rable degree of certainty, by the foregoing method. There doubt. less are instruments much better adapted to the work, both for speed and accuracy, than the dial; and it is matter of surprise that they have not been more generally introduced in our mines: of these instruments the Theodolite certainly stands unrivalled for taking both horizontal and vertical angles.

It is not our design to enter into controversy on this subject ; those who imagine the sextant or quadrant graduated on the cover of the dial well calculated for the purpose, let them continue to use it; only we would especially note, that should an error ensue, it ought by all means to be attributed to the real cause, and to that only: for, as in all trigonometrical questions, the angle and side are always given to find the other parts of the triangle, consequently the sum of the one and length of the other are presupposed to have been correctly ascertained, previous to the commencement of any other operation.

Finally, for the learner's sake, we observe, that as the tables exhibit only the relative proportions to the radius of one fathom, or six feet, and are wrought out to five places of decimals to an inch, it becomes necessary that every one who would use this work successfully should have some knowledge of decimated arithmetic; because he will have, in most cases, to multiply for the whole numbers, and take parts for the fraction of the fathom. For example: suppose the given side to be the hypothenuse, measuring 16 fathoms, 3 feet, and 6 inches, he will then have to take out the numbers opposite the given angle in the tables, and multiply them by 16, for the base and perpendicular respectively, then divide half the tabular measure for the 3 feet, and one-sixth of the remainder for the 6 inches, and add them together for the sum of the required sides of the triangle. We have therefore introduced the following rules and examples in decimals, which are sufficient to enable any one, hitherto unacquainted with this branch of arithmetic, to use the tables with the greatest facility.

REDUCTION OF DECIMALS.

Rule.-Multiply the decimal by the number of parts in the next less denomination, and cut off as many places to the right hand as there are places in the given decimals.

Rule.-Place the numbers so that the decimal points may stand directly under each other, add up as in simple addition, and cut off for decimals, as many figures to the right as there are decimals in the greatest given number.

EXAMPLE.

What is the sum of 3·72 and 14·7368 and 146·2 and ·728 and 5.034?

3.72 14.7368

146-2

.728 5.034

170-4188

Rule. Arrange and cut off the decimals as in addition.

Rule.-Multiply as in whole numbers, and cut off as many figures from the product as there are decimals in the multiplier and multiplicand.

DIVISION OF DECIMALS.

Rule.-Divide as in whole numbers, and cut off as many figures in the quotient as the decimal places in the dividend exceed those of the divisor.

It has been observed that the radius in every case is 6 feet or 1 fathom, consequently the number of fathoms in the given side, whether that side be hypothenuse, perpendicular, or base, will be the multiplier of the tabular numbers, and should there be a fraction in the multiplier, the multiplicand must be divided by that fraction agreeably with the rule of practice. The table of aliquot parts of a fathom, in the adjoining page, will be found useful in facilitating this part of the process.

In some of the following examples the product has been obtained in fathoms and parts; but we would recommend the learner to carry on the work in feet (except in cases where the answer is required in fathoms), as he will find it more simple and expeditious; we speak of the multiplicand or number multiplied; the multiplier must invariably be fathoms, and should the given side be nominated in feet, it must be divided by 6, to bring it

into fathoms, before the operation is begun by the foregoing

cases.

It may be further noticed that when any of the given sides in the tables amount to 6 feet, they are expressed in fathoms, &c.; but whenever it may be required to produce the answer in feet, &c., the numbers should be reduced to that measure before they are multiplied, and this can be done by mere inspection; viz.: