Page images
PDF
EPUB

(5.) Required the product of 561 × 7 × 19 ×

[blocks in formation]

Ans. Logarithm of the product is 1.34735, and the product

= 22.251

(6.) Required the product of '05'94405' x 583' x '0322916 x 4'28571' x 1's'. Ans. Log. of the product is and the product = '000087247.

(7.) Divide 0565 by 25. Quotient =.226.
(8.) Divide 00375 by 0678.
(9.) Divide 54498 by '093.
(10.) Divide 83 by 1528.

I 3

Quotient = 05531.
Quotient = 586000.
Quotient
Quotient 0041963.

5.94075,

(11.) Involve 1:05 to the 40th power. Ans. 70399.7·0% (12.) Required the 3.75 power of 1479; or find the v

of 14-79). Ans. 24399'5.

(13.) Required the 34'54' power of 94.75'; or find the

3451
9990

value of 94.75 Ans. 481736.

[ocr errors]

(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 23.12.37′′; 24 hours; and 7.59'.34". Ans. 8h.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 in 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

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 logarithmical sine and cosine of 39°. 42'. Natural sine of 39°. 42′ 63877, cosine = 76940. Logarithmical sine of 39°. 42′ = 9.80534, cosine = 9.88615. Required the natural and logarithmical sine and cosine of 73°.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, co-tangent, and co-secant of 137° .29'.

180°
137.29'

rem. 42°. 31' sine 9:82982, cosine = 9.86752, tangent= 9.96231, co-tangent = 10.03769, secant 1013248, co-secant 1017018, and these are respectively equal to the co-sine, sine, co-tangent, tangent, co-secant, and secant of 47°.29′ = 137°.29'-90°,

PROPOSITION II.

=

(S) To find the logarithmical 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 logarithmical sine, tangent, and secant of 35°.44'.24". Log.sine 35°. 449.76642 tangent=9.85700 secant=10·09058 next greater sine = 9.76660 tangent=9·85727 secant=10·09067

[blocks in formation]

14

THE USE OF THE TABLE OF LOGARITHMICAL

Book I.

Then,sine 35°. 44'. 24" 9.76649,tang.-9.85710, sec. 10′09061. In the same manner the natural sine is found, being ⚫58410. Required the logarithmical co-sine, co-tangent, and co-secant of 35°. 44'. 24".

Log.cosine35°.44'-9.90942co-tan.-10.14300 co-sec.-10-23358 next less cosine =9'90933co-tan.= 10.14273 co-sec.-10.23340

[blocks in formation]

Subtract prop. part

3

10

7

Then cosine 35°. 44′.24′′ 9·90939, co-tangent = 10-14290,

co-secant 10.23351.

In a similar manner the natural cosine is found, being ⚫81167.

PROPOSITION III.

(T) To find the degrees, minutes, or degrees, minutes, and seconds, corresponding to any given logarithmical 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 one 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 co-sine, 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 logarith

mical sine 9.43299.

Nearest sine less than the given one 9.43278
Nearest sine greater than the given one 9.43323

21

60

First difference 45 45 | 1260

Given sine 9.43299

Nearest less 9.43278 answering to 15°. 43'

[blocks in formation]

28 quot.

The

The same manner of proceeding must be observed in finding tangent, secant, or natural sine.

Find the degrees, minutes, and seconds, corresponding to the logarith mical 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

Nearest less 9.43278 answering to 74°.17'.

25quot.

Second difference 19

Therefore the required arc is 74°. 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 sine Required the natural versed sinė of 65°. 45'.

Radius =1

Nat. cosine 65°. 45′ = 41072

versed sine 65°. 45′ = 58928

of 1159.35'.

Natural cosine 115°. 35' or cosine 64°. 25′ = *43182 To which add 1•

vers. sine of 115°.35′=1·43182

Required the logarithm. versed Required the log. versed sine of

sine of 72°. 14 Logarithmical sine 36°.7′ half arc

➡9.77043

37°.53'

2

Here half the arc is 18°.56%
Log. sine 18°.56' 9.51117
Log. sine 18°.57′ =9•51154

19.54086

9.69897

double sine 18°.56-19.02271

9.69897

Log.vers.sine72°.149-84189 Log.vers. sine 37°53′=9.32374

CHAP.

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 the logarithms of these numbers.

(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 A1, A2, A3, &c. to the straight line AD, and it will be a line of rhumbs containing eight points of the compass, or onefourth of the whole circumference of the compass. By subdividing each of the divisions A1; 1, 2, &c. into four equal parts, and transfering them in the same manner to the line AD, it will contain the points, and half and quarter points.

(Z) Chords. Divide the arc BD 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, B 10; 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 CB will be divided into a line of sines reckoning from c to (for CG is the cosine of the arc B 80, or the sine of the arc D 80, 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 line of tangents, to the line DF, and it will become a line of secants which must be numbered from D towards F, aý 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

« PreviousContinue »