NOTE In the firft ftating, where the hypothenufe is made radius, the fum of the logarithms of the fecond and third terms are 12,30863, from which it is easy to fubtract the logarithm of the firft term; for you may either cancel it, or leave it out; and then caft off the firit figure towards the left hand, and it will leave the logarithm 2.30863, the fame as if 10.00000 had been fet down and fubtracted from it; and, indeed, the five cyphers may be always omitted in the radius, and only the index 10 fet down. It will greatly expedite the working the proportions by logarithms, if the two or all the ftatings be firft made, and then the fines, tangents, or fecants, may be taken out at one opening of the book; for if one angle of a right-angled triangle be given, the logarithm of its complement, or the other angle, whether fine, tangent, or fecant, is found in the adjoining column, without being at the trouble of fubtracting the given angle from 90°. If the given angle be less than 45 degrees, it is found at the top of the table, and the minutes in the left-hand column reckoned down. wards, and its complement is found at the bottom, and the minutes on the right-hand column, On the contrary, if the given angle is found at the bottom, its complement, or the other angle, will be at the top of the table, and the minutes in the left-hand column, againft which is the log. fine, tangent, or fecant, corresponding to it. By GUNTER'S SCALE, In all proportions wrought by Gunter's Scale, when the first and fecond terms are of the fame kind, then the extent from the first term to the second, will reach from the third to the fourth; Or when the first and third terms are of the fame kind, The extent from the firft term to the third will reach from the fecond to the fourth; that is, fet one point of the compaffes on the divifion expreffing the fecond term, then, without altering the opening of the compaffes, fet one point on the divifion representing the third term, or fecond term, and the other point will fall on the divifion fhewing the fourth term or answer. Now, in this laft cafe, it will run thas: Extend from radius, or 90°, to 54° 30', on the line of fines, that extent will reach from 250, the hypothenufe, to 203,5, the bafe, on the line of numbers; and the extent from radius, or fine of 90°, to 35° 30' on the line of fines, will reach from 250 to 145 on the line of numbers. Obferve the like in all that follows, except in thofe proportions where the word fecant is mentioned, which may be readily wrought by confidering the hypothenufe radius, as in the last cafe; there being no line of fecants on Gunter's Scale. NOTE. NOTE. The radius, according to the nature of the proportion, may be any of these : 8 Points on the line of Rhumbs. | 90° On the line of Sines. 4 Points on the line of Tan. Rhbs. 45° On the line of Tangents. CASES II. and III. The Angles and one Leg given, to find the Hypothenuse and other Leg. The angle ACB 33° 15', the leg BC 325 miles given, to find the hypothenuse and the other leg. By CONSTRUCTION. Draw the line BC, which make equal to 325 miles; on B A erect the perpendicular BA; on C defcribe an arch with the chord of 60°, and make the angle C=33° 15', through where that cuts the arch draw AC to cut AB in A, and it is done'; for BA being measured on the fame scale that BC was, will be B 213,1, and AC 388,6 miles. 213.1 90. 33.15 50.45 388.6 33.15 325 By making the Hypothenuse AC Radius, it will be, To find the hypothenufe AC. 9.92235 To find the perpendicular AB. 2.51188 10.00000 12.51188 9 92235 To the perpen. AB 213,1 2.32854 To the hypoth. AC 388,6 2.58953 By making the Base BC Radius, it will be, To find the perpendicular AB. To find the hypothenufe AC. As radius 90° Is to the base BC 325 So is tang. ang. C 33° 15′ To the perpen. AB 213,1 2.32854 To the hypoth. AC 388,6 2.58953 To find the perpendicular AB. By making the Perpendicular AB Radius, it will be, As tang. ang. A 56° 45' Is to the base BC 235 2.51188 10.00000 So is fec. ang. A 56° 45' 10.26099 To the perpen. AB 213,1 2.32854 | To the hypoth. AC 388,6 2.58953 By GUNTER. • Extend from 56 degrees 45 minutes, to 33 degrees 15 minutes, on the line of fines, that extent will reach from the base 325, to the perpendicular 213,1, on the line of numbers. 