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

Answer, 12 by 20 rods.

10. The area of a parallelogram is five acres one quarter and thirty-five rods, and the diagonal is 43 rods. Required the length of the sides.

Answer, 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.

Answer, 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. Answer, 130.93 by 3051 rods.

13. John and Jonathan divided a lot of land equally as to value, which contained 600 acres, and which was appraised at $2 per acre. John's division was valued at 50 cents per acre more than Jonathan's. What was the contents and price per acre of each division. Answer. John's division contained 263 acres and 12 rods, and the price per acre was $2.28, nearly.

Jonathan's division contained 336 acres, 3 roods, and 28 rods, and the price per acre was $1.78 nearly.

14. The state of Connecticut contains a little more than 4,828 square miles, or 3,090,000 acres, including rivers, harbors, creeks, roads, &c. If this quantity is laid in a square, what will be the length of a side?

A. Q. R.

Answer, 69 1 75.11.

One word of advice to the young surveyor, who is coming forward to be useful in his occupation, will close the appendix.

In the choice of assistants to perform practical operations, never call to your aid Sir Richard Rum. He frequently changes his name to brandy, gin, whiskey, &c. He is treacherous, and he causes the head to whirl, the body to reel, and the foot to stumble.

By his might, the strong man has fallen, and the promising youth has been brought to an untimely grave.

If you employ Sir Richard, your columns of latitude may differ too much for correct work, and your columns of departure may be still worse. You can place no confidence in him, and it is hoped that you will too highly respect your own character to be found in such company.

NOTE.-Such theory as is more curious than useful, how. ever correct it may be, has been excluded from the appendix. The plainest methods have been selected. On some points, the Author has been more minute on account of the fact that many copies of this book are bought by men who do not ex. pect to be surveyors, and who do not place themselves under instructors.

Hebron, (Conn.) August, 1832.

LOGARITHMS.

Let there be a series of numbers, increasing by a common difference, as for instance, by 1, viz.

0, 1, 2, 3, 4,

5,

6, &c. and another, viz. 1, 10, 100, 1,000, 10,000, 100,000, 1000,000, &c. increasing by a common multiplier, 10. The former are the LOGA. RITHMS of the latter.

It will be seen that 1 is added in the upper series, as often as 10 is made a multiplier in the lower. If two logarithms, then, be added, and the numbers below them multiplied, the sum of the logarithms will be the logarithm of the product.

Thus, if the logs. 1 and 2 be added, and the corresponding numbers, 10 and 100 be multiplied, the sum will be 3, and the product 1,000. These numbers may be seen to correspond to each other, in the two series above; there, 3 stands over 1,000, and is, of course, its logarithm. So, if 2 and 4 be added, and 100 and 10,000 multiplied, the sum will be 6, and the product, 1,000,000, which may be likewise seen above to correspond.

Furthermore, if, from a greater logarithm a less be taken, and, at the same time, the number corresponding to the greater be divided by the number corresponding to the less, the remainder will be the logarithm of the quotient.

Thus, if from the log. 5, the log. 3 be subtracted, and, at the same time, 100,000, the number corresponding to 5, be divided by 1,000, the number corresponding to 3, the remainder will be 2, and the quotient 100, which may be seen above to correspond to each other.

With a set of logarithms, then, calculated, not merely for the numbers 10, 100, 1000, &c., but for all numbers whatever, it is plain that we might perform the operations of multiplication and division, by addition and subtraction only. Such logarithms are calculated and arranged in tables for use.

By observing the two ranks of numbers at the head of this article, it will be perceived, that the logarithms increase by a constant addition, (of 1,) and the corresponding numbers by a constant multiplication, (by 10,); and therefore, that when the logarithms are in arithmetical progression, the numbers are in geometrical progression. On this account, logarithms have been defined to be a series of numbers in arithmetical progression, corresponding to another series in geometrical progression.

But it will be seen that the numbers, 10, 100, 1,000, 10,000, &c. are the first, second, third, fourth, &c. powers of 10, and that their corresponding logarithms, 1, 2, 3, 4, &c. are the indices of these powers. Hence a better definition is,

LOGARITHMS OF NUMBERS ARE THE INDICES, EXPRESSING THE POWERS TO WHICH A GIVEN NUMBER MUST BE RAISED, TO PRODUCE THOSE NUM. BERS.

