as positive, and in the second semi-circle as negative; and the cosines in the first and fourth quadrant are considered as positive, and in the second and third quadrants as negative; they lying, in the second case, on the contrary side of the diameter passing through the point of ninety degrees, to what they do in the former. Hence it easily follows, that, the tangent of any arch and of its supplement to the whole circumference, or 360 degrees, are equal and contrary to one another, or the one negative of the other.

t u

tu

Let t, u, w, be put for the tangents of the three arches A, B, C respectively, and r for, the radius, and o for the whole circumference. Then A+B+Co, and C=0¬A†B. By trigonometry, t, A+B="+", and the tang. C= tang. (0 —A+B) =—tang. AB, by what has been said above. Therefore t, A+t, B+t, Cort+u+w=t+u- r2-tu ; but t and u are the expressions for the tangents of A and B respectively, and *x+ is the expression

=⋅tux

rxt+ u
Jtu

r-tu

r2xitu

for the tangent of C, or for w. Therefore, rxt+u+w, or the square of the radius multiplied into the sum of the three tangents of A, B, and C = tuw, or the product of the tangents. Q.E.D.

X. An Account of the Application of the Gas from Coal to economical Purposes. By Mr. William Murdoch. Communicated by the Right Hon. Sir Joseph Banks, Bart. K. B. P. R. S.

Read February 25, 1808.

THE facts and results intended to be communicated in this Paper, are founded upon observations made, during the present winter, at the cotton manufactory of Messrs. PHILIPS and LEE at Manchester, where the light obtained by the combustion of the gas from coal is used upon a very large scale; the apparatus for its production and application having been prepared by me at the works of Messrs. BOULTON, WATT, and Co. at Soho.

The whole of the rooms of this cotton mill, which is, I believe, the most extensive in the United Kingdom, as well as its counting-houses and store-rooms, and the adjacent dwellinghouse of Mr. LEE, are lighted with the gas from coal. The total quantity of light used during the hours of burning, has been ascertained, by a comparison of shadows, to be about equal to the light which 2500 mould candles of six in the pound would give; each of the candles, with which the comparison was made consuming at the rate of 4-10ths of an ounce (175 grains) of tallow per hour.

The quantity of light is necessarily liable to some variation, from the difficulty of adjusting all the flames, so as to be perfectly equal at all times; but the admirable precision and