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 fig ures 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.:

PRELIMINARY CHAPTER

TO THE

PRACTICAL DIALLING EXAMPLES.

IT must have been matter of regret to every reflecting well-informed, and interested person, that (previous to the present work) nothing has ever been published with a design to assist the miner in his subterraneous operations; and while the press has teemed with publications, distinctly and exclusively adapted to benefit the navigator, the architect, the sculptor, the surveyor, and even the mechanic and artisan, not a single effort has ever been made to extricate the miner from the disadvantages under which he has ever labored, (solely for the want of a plain, concise, technical, and scientific treatise on dialling, accompanied with appropriate tables,) although his profession yields to none in importance and utility; in fact, it may be said, in a certain sense, to be the parent of every art and science in the world; the use of metallic substances, in some shape or other, being indispensable in every one of them: nevertheless this highly essential art has hitherto been totally disregarded by all classes of mathematicians, and while the famous invention of logarithms has caused the science of trigonometry to soar to the very skies, and traverse old ocean's vast and unfathomable expanse, the unsupported miner has been left to struggle under the greatest disadvantages, with nearly as little obligation to geometrical science, as his antediluvian progenitors; and although he has done every thing that deep thought, strong natural understanding, unwearied perseverance, and inventive genius (unassisted by trigonometrical demonstration) could possibly accomplish, yet, for the want of mathematical light, his exertions have been ineffectual and insufficient to disentangle him from the difficulties with which he has been encircled; hence his avocation has, in general, been replete with toil, anxiety, apprehension, dissatisfaction, and disappointment.

How far the present work is adapted to answer the great end in contemplation, must be left for the judgment of the mining world to decide; and we doubt not but the defects (real or imaginary) which may be considered to exist in the application, will be passed over and excused by every liberal man, on the grounds already stated in the preface; having an unshaken confidence that the fundamentals of the work, comprised in the trig onometrical tables, will be found plain, true, and unexceptionable.

Definition of Right-angled Triangles.—In order to use the following tables with due effect, there is no necessity that the reader should understand any thing of the science of trigonometry, that part of the work having been accomplished already to his hand; so that, by the help of a few of the common rules of arithmetic, he may obtain, with the greatest ease and certainty, every thing required to be known in the geometrical part of mining.

Previous to an elucidation of the simple method of working by the tables, it may be satisfactory to introduce the operation by a few preliminary observations and extracts on the nature and properties of right-angled triangles.

Plane trigonometry is the art of measuring the sides and angles of triangles described on a plane surface, or of such triangles as are composed of straight lines.

The theory of triangles is the very foundation of all geometrical knowledge, for all straight-lined figures may be reduced to triangles. The angles of a triangle determine only its relative species, and are measured in degrees, minutes, and seconds; but the sides determine its absolute magnitude, and may be expressed in fathoms, yards, feet, or any other lineal measure.

Theorems.—A right-angled triangle (the only kind generally necessary to be treated of for mining purposes) is that which has one right angle in it; the longest side, or that opposite to the right angle, is called the hypothenuse, the other two are called the legs or sides, or the base and perpendicular: or, by Euclid's definition, "In a right-angled triangle, the side opposite to the right angle is called the hypothenuse, and of the other sides, that upon which the figure is supposed to stand is called the base, and the remaining side the perpendicular.'

The three angles of every triangle are together equal to two right angles, or 180 degrees.