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19. The diameter of a circular enclosure is 89 yards: how many acres does it contain?

Ans. 1 ac. 1 ro. 5.7 — po. 20. What is the area of a circle whose circumference is 32.25? Ans. 82.77.

21. Find the solid content of a cylinder whose length is 42 inches, and diameter 15 inches. Ans. 4.295 + c. feet. 22. Compute the solid content of a regular triangular pyramid, whose perpendicular height is 6.99, each side of the base being 4.18. Ans. 17.63-. 23. The fixed axis of a prolate spheroid is 7·74 inches, and the revolving axis 5.88 inches. What is its solid content? Ans. 140.1 + c. inches. 24. If a cannon ball, 3 inches in diameter, weigh 6 lb., what will be the weight of another ball of the same metal, 6.7 inches in diameter? Ans. 42.09-lb.

CHAPTER IV.

OF LOGARITHMIC SCALES.

The Logarithmic Scales, of various kinds, are merely rules on which the lengths of logarithms, taken from a table, are set off by measurement, marked with the corresponding natural numbers.* Thus, the figure 6 is marked at the termination of the length of the logarithm of 6. All the lengths commence at one point, which is marked with a unit, since the logarithm of 1 is 0.

Consequently, logarithms of numbers are added together by uniting their lengths upon the scale; and one is subtracted from another by measuring the length of the former backward from the termination of the latter.

Consequently, also, natural numbers are multiplied together by uniting the lengths of their logarithms on the scale, and observing the marked number to which their united length reaches; and one number is divided by another, by measuring the length corresponding to the latter backward from the termination of the length corresponding to the former.

Sliding Rules consist of two such logarithmic scales

Except the Trigonometrical scales of sines, tangents, &c., which are not marked with the natural numbers, but with the corresponding degrees.

sliding along each other, with their graduated edges in close contiguity.

In using the single logarithmic scale, one of the two given lengths requires to be measured with the compasses; but the sliding rule renders this unnecessary, since the one length is added to the other or subtracted from it by mere position, that is, by bringing the commencement of the former to the termination of the latter in the one case, and, in the other, by placing the two terminations together.

For scales, Briggs's system of logarithms alone is used, in order that the length from 10 to 100, with its divisions, may be an exact counterpart of that from 1 to 10; since, in that system, as before explained, the logarithm of any number is the same as the logarithm of ten times that number, except that the index differs by a unit.

Logarithmic scales in their usual and simplest form, are called, from their inventor, Gunter's Scales.

PROBLEM I.

To find the Product of two Numbers by Gunter's Scale.

RULE. Extend the compasses from 1 to the smaller of the two factors on the line of Numbers. Keep the distance unaltered in the compasses, and it will extend forward from the other factor to the number indicating the product.*

EXAMPLE 1. Multiply 9 by 6.

Operation. Place the compasses on the scale with the left foot on 1 and the right foot on 6. Remove the compasses, without altering the distance, and place the left foot on 9; the right foot will reach to 54, which is the Answer.

NOTE. We cannot measure a number to greater nicety than three figures on the best scales, and scarcely even so

* The numbers on the first half of the scale must be regarded as units, and those on the second half as tens. When the first figures of the given numbers are not units (or tens in division) the operation requires particular rules, which will be found in the Author's "Treatise on Logarithms and Plane Trigonometry."

far if the first figure is large. If, therefore, the given numbers consist of more than three figures, we must take the third to the nearest figure, whether more or less, and omit the subsequent figures. Thus, for 7-942 we take 7.94, and 34291 we take 34300. In the results we must also be satisfied with three figures, filling up the places of the remainder with ciphers when necessary.

EXAMPLE 2. Multiply 3.811 by 7.428.

Operation. 3.81 x 7.43 28.3.

=

Ans. 28.3.

COR. The product of two numbers is found by the Sliding Rule thus :-Calling one of the two scales A and the other B, place 1 of scale B adjacent to one of the factors on scale A: then the other factor on scale B will be adjacent to the product on scale A.

EXERCISE 1. Multiply 3 by 7, and 9 by 9.

Answers: 21 and 81.

2. Multiply 2.5 by 5.4, and 1.25 by 9.6.

Answers: 13.5 and 12.

3. What is the product of 3.85 by 7.62, and of 6.542 by 3.669? (See Note). Answers: 29.3 and 24.

PROBLEM II.

To divide one Number by another by Gunter's Scale.

RULE. Extend the compasses on the scale from 1 to the divisor. Keep the distance unaltered in the compasses, and it will extend backward from the dividend to the quotient.

EXAMPLE 1. What is the seventh part of 63 ?

Operation. Place the left foot of the compasses on 1 and the right on 7. Remove the compasses without altering the distance, and place the right foot on 63. The left will reach 9, which is the Answer.

COR. The same operation is performed by the Sliding Rule, thus-Place the divisor on scale B adjacent to the

dividend on scale A. The quotient will be on scale A adjacent to 1 on scale B.

EXERCISE 1. Divide 84 by 7, and 7·02 by 3.9.

Answers: 12 and 1.8.

Answers: 341, and 8.4.

2. Divide 7.9794 by 2.34, and 66.36 by 7.9.

PROBLEM III.

To find the Fourth Term of a Proportion by Gunter's Scale.

RULE. Extend the compasses on the scale from the first term to the second. The same stretch will reach from the third to the fourth.

Or:-Extend the compasses from the first term to the third. That extent will reach from the second to the

fourth.

EXAMPLE 1. What number bears the same proportion to 5, that 12 bears to four?

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Place the left foot on 4 and the right on 12. Keeping the same distance in the compasses, place the left on 5: the right will reach 15.

Or:-Place the left foot on 4 and the right on 5. That extent will reach from 12 to 15.

EXAMPLE 2. Find the fourth term of the proportion, 21: 27 :: 14:

Operation. As 21: 27 :: 14: 18.

Ans. 18.

COR. The same operation is performed by the Sliding Rule, thus:

Place the first term on B adjacent to the second on A: then the third on B will be adjacent to the fourth on A. Or,

Place the first term on B adjacent to the third on A: then the second on B will be adjacent to the fourth on A.

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2. Some triangles are delineated on a plane, others on a spherical surface; and, accordingly as Trigonometry relates to the former or to the latter class of triangles, it is distinguished by the names of Plane and Spherical Trigonometry.

3. Plane Trigonometry is not exactly what might be inferred from the etymology of its name. Properly explained, it is that department of Mathematics which treats of the practical computation of the sides and angles of plane triangles from each other. It does not include the computation of the areas of triangles, which belongs to Mensuration of Surfaces; and a few problems relating to the sides and altitudes of triangles, without reference to the angles, may be assigned either to Trigonometry or to Mensuration of Lines, more properly perhaps to the latter; while many theorems of similar reference are invariably assigned to Geometry.

4. The circumference* of every circle is supposed to consist of 360 equal parts, called Degrees; a degree, to consist of 60 Minutes; a minute of 60 Seconds; a second of 60 Thirds, &c.

With the common geometrical terms the scholar ought to be acquainted before commencing this volume.

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