to be had of the Publisher of this Effay, Price Sixpence. 30. The common Cafes of right-angled As are: Cafe 1. Given the Angle at the Bafe, and the Hypothenufe, to find the Bafe and the Perpendicular. EXAMPLE. Let the Angle at A be 40 d. 15', and Pl. II. the Hypothenufe 130 Feet: Quære the other two Fig. 8. Sides? Conftruction 1. Draw a Line, AB, at Pleasure. 2. Make an A= the given 2 40 d. 15', and draw AC, which make equal to the Hypothenufe, 130 Feet. 3. From C let fall the Perpendicular CB; then it is manifeft that the right-angled Triangle ABC is that required. 37. If AB and BC be measured on the fame Scale of equal Parts as AC was fet off from, you will have nearly the Lengths of the Bafe and Perpendicular; but more accurately by Calculation, as follows. CANON S. 38. ft. Making the Hypothenule Radius. To find the Base. Is to AC 130 Feet 2.1139433 To AB 99.22 Feet 1.9966001 To find the Perpendicular. As Radius 90° 10.0000000 Is to AC 130 Feet 2.1139433 To BC 84 Feet 1.9242592 By Natural Sines thus. By the Table, at the End of this Effay, the Sine of 49° 45′ is .7632, and the Sine of 40° 15′ is .6461; therefore fay, by the Golden Rule, As Radius : 130 :: .7632: .7632 × 13099.2, nearly, the required Bale; and as 1 : 1306461 X 130 84, nearly, the Perpendicular which was to be found. 39. Note, LC 49 d. 45′ was found by fubtracting A 40 d. 15' from 90 d. If the Learner is in clined, clined, he may fave himself the Trouble of fubtracting, by taking out the Cofine of 40 d. 15′; because, the Angles A and C being always = 90 d. the Cofine of the 4 A must be equal to the Sine of the Angle C. 40. In the above Canons, the firft Logarithm is to be fubtracted from the Sum of the two middle Logarithms; which, as the firft Logarithm is 10, is here done by only cutting off 1, in the Place of Tens, in the Sum. 41. 2. Making the Base Radius. AsSec. 4A 40°15'10.1173432 | As S. ZA 40° 15' 10.1173432 Feet 2.1139433 12.0416023 10.1173432 12.1139433 To AB 99.22 Feet 1.9966001 To BC 84 Feet 1.9242591 Or thus, by the Table of Natural Secants, &c. By the Table, the Natural Secant of 40° 15′ = 1.309 And the Natural Tangent of 40 15.8466 * If the Table of Logarithms we are ufing has not a Table of Secants, take out the Log. Cofine of 40° 15, viz. 9.8826568, and, fubtracting it from 20, we readily get 10.1173432, the required Log. Secant of 40° 15'. And in the fame Manner the Secant for any other Number of Degrees may be found 42. 3. Making the Perpendicular Radius. As Sec. 4C 49°45' 10.1896841 | As Sec. LC 49°45′ 10.1896841 The Natural Secant of 49° 45′ being 1.547 And the Natural Tangent of 49 45 being 1.181 We have, As 1.547: 130 1.181: 99.2, nearly, the required Bafe. And, laftly, As 1.547 130 :: 1 : 1.181X130 1.547 130X1 1.547 84, nearly, the Perpendicular which was re= quired. 43. In like Manner may the following Cafes be folved by Natural Sines, Tangents, &c. as well as by Logarithms; but as, in Order to be very nearly accurate, Tables must be given, calculated to every Minute of the Quadrant or 90 Degrees, it would take up more Room than we could fpare; and, as they must be computed to 6 or 7 Places of Decimals, the Operations of multiplying and dividing would become more tedious than by Logarithms: We fhall, therefore, for the Future, give only the Solution by Logarithms, it being fufficient to have fhewn the Method of Solution by Natural Numbers. N. B. The Natural Sines are given to 7 Places of Figures, for every Minute of the Quadrant, in our British Mariner's Affiftant. 44. Cafe Pl. II. Fig. 9. 44. Cafe 2. Given the Angle at the Bafe, and the Bafe, to find the Perpendicular and Hypothenufe. EXAMPLE. Let the Angle at A be 50 d. and the Base 86 Yards; to find the Perpendicular and Hypothenuse. Conftruction. Make A 50 d. AB Base, 86 Yards: On the Point B erect the Perpendicular BC, to interfect AC in C: Then is ABC the Triangle required. To BC 102.5 Yards 2.0106849 To AC 133.8 Yards 2.1264309, 46. 2. Making the Perpendicular Radius. As Tang. C 40° 9.9238135 | As Tang. 4C 40° 9.9238135 To BC 102.5 Yards 2.0106849 To AC 133.8 Yards 2.1264309. 47. 3. Making the Bafe Radius." As Radius 90° 10.0000000 Is to AB 86 Yards 1.9344984 To BC 102.5 Yds 12.0106849 -2,10% As Radius goo 10.0000000 Is to AB 86 Yards 1.9344984 48. Cafe 3. Given the Perpendicular and the Angle at the Bafe, to find the Bafe and Hypothenufe. EXAMPLE. Let the at A be 33 d. 45', and the 175 Féét; required the Base and Hypothenuse. Conftruction. Make A 33 d. 45', and draw AD, then, 11 to AB, at a Diftance 75 Feet, draw EF, ]] to interfect AD in C; from C let tall the CB: Then it is evident the AABC is that required. Otherwife, ift, By fubtracting the 2 A 33 d. 45 from go d. find the C56 d. 15'. 2. Affume a Point, B, in the Line AG, and raife the BC 75 Feet. 3. Make ZC568. 15′, and draw CA, to interfect HG in A and the A is conftructed. Pl. II. Fig. 10. |