2dly. Extend from 50 degrees 45 minutes to radius on the line of fines, that extent will reach from the base 325, to the hypothenufe 388,6 on the line of numbers.' CASES IV. and V. The Hypothenufe and one Leg given, to find the Angles and the other Leg. The Leg AB 91, the hypothenufe 170 given, to find the angle ACB, or BAC, and the leg BC. By making the Hypothenufe Radius, it will be, To fine ang. C 32° 22′ 9.72859 To the base 143,6 12.15712 10.00000 2.15712 By PROPOSITION IV.-In every plane triangle, whether right or oblique, the three angles are equal to two right angles, or 180°. In the triangle AGB draw CD parallel to AB through the point G; on which point, with the chord of 60°, or any convenient radius, defcribe a circle; and, with the fame radius, on A and B defcribe arches; now, by the laft propofition, the angle AGB will be equal to the angles FGE, and the angle ABG will be equal to the angles CGE, and the angle BAG is equal to the angle DGF: now, fince the oppofite angles are equal, the angles DGF, FGE, and EGC, together, make a femicircle, or 180°; therefore it is plain that the three angles of a plane triangle, whether right, acute, or obtufe, together, are equal to two right angles or 180°; hence it follows that, as the right angle BAG, Fig. 2, is 90°, the other two acute angles, ABG, and AGB, taken together, can be no more than 90°; therefore, if one of the acute angles, in a right-angled triangle, be given, the other is found by fubtracting the given angle from 90°. And in any oblique-angled triangle, if one of the angles be given, the fum of the other two is found by fubtracting the given angle from 180°; and if two angles are given, the third is found by fubtracting the fum of the two angles from 180°. PROPOSITION V.-In every plane triangle, if one of its fides be produced, the outward angle will be equal to the two inward oppofite angles. Let ABC be the triangle, and CD the fide produced, with the chord of 60°, or any other radius, describe arches on AB and C, draw CE parallel to A B; then, by the third propofition, the angle ACE must be equal to the angle B BAC, and the angle DCE equal to the 155 05 65 E angle CBA; therefore the outward angle DCA is equal to the two inward oppofite angles ACB, and BAC; which may be easily proved by measuring the angles by the line of chords on the plane fcale. NOTE. I hope the learned Mathematician will excufe the method here taken of demonftrating the above propofitions in a mechanical manner, judging it beft adapted to the capacity of those for whose use this book is intended, not doubting but the Teacher will, as I always do, demonftrate them in a inore geometrical manner to those who are capable of receiving fuch. P TRIGONOMETRY." LAIN Trigonometry is the art of measuring plane triangles, by comparing the fides and angles together by known analogies; whereby three things being given, a fourth may be found, on condition that one of them be a fide: but as angles are measured by the arch of a circle, defcribed upon their angular points, and the proportions that these arches bear to right lines cannot be exactly found; therefore the writers on Trigonometry have applied right lines to these arches, that the proportion they bear to the fides of a plane triangle may be found, The right lines applied to a circle are: Ift. A CHORD, or the fubtense of an arch, is a right line that divides the circle into two unequal parts, and is a chord to them both, as DH is the chord of the arches DH and DAH. 2d. A RIGHT SINE of an arch is, a right line drawn from one end or termination of an arch perpendicular to the radius; or it is half the chord of twice the arch; so that RS is the fine of the arch AS, and 6Z the co-fine. 3d. A VERSED SINE is that of the diameter contained between the right fine, and the part Di Chord Rad HCotan Cosine Sine Sec Ver-Sin I arch, as RA and RCD, is the verfed fine of SHD, or DEP, its equal. 4th. A TANGENT of an arch is a right line drawn perpendicular to the end of the diameter, juft touching the arch, as AT is the tangent of the arch AS, and HG the co-tangent. 5th. A SECANT of an arch is a right line drawn from the centre through the circumference, and produced until it cuts the tangent as CT. NOTE -The fine, tangent, and fecant of the complement of an arch, is called the co-fine, co-tangent, and co-fecant of that arch. The fines, tangents, and fecants of an arch, are faid to be the measure |