cube root, &c. 66 difference between two numbers, when it is not known which is the greater. Thus, 5+3, denotes that 3 is to be added to 5. 7 × 3, or 7. 3, denotes that 7 is to be multiplied by 3. 2:34:6, shows that 2 is to 3 as 4 is to 6. 6 +4 = 10, shows that the sum of 6 and 4 is equal to 10. √3, or 3*, denotes the square root of the number 3. 13/5, or 53, denotes the cube root of the number 5. 72, denotes that the number 7 is to be squared. 83, denotes that the number 8 is to be cubed. Et cetera. THE word Surveying, in the mathematics, signifies the art of measuring land, and of delineating its boundaries on a map. The Surveyor, in the practice of this art, directs his attention, at first, to the tracing and measuring of lines; secondly, to the position of these lines in respect to each other, or the angles formed by them; thirdly, to the plan, or representation of the field or tract which he surveys; and fourthly, to the calculation of its area, or superficial content. When this art is employed in determining the variation of the compass, in observing and delineating coasts and harbours, their latitude, longitude, and soundings, together with the bearings of their most remarkable plaçes from each other, it is usually denominated Maritime Surveying. This branch of Surveying, however, demands no other qualifications than those which should be thoroughly acquired by every land-surveyor who aspires to the character of an accomplished and skilful practitioner. Surveying, therefore, requires an intimate acquaintance with the several parts of the mathematics which are here inserted as an introduction to this treatise. PART I. Containing Decimal Fractions, Involution and Evolution, the Nature and Use of Logarithms, Geometry, and Plane Trigonometry. SECTION I. DECIMAL FRACTIONS. If we suppose unity or any one thing to be divided into any assigned number of equal parts, this number is called the de nominator; and if we choose to take any number of such parts less than the whole, this is called the numerator of a fraction. The numerator, in the vulgar form, is always written over the denominator, and these are separated by a small line thus 2, or §; the first of these is called three-fourths, and the latter five-eighths, of an inch, yard, &c., or of whatever the whole thing originally consisted: the 4 and the 8 are the denominators, showing into how many equal parts the unit is divided; and the three and the five are the numerators, showing how many of those parts are under consideration. Fractions are expressed in two forms, that is, either vulgarly or decimally. All fractions whose denominators do not consist of a cipher or ciphers, set after unity, are called vulgar; and their denominators are always written under their numerators. The treatment of these, however, would be foreign to our present purpose. But fractions whose denominators consist of a unit prefixed to one or more ciphers, are called decimal fractions; the numerators of which are written without their denominators, and are distinguished from integers by a point prefixed; thus, 42, 12, in the decimal form, are expressed by .2, .42, .172. 10009 300 The denominators of such fractions consisting always of a unit prefixed to as many ciphers as there are places of figures in the numerators, it follows, that any number of ciphers put after those numerators, will neither increase nor lessen their value: for,, and are all of the same value, and will stand in the decimal form thus .3, .30, .300; but a cipher or ciphers prefixed to those numerators lessen their value in a tenfold proportion: for,, and, which in the decimal form we denote by .3, .03, and .003, are fractions, of which the first is ten times greater than the second; and the second, ten times greater than the third. Hence it appears, that as the value and denomination of any figure, or number of figures, in common arithmetic is enlarged and becomes ten, or a hundred, or a thousand times greater, by placing one, or two, or three ciphers after it; so in decimal arithmetic, the value of any figure, or number of figures, decreases and becomes ten, or a hundred, or a thousand times less, while the denomination of it increases, and becomes so many times greater, by prefixing one, or two, or three ciphers to it and that any number of ciphers before an integer, or ADDITION OF DECIMALS. Write the numbers under each other according to the value or denomination of their places; which position will bring all the decimal points into a column, or vertical line, by themselves. Then, beginning at the right-hand column of figures, add in the same manner as in whole numbers, and put the decimal point in the sum directly beneath the other points. EXAMPLES. Add 4.7832, 3.2543, 7.8251, 6.03, 2.857, and 3.251 together. Place them thus, Add 6.2, 121.306, .75, 2.7, and .0007 together. What is the sum of 6.57, 1.026, .75, 146.5, 8.7, 526., 3.97, and .0271? Answer, 693.5431. |