subtended an angle of 2'.-Required the distance from the last place of observation, also the height and diameter of it. M. Q. R.

Answer.-The distance, 5. 3.. 60

Height, 3 0 .. 70
Diameter,

18 feet 2 inches.

3. From the top of a steeple 165 feet high, the angle of depression of the nearest bank of a river is 11° 15', that of the opposite bank is 6° 15'. Required the width of the river. Answ. 41.13 rods.

4. What length of cart-tire will it take to band a wheel 5 feet in diameter? Answ. 15 feet 8 1-2 inches.

5. A gentleman laid out a garden in a circle, containing one acre, one quarter, and one rod, with a gravelled walk on the outer side of it within the circle which took up twelve rods of ground. What is the diameter of the circle, and what is the width of the walk?

Answ. The diameter 16 rods-Width of the walk 4 feet. 6. Neptune laid out 1000 square miles of the surface of the sea in a circle, and sold to Aeolus all that part of it which lies without a concentric circle of ane third of the diameter. What is the diameter, and how much was sold? Answ. The diameter 35.68 miles. The quantity sold 888.92 square miles. 7. A Farmer laid out an elliptical orchard, the longest diameter of which was 30 rods, and the shortest was 20 rods, and surrounded the same with a wall two feet thick, within the figure. What is the quantity within the wall, and how much is covered by it? A. Q. R. Within the wall 2 3 22 Covered by the wall, 9.3 rods.

Answ.

8. From a point in an equilateral triangle, I measured the distances to each corner, and found them 20, 29, and 30 rods. Required the area and the length of the sides.*

A. Q. R.
Answ. The Area 5 1
..33
Length of each side 45 rods.

9. Required the dimensions of a parallelogram, containing one acre and a half, bounded by 64 rods of fence. Answ. 12 by 20 rods.

10. The area of a parallelogram is five acres one quarter and thirty-five rods, and the diagonal is forty-three rods. Required the length of the sides. Answ. 35 by 25 rods.

11. Required the dimensions of a parallelogram containing twenty-six acres one quarter and twenty-four rods, when the length exceeds the breadth by fifty-two rods. Answ. 44 by 96 rods.

12. Required the dimensions of a parallelogram containing 250 acres, when the sides are in the proportion of 7 to 3. Answ. 130.93 by 305 1-2. 13. The state of Connecticut contains a little upwards of 4828 square miles, or 3,090,000 acres, including rivers, harbours, creeks, roads, &c. if this quantity of land is laid in a square, what will be the length of each side? M. Q. R. Answ. 69.. 1 75.11

Note. In the Preface, it is observed that the Traverse Table in this book is calculated for any distance up to 50. After the Preface was printed, it was thought best to extend that Table to 70. The table of Logarithms is also much more extensive, than is noticed in the Preface.

MATHEMATICAL TABLES.

VIZ:

I. A Table of Logarithms for Numbers.

II. A Table of Logarithmic or Artificial Sines, Tangents, and Secants.

III. A Traverse Table, or Table of Difference of Latitude and Departure.

IV. A Table of Natural Sines.

I. A TABLE OF LOGARITHMS FOR NUMBERS.

Logarithms are Numbers in Arithmetical Progression, corresponding to other Numbers in Geometrical Proportion.

As,

0.
1.

1. 10.

2. 3. 4. Logarithms. 100. 1000. 10000. Numbers.

The Logarithm for any Number less than 10 is a certain number of Decimals; for any Number between 10 and 100, it is 1 with Decimals; for any Number between 100 and 1000, it is 2 with Decimals, &c. The whole Number in Logarithms, or the Number which stands at the left hand of the Decimal point, is called the Index; and is always a unit less than the places of figures in the whole Number for which it is the Logarithm: Thus,

The Log. of a Decimal Fraction is the same as that of an Integer, only the Index is negative, and is distinguished from a positive one, by placing a Point, or a negative Sign before it : Thus,

Note. In the following Table the Index is not prefixed. It may be easily supplied as it is always a unit less than the number of figures in the corresponding natural whole number.

To find the Logarithm of any Number.

If the Number is less than 100, its Log. is found in the first page of the Table, directly opposite thereto : Thus, the Log. of 34 is 1.53148.

If the Number consists of three figures, find it in the first column of the following part of the Table, opposite to which, and under 0, is the Log. Thus the Log. of 346 is .53908 to which prefix 2 for the Index, because there are three places of figures in the whole Number.

