The operation of subtracting 10 is always performed mentally. NOTE 1.—In finding any term of a proportion by logarithms -observe that, 1. The sum of the logarithms of the extremes, is equal to the sum of the logarithms of the means : 2. The logarithm of the fourth term, is equal to the arithmetical complement of the logarithm of the first term added to the logarithms of the mean terms: 3. The logarithm of either mean term, is equal to the arithmetical complement of the logarithm of the other mean added to the logarithms of the extremes. a : с : NOTE 2. If any proportion, as X, be changed into the form of an equation, thus, bc, then AX = = The arithmetical complement of the logarithm of the multiplier of the unknown factor, plus the logarithms of the two other factors, minus 10, is equal to the logarithm of the unknown factor. NOTE 3.-If the logarithm, whose arithmetical complement is taken, exceeds 10, subtract it from 20, and reject 20 in the final operation. RAISING TO POWERS BY LOGARITHMS. 18. To raise a number to any power. From the principle proved in (Art. 7), we have the following RULE.— Find the logarithm of the number, and multiply it by the exponent of the power ; then find the number corresponding to the resulting logarithm, and it will be the power required. EXTRACTING ROOTS BY LOGARITHMS. 19. To find any root of a number, from the principle proved in (Art. 8), we have the following: RULE.—Find the logarithm of the number, and divide it by the index of the root; then find the number corresponding to the resulting logarithm, and it will be the root required. EXAMPLE. 1. Find the cube root of 4096. The logarithm of 4096 is 3.612360, and one-third of this is 1.204120. The corresponding number is 16, which is the root sought. SECTION II. GEOMETRICAL CONSTRUCTIONS. 20. Before explaining the method of constructing geometrical problems, we shall describe some of the simpler instruments and their uses. 21. The dividers is the most simple and useful of the instruments used for drawing. It consists of two legs ba, bc, which may be easily turned around a joint at b. One of the principal uses of this instrument is to lay off on a line, a distance equal to a given line. For example, to lay off on CD, a distance equal to AB. B For this purpose, place the forefinger on the joint of the dividers, and set one foot at A: then extend, with the thumb cland other fingers, the other leg of the dividers, until its foot reaches the point B. Then raise the dividers, place one foot at C, and mark with the other the distance CE: this will evidently be equal to AB. RULER AND TRIANGLE. 22. A Ruler of convenient size is about twenty inches in length, two inches wide, and a fifth of an inch in thickness. It should be made of a hard material, perfectly straight and smooth. The hypothenuse of the right-angled triangle, which is used in connection with it, should be about ten inches in length, and it is most convenient to have one of the sides considerably longer than the other. We can solve, with the ruler and triangle, the two following problems. I. To draw through a given point a line which shall be parallel to a given line. C B 23. Let C be the given point, and AB the given line. Place the hypothenuse of the triangle against the edge of the ruler, and then place the ruler and triangle on the paper, A so that one of the sides of the triangle shall coincide exactly with AB: the triangle being below the line. Then, placing the thumb and fingers of the left hand firmly on the ruler, slide the triangle, with the other hand, along the ruler, until the side which coincided with AB reaches the point C. Leaving the thumb of the left hand on the ruler, extend the fingers upon the triangle and hold it firmly, and with the right hand, mark with a pen or pencil, a line through C: this line will be parallel to AB. II. To draw through a given point a line which shall be perpen dicular to a given line. 24. Let AB be the given line, and D the given point. Place the hypothen use of the triangle against the edge of the ruler, as before. Then place the ruler and triangle so that one of the sides of the triangle shall coincide exactly with the line AB. Then slide the triangle along the ruler until the other side reaches the point D: then, draw through D, a right line, and it will be perpendicular to A B. А B 25. A scale of equal parts is formed by dividing a line of a given length, into equal portions. If, for example, the line ab, of a given length, say one inch, be divided into any number of equal parts, as 10, the scale thus formed is called a scale of ten parts to the inch. The line ab, which is divided, is called the unit of the scale. This unit is laid off several times on the left of the divided line, and the points marked 1, 2, 3, &c. The unit of scales of equal parts is, in general, either an inch, or an exact part of an inch. If, for example, ab, the unit of the scale, were half an inch, the scale would be one of 10 parts to half an inch, or of 20 parts to the inch. If it were required to take from the scale a line equal to two inches and six-tenths, place one foot of the dividers at 2, on the left, and extend the other to .6, which marks the sixth of the small divisions: the dividers will then embrace the required distance. |