contain the divisor an exact number of times, increase it by so many as are necessary to make it do so, and carry the number so borrowed, as so many tens to the first figure of the decimal. Ex. 1. Extract the fourth root of 56.372. 56.372 Result, 2.7401 log. 4)1.751063 .437766 Ex. 2. Extract the fifth root of .000763. .000763 Result, .23796 log. 5)-4.882525 Ex. 3. What is the fifth root of .00417 ? Ex. 7. Required the fifth root of .84392. Ans. .3342. Ans. .80455. Ans. 2.479. Ans. .6449. Ans. .96663. Ans. .40429. 17. Arithmetical Complements. When several numbers are to be added, and others subtracted from the sum, it is often more convenient to perform the operation as though it were a simple case of addition. This may be done by conceiving each subtractive quantity to be taken from a unit of the next higher order than any to be found among the numbers employed; then add the results with the additive numbers, and deduct from the result as many units of the order mentioned as there were subtractive numbers. The difference between any number and a unit of the next higher order than the highest it contains is called the arithmetical complement of the number. Thus, the arithmetical complement of 8765 is 1235. It is easily obtained by taking the first significant figure on the right from ten, and each of the others from nine. This may be done mentally, so that the arithmetical complements need not be written down. Thus, suppose A started out with 375 dollars to collect some bills and to pay sundry debts. From B he received $104, to D he pays $215, to E he pays $75, from F he receives $437, and, finally, pays to G $137. How much has deducting 3000 from the final result 3489, because there were three subtractive quantities. The arithmetical complements of logarithms are generally employed where there are more subtractive logarithms than one. To give symmetry, to the result, it would be neater to employ them in all cases. To a person who has much facility in calculation, it is most convenient to write down the logarithm as taken from the table, and obtain the arithmetical complement as the work is carried on. Thus, in the example above, the numbers could be written as in the first column; but in the addition, instead of employing the figures as they appear in the subtractive number, the complement of the first significant figure to ten, and of the others to nine, should be employed. As an example of the use of the arithmetical complements of the logarithms of numbers, let it be required to 27 475 work by logarithms the proportion as : :: 125: x. 55 17 Here, as the first term is a fraction, it will have to be inverted; and the question will be the same as finding the 55 x 475 x 125 value of 27 × 17 (1.431364) which are (A. C. 8.568636 added as A. C. 8.769551 2.676694 they were written 1.740363 2.676694 2.096910 3.852154 deducting 20, because there were two arithmetical complements employed. In the examples wrought out in the subsequent part of this work, the arithmetical complements of the logarithms of the first term of every proportion are employed. CHAPTER II. PRACTICAL GEOMETRY. SECTION I. DEFINITIONS. 18. THE practical surveyor will find a good knowledge of Algebra and of the Elements of Geometry an invaluable aid not only in elucidating the principles of the science, but in enabling him to overcome difficulties with which he will be certain to meet. In fact, so completely is Surveying dependent on geometrical principles, that no one can obtain other than a mere practical knowledge of it, without first having mastered them; and he who depends solely on his practical experience will be certain to meet with cases which will call for a kind of knowledge which he does not possess, and which he can obtain only from Geometry. Every student, therefore, who desires to become an intelligent surveyor, should first study Euclid, or some other treatise on Geometry. He will then have a key which will not only unlock the mysteries contained in the ordinary practice, but which will also open the way to the solution of all the more difficult cases which occur. To those who have taken the course above recommended, the problems solved in the present chapter will be familiar. They are inserted for the benefit of those who may not be thus prepared, and also as affording some of the most convenient modes of performing the operations on the ground. 19. Geometry is the science of magnitude and position. 20. A solid is a magnitude having length, breadth, and thickness. All material bodies are solids, and so are all portions of space, whether they are occupied with material substances or not. Geometry, treating only of dimension and position, has no reference to the physical properties of matter. 21. The surfaces of solids are superficies. A superficies has, therefore, only length and breadth. 22. The boundaries of superficies, and the intersection of superficies, are lines. Hence, a line has length only. 23. The extremities of lines, and the intersections of lines, are points. A point has, therefore, neither length, breadth, or thickness. 24. A point, therefore, may be defined as that which has position, but not magnitude. 25. A line is that which has length only. 26. A straight line is one the direction of which does not change. It is the shortest line that can be drawn between two points. 27. A superficies has length and breadth only. 28. A plane superficies, generally called simply a plane, is one with which a straight line may be made to coincide in any direction. 29. A plane rectilineal angle, or simply an angle, is the inclination of two lines which meet each other. (Fig. 1.) Fig. 1. B 30. An angle may be read either by the single letter at |