43 Х 47 (4.) Find the logarithm of .0084'97133'. Ans. — 3.92927. (5.) Required the product of 564 x 07 x tox 443 684 Ans. Logarithm of the product is 1.34735, and the product = 22.251. (6.) Required the product of .05'94405' * •583' * *0322916' x •4/28571' * •l'8'. Ans. Log. of the product is 5.94075, and the product = .000087247. (7.) Divide •0565 by .25. Quotient = .226. (12.) Required the 3.75 power of 14•79; or find the value of 14:79. Ans. 24399.5. (13.) Required the •34'54' power of 94.75'; or find the value of 94.793413. Ans. 4•81736. (14.) Involve 09475' to the '34'54' power. Ans. •44307. (15.) Find the cube-root of .000381078. Ans. 0725. (16.) What is the .625 root of .027588 ? Ans. •0032. (17.) Find a fourth proportional to 58'. 13"; 11":75; and 24 hours. Ans. 4'.50":6. (18.) Find a fourth proportional to 231. 12. 37"; 24 hours; and 7h. 59!. 34". Ans. 86. 15'. 53". CHAP. III. THE USE OF THE TABLES OF SINES AND TANGENTS. PROPOSITION I. (R.) To find the natural sine or cosine of an arc, also the logarithmical sine, tangent, secant, &c. RULE. If the degrees in the arc be less than 45, look for them at the top of the table, and for the minutes (if any) in the left hand column marked M, against which, in the column signed at the top of the table with the proposed name, viz. sine, cosine, &c. stands the sine, cosine, &c. required. If the degrees are more than 45, they must be found at the bottom of the table, and the minutes (if any) must be found in the right hand column. The name in this case, viz. sine, tangent, &c. must be taken at the bottom of the table. To find the secants see the first page of Table III. * The construction of these tables will be found at the end of Book II. Chap. V. Before the student reads this and the following chapter, it will be proper for him to read the definitions, &c. in Book II. Chap. I. The natural sines must be looked for in the table entitled natural sines; and the logarithmical sines in the table entitled logarithmical sines and tangents.. Required the natural and logarithm. sine and cosine of 390.42'. Natural sine of 39° . 42 = 63877, cosine = •76940. Logarithmical sine of 39o. 42' = 9.80534, cosine = 9.88615. Required the natural and logarithm. sine and cosine of 730.27'. Natural sine of 73°.27' = .95857, cosine = 28485. Logarithmical sine of 73°.27' = 9.98162, cosine = 9:45462. If the sine, tangent, &c. be wanted to any number of degrees above 90; subtract those degrees from 180° and find the sine, tangent, &c. of the remainder: or subtract 90° from the given number of degrees, and find the cosine, co-tangent, &c. of the remainder, which is the same thing. Required the logarithmical sine, tangent, secant, cosine, cotangent, and co-secant of 1370.29'. 180° rem. 42o. 31' sine = 9.82982, cosine = 9.86752, tangent = 9.96231, co-tangent = 10:03 769, secant = 10:13248, cosecant = 10:17018, and these are respectively equal to the cosine, sine, co-tangent, tangent, co-secant, and secant of 479.29' = 137° . 29 - 90°. PROPOSITION II. (S) To find the logarith. sine, cosine, &c. of an arc to seconds. Find the logarithm to the degrees and minutes as in Proposition I. take the difference between this logarithm and the next greater or less in the same column, according as you want a sine or cosine, tangent or co-tangent, &c. multiply this difference by the number of seconds given, and divide the product by 60; add the quotient to the given logarithm if it be a sine, tangent, or secant, but subtract the quotient from the given logarithm if it be a cosine, co-tangent, or co-secant, and the sum, or remainder, will be the logarithm required. Required. the logarith. sine, tangent, and secant of 350.44.24". Log. sine 350.44' = 9•76642 tang. = 9.85700 sec. = 10:09058 next greater sine = 9.76660 tang. = 9.85727 sec. = 10·09067 Then, sine 359.441.24" = 9•76649. tan. = 9.85710, sec. = 10.09061. In the same manner the natural sine is found, being •58410. Required the logarithmical cosine, co-tangent, and co-secant diff. 18 24 60 | 216 60 | 648 60 | 432 Subtract prop. part 3 7 Then cosine 350.44'. 