If the given Number contains four figures, the first three are to be found, as before, in the side column, and under the fourth at the top of the table is the Log. to which the Index 3 is to be prefixed, if the given Number is an Integer: Thus the Log. of 3467 is .53995 to which prefix 3 for the Index.

If the given Number exceeds four figures, find the difference between the Log. of the first four figures, and the next following Log. Multiply this difference by the remaining figure or figures in the given Number; point off as many figures to the right hand as there are in the multiplier; and the remainder, added to the Log. of the first four figures, will be the required Log.

To find the Number corresponding to any given Logarithm.

Find the next less Log. to that given in the column marked O at the top, and continue the sight along that horizontal line, and the Log. the same as that gives, or very near it, will be found; then the first three figures of the corresponding Number will be found opposite, in the first side column, and the fourth figure directly above, at the top of the page. If the Index of the given Log. is 3, the four figures thus found are whole numbers; if the Index is 2, the first three figures are whole numbers, and the fourth is a Decimal, and so on.

To find the nearest number corresponding to any Log. for more than four figures, find the Log. next less than the given one, and take the difference between that and the given one; also take the difference between the next greater and the next less Log. than the given one; divide the former difference by the latter, according to the Rule in Division of Decimals for dividing a less number by a greater; add the Quotient to the number answering to the Log, next less than the given one, and you will have the required Number; whether a whole, or a mixed Number will be determined by the Index.

The addition and subtraction of Logarithms answers the

ponding Numbers: That is, the Log. of any two Numbers being added, their sum will be the Log. of the Product of those Numbers; and the Log. of one Number being subtracted from the Log. of another Number, the Remainder will be the Log. of the Quotient of one of those Numbers divided by the other. Again, the Log. of any Number being doubled will produce the Log. of the Square of that number; and one half the Log. of any Number is the Log. of the Square Root of that Number.

II. Of the Table of Logarithmic or Artificial Sines, Tangents, and Secants.

To find the Logarithmic Sine, &c. for any number of Degrees and Minutes, within the compass of the Table.

If the Degrees be less than 45, look for them at the top of the columns, and under Sine, Tangent or Secant, whichever is wanted, and for the Minutes at the left hand; but if more than 45, look for the Degrees at the bottom over Sine, &c. and for the Minutes at the right hand; under or over the Degrees and against the Minutes will be the required Log. Sine, &c.

To find the Degrees and Minutes corresponding to a given Logarithmic Sine, &c.

Look in the proper column for the nearest Log. to the given one; and the Degrees and Minutes standing over or under and against it, are those required.

Note. When the Log. Sine, &c. for more than 90° is re- quired, subtract the given number of Degrees from 180°, and make use of the Remainder.

It will be observed that this Table is calculated only for every 5 Minutes. This was thought sufficient for Surveyors, as few Compasses will take a course to greater exactness. If, however, a Question is to be solved where greater accuracy is required, work by natural Sines. Or,

The Log. Sine, &c. for any Minute may be found as follows: Look in the Table for the Log. of the nearest number of Minutes greater than the given one, and from this subtract the next less Log. contained in the Table: Then say, as 5 Minutes, is to this difference; So is the excess of the given Minutes above 5, 10, 15, 20, 25, &c.; To a fourth number, which add to the Log. of the Minutes next less than the given number, and

To find the nearest Minutes corresponding to a given Logarithmic Sine, &c.

Look in the Table, in the proper column, for the Log. next less than the given one, and take the difference between that and the given one; also take the difference between the next greater and the next less Log. than the given one; Then say, As the latter difference; is to 5 Minutes; so is the former difference; to the number of Minutes to be added to the Minutes of the Log. next less than the given one.

EXAMPLE.

Required the Degrees and Minutes corresponding to the Logarithmic Tangent 9.73597.

The Degrees and Minutes for the Log. next less than the given one are 28° 30' to which add 4' and it makes 28° 34'.

Note. As after the most careful attention of the Printers, some figures in the Table may be wrong; and as some may be so blurred as to be illegible, let it be observed, that the Sines and Co-Secants, the Co-Sines and Secants, and the Tangents and Co-Tangents, standing against each other respectively, being added together, will amount to 20.00000, if the Tables are accurate. Thus against 28°