This "given number" is called the RADIX, Or BASE, of the system, and may be any number whatever. The number 10, however, is most con

The logarithm of 1 is 0, and that of 10 is 1. Hence it is evident, that, for numbers between 1 and 10, the logarithm will be between 0 and 1. Of course, it will be a fraction; and, when placed in the tables, it is expressed by a decimal. For numbers between 10 and 100, the logarithm is between 1 and 2; for numbers from 100 to 1,000, between 2 and 3; from 1,000 to 10,000, between 3 and 4, and so on, consisting, of course, in each case, in part of a whole number, and in part of a decimal. From the above, it will be sufficiently evident, that

THE INTEGRAL (OR WHOLE NUMBER) PART OF ANY LOGARITHM, IS A UNIT LESS THAN THE NUMBER OF INTEGRAL PLACES IN THE CORRESPONDING

NUMBER. OR, IT SHOWS HOW FAR THE HIGHEST FIGURE OF THE NUMBER

THE INTEGRAL PART OF A LOGARITHM IS CALLED THE INDEX, OR CHARACTERISTIC, OF THAT LOGARITHM.

As this characteristic is always a unit less than the number of integral places in the corresponding number, it is evident that when this number has but one integral place, the characteristic will be 0; which is the case with the last log. above: If, then, the number be a proper frac tion, or a decimal, and have, therefore, no integral place, or places, the characteristic of the log. on this principle, ought to be less than nothing. Now, since we cannot diminish nothing, so as to render it less than nothing, we employ, whenever it is necessary to use logarithms of this kind, à characteristic with a mark over it, thus, T, 7, 3, 4, &c. The mark shows that the characteristic ought to be less than nothing, and the number, over which it is drawn, informs us, how much it should be less than nothing. In all cases, then, where it is necessary to add the logarithm, we must subtract this characteristic; and in all cases where it is necessary to subtract the logarithm, we must add the characteristic. For, since adding nothing to a number does not alter it, adding less than nothing ought to diminish it. And, since subtracting nothing from a number does not alter it, subtracting less than nothing ought to increase it.*

It will be seen above, that for every division of the number by 10, (or, every removal of its decimal point towards the left) the character. istic becomes a unit less.

* An apology may perhaps be considered necessary for the language here employed. The writer would not be thought to advocate the absurdity that a number may actually be less than nothing; but the phrase is so concise and expressive, and notwithstanding its absurdity, conveys to the mind of one unacquainted with the nature of negative quantities, the idea intended, so much more perfectly, than any explanation we should here have room to make, could do, that it has been thought advisable to employ it. This treatise is intended for practical men, and not for metaphysicians nor scholars. To those who have attended to

By observing these logarithms, it will be seen, that

THE NEGATIVE CHARACTERISTIC OF A LOGARITHM, SHOWS HOW FAR THE FIRST SIGNIFICANT FIGURE OF ITS CORRESPONDING DECIMAL IS DISTANT FROM THE UNIT'S PLACE.

EXPLANATION OF THE TABLE.

When the logarithms of numbers, from 1 upward to any other given number, are calculated, and arranged in a table, they constitute a table of logarithms. Tables of logarithms of great extent have been calculated, but those in common use extend to numbers no higher than 10,000. This is far enough for the purposes of ordinary calculation.

In the first column on the left of each page, are arranged the natural numbers, and to distinguish this column from the others, the letter N is placed over it. Opposite these numbers in the next ten columns, are arranged the logarithms. For the first 100 numbers, however, on the first page of the table, there is but a single row of logarithms, opposite each column of numbers, and there are four rows of numbers on the page.

TO FIND THE LOGARITHM OF ANY WHOLE NUMBER.

If the number be less than 100, look for it in the column headed N, and directly opposite to it, in the column headed Log. will be found its logarithm.

If the number be greater than 100, but less than 1,000, find it as before, in the column headed N, and directly opposite, in the column headed 0, will be found its logarithm. It will be seen, that when the first two figures of several successive logarithms are alike, they are omitted in the table, after having been once inserted, and only four figures are retained. When, therefore, there are but four figures in the logarithm opposite the given number, cast the eye up the blank till you find two more, which prefix to those already found. It will likewise be observ. ed, that for the logarithms of numbers above 100, no characteristic is inserted in the table. But this may easily be supplied, since it is always a unit less than the integral places in the number. For numbers between 100 and 1,000, then, it is 2.

Find the logarithm of 868. Opposite this number in the table, are found the figures 8520. Casting the eye up the blank we meet with 93, which we prefix to 8520, making 938520. The characteristic, 2, being then prefixed, we have the complete logarithm, 2.938520.

If the number be greater than 1,000, and less than 10,000, find the first three figures in the column, headed N, and the fourth at the head of the page. Then, exactly opposite the first three, and in the column headed by the fourth, will be found the last four figures of the required logarithm. To these must be prefixed two others, found as above, in the column headed 0.