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given straight line which is to be taken from BC, and let the ratio which the remainder is required to have to BD be the given ratio of HG to GL, and place GL at right angles to FH, and join LF, LH: next, as HG is to GF, so make HK to AE; produce AE to N, so that AN be the straight line to the square of which the sum of the squares of AB, BD, is required to be equal; and make the angle NED equal to the angle GFL; and from the centre A, at the distance AŃ, describe a circle, and let its circumference meet ED in D, and draw DB perpendicular to AN and DM making the angle BDM equal to the angle GLH. Lastly, produce BM to C, so that MC be equal to HK; then is AB the first, BC the second, and BD the third of the straight lines that were to be found.
For the triangles EBD, FGL, as also DBM, LGH, being equiangular, as EB to BD, so is FG to GL; and as DB to BM, so is LG to GH; therefore, ex æquali, as EB to BM, so is (FG to GH, and so is) AE to HK or MC; wherefore *, AB is to BC, as AE to HK, that is, as FG to GH, that is, in the given ratio : and from the straight line BC taking MC, which is equal to the given straight line HK, the remainder BM has to BD the given ratio of HG to GL: and the sum of the squares of AB, BD, is equal* to the square of AD or AŃ, which is the given space. Q. E. D.
I believe it would be in vain to try to deduce the preceding construction from an algebraical solution of the problem.
* 47. 1.
END OF THE NOTES TO THE DATA,
PLANE AND SPHERICAL
PRINTED FOR J. COLLINGWOOD,
And the rest of the Proprietors.
LEMMA I. Fig. ). LET ABC be a rectilineal angle: if about the point B as a centre, and with any distance BA, a circle be described, meeting BA, BC, the straight lines including the angle ABC in A, C; the angle ABC will be to four right angles, as the arch AC to the whole circumference.
Produce AB till it meet the circle again in F, and through B draw'DE perpendicular to AB, meeting the circle in D, E.
By 33. 6. Elem. the angle ABC is to a right angle ABD, as the arch AC to the arch AD; and quadrupling the consequents, the angle ABC will be to four right angles, as the arch AC to four times the arch AD, or to the whole circumference.
LEMMA II. Fig. 2.
Let. ABC be a plane rectilineal angle a's before: about B as a centre with any two distances BD, BA, let two circles be described meeting BA, BC, in D, E, A, C; the arch AC will be to the whole circumference of which it is an arch, as the arch DE is to the whole circumference of which it is an arch.
By Lemma 1. the arch AC is to the whole circumference of which it is an arch, as the angle ABC is to four right angles; and by the same Lemma 1. the arch DE is to the whole circumference of which it is ån arch, as the angle ABC is to four right angles; therefore the arch AC is to the whole circumference of which it is an arch, as the arch DE to the whole circumference of which it is an arch.
DEFINITIONS. Fig. 3.
Let ABC be a plane rectilineal angle; if about B as a centre, with BA any distance, a circle ACF be de
scribed, meeting BA, BC, in A, C; the arch AC is called the measure of the angle ABC.
The circumference of a circle is supposed to be divided
into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, &c. And as many degrees, minutes, seconds, &c. as are contained in any arch, of so many degrees, minutes, seconds, &c. is the angle, of which that arch is the
measure, said to be. Cor. Whatever be the radius of the circle of which the
measure of a given angle is an arch, that arch will contain the same number of degrees, minutes, seconds, &c. as is manifest from Lemma 2.
III. Let AB be produced till it meet the circle again in F;
the angle CBF, which together with ABC is equal to two right angles, is called the Supplement of the angle ABC.
A straight line CD drawn through C, one of the ex
tremities of the arch AC, perpendicular upon the diameter passing through the other extremity A, is called the Sine of the arch AC, or of the angle ABC,
of which it is the measure. Cor. The Sine of a quadrant, or of a right angle, is
equal to the radius.
The segment DA of the diameter passing through A,
one extremity of the arch AC, between the sine CD, and that extremity, is called the Versed Sine of the arch AC, or angle ABC.
A straight line AE touching the circle at A, one ex
tremity of the arch AC, and meeting the diameter BC passing through the other extremity C in E, is called the Tangent of the arch AC, or of the angle ABC.