24" = 9.90939, co-tan. = 10.14290, co-secant=10.23351. In a similar manner the natural cosine is found, being •81167. PROPOSITION III. one (T) To find the degrees, minutes, or degrees, minutes, and seconds, corresponding to any given logarithm, sine, tangent, &c. RULE. Find the nearest logarithm to the given one in the table, and the degrees answering to it will be found at the top of the column if the name be there, and the minutes on the left hand; but if the name be at the bottom of the table, the degrees must be found at the bottom of the table, and the minutes on the right hand. To find the arc to seconds, take the difference between the two nearest logarithms to the given which you can find in the table, also the difference between the given logarithm and the nearest less. Multiply the second difference by 60, and divide the product by the first difference, the quotient will give a number of seconds, which must be added to the degrees and minutes corresponding to the nearest less number in the tables, if your given logarithm be a sine, tangent, or secant; but if your given logarithm be a cosine, co-tangent, or co-secant, the number of seconds must be subtracted from the degrees and minutes corresponding to the nearest less number in the tables. Find the degrees, minutes, and seconds, corresponding to the logarithmical sine 9.43299. Nearest sine less than the given one 9.43278 21 Nearest sine greater than the given one 9•43323 60 Second diff. 21 The same manner of proceeding must be observed in finding a tangent, secant, or natural sine. Find the degrees, minutes, and seconds, corresponding to the logarithmical cosine 9.43297. Nearest cosine less than the given one 9.43278 Nearest cosine greater than the given one 9.43323 19 First difference 45 60 Given cosine 9.43297 45 | 1140 25quot. Second difference 19 Therefore the required arc is 74o.16'. 34". PROPOSITION IV. (U) To find the natural or logarithmical versed sine of an arc, by the help of a table of natural or logarithmical sines. To find the natural versed sine; subtract the natural cosine from an unit if the arc be less than 90°, but if greater than 90, add it to an unit. To find the logarithmical versed sine; find the logarithmical sine of half the arc, double it, and subtract 9.69897 from the product. Required the natural versed | Required the natural versed sine of 65°,45'. sine of 1150.35'. Radius =1 Natural cosine 1150.35' or Nat. cosine 65°.45' = 41072 cosine 640.25' = 43182 To which add 1: versed sine 65°,45'. = 58928 vers.sine of 115°.35'=1.43182 Required the logarithm. versed Required the log. versed sine of sine of 720.14'. 370.53'. Logarithmical sine of 36o.7' Here half the arc is 18°.56 half arc = 9:77043 Log. sine 18°.56' =9.51117 2 Log. sine 180.57'=9:51154 19.54086 doublesine189.561'=19.02271 9•69897 CHAP. IV. THE CONSTRUCTION AND USE OF THE PLAIN SCALE. (W) The Plain Scale is a mathematical instrument of extensive use. The scale generally used at sea is a ruler of two feet in length, having drawn upon it equal parts, chords, sines, tangents, secants, &c. These are contained on one side of the scale, and the other side contains their logarithms. (X) Describe a semicircle with any convenient radius CB (Fig. I. Plate II.); from the centre c draw cd perpendicular to AB, and produce it to F, &c.; draw BE parallel to CF, and join ad and BD. (Y) Rhumbs. Divide the quadrantal arc ad into eight equal parts, with one foot of the compasses in A, transfer the distances al, A2, A3, &c. to the straight line AD, and it will be a line of rhumbs containing eight points of the compass, or one-fourth of the whole circumference of the compass. By subdividing each of the divisions al; 1, 2, &c. into four equal parts, and transferring them in the same manner to the line AD, it will contain the points, and half and quarter points. (Z) Chords. Divide the arc ed into nine equal parts, with one foot of the compasses in B and the distances B10, B20, B30, &c.; transfer them to the straight line BD, which will be a line of chords constructed to every ten degrees. The single degrees are constructed by subdividing the arcs, B10; 10, 20; &c. into ten equal parts, and transferring the divisions in the same manner to the line BD. (A) Sines. Through each of the divisions of the arc BD draw lines parallel to cd, such as 80, 10; 70, 20; &c. and the line ce will be divided into a line of sines reckoning from c to B (for cg is the cosine of the arc B80, or the sine of the arc D80, which is ten degrees); if this line be numbered from B towards c, it will become a line of versed sines. (B) Tangent. From the centre c draw straight lines through the several divisions of the quadrantal arc BD, to touch the straight line BE, which will become a line of tangents. (C) Secants. Transfer the distances between the centre c and the divisions of the line of tangents, to the line DF, and it will become a line of secants which must be numbered from D towards F, as in the figure. (D) Semi-tangents. From a draw lines through the several divisions of the arc BD, and they will divide the line cd into |