m m m Untit m 1. To supply the deficient terms of the regular series. Suppose m of the terms given, then the mth difference, as derived from these, must be taken equal to zero, in conformity with the principle of the last solution. But in general m(m - 1) Antluc = Untm Usmalt Ustm-2 土 1 1.2 and if in this we make x equal to 1, 2, (n - m) we shall have, putting A”u, = 0, Aou, = 0, ... Aoun-- = 0, the equations requisite for the determination of the n m unknown terms, viz. : mam — 1) A"u, = Unti un + +u, = 0 1.2 mm — 1) A" Ug = Unte + Ug = 0 1.2 m(m - 1) Aug = Umts Unti + Ug = 0, 1.2 and so on, till we obtain as many equations as there are unknown or absent terms of the series. For illustration, suppose w, and were given to find Uz. Then, as there is but one deficient term, we have simply A’u, = 4, — 24, + = 0, or u, = } {u, + w }. Again, if u, Ugo Un, Uz, were given to find uz: then 1 A*4 = Us — 4u, + 64, 4u, + 4 = 0, or us = {4{u, + wa) — (+ .)}. 6 Thirdly, suppose th, U2, U3, Us, were given to find uz and Ug. Here Atu, = 0 and A^u2 = 0; or putting their values, Ug 4u4 + 6112 — 4u+ U = 0; Up - 4ug + 64, — 4U, + Ug = 0, which two equations resolved for ug and Un, we obtain , = io to {– 34 + 104, + 5us – 246 }; wg = - 24. + 5u, + 10u, — 3uo). 10 • Untat = = EXAMPLES. u Ex. 1. Given the logarithms of 101, 102, 104, 105, to find that of 103. In this, w, = 2:0043214, Ug = 2:0086002, w = 2 0170333, Ug = 2:0211893. By the method explained we have 1 = Ex. 3. Given the square roots of 1, 2, 3, 5, 6, 7, 8, to find the square roots of 4 and 9. Likewise determine the cube of 10 from those of 7, 8, 9, 11, 12; and from those of 8, 9, 11, 12, 13. Ex. 4. Given the logs. of 10, 11, 12, 14, 16, and 19, to interpolate the logs. of 13, 15, 17, 18. Ex. 5. The expression which gave the following values has its terms at the asterisks deficient: it is required to supply them, as nearly as can be done, by interpolation. They are 3.9956352, 3.9956396, *, 3.9956527, 3.9956571, *, 3.9956659, 3.9956703, 3 9956659. 2. When the terms to be inserted are not those belonging to the equidistant values of a. The value will be approximately given by the general formula 20 (x - 1) x(x 1) (0 — 2) = U Δυ, + 42u + A3u, to... 1.2 1.2.3 the series being continued till A"Uz = 0 occurs as before. For this is only assuming that the series preserves the same law in passing through the intermediate stages between any two terms, that it does in passing from term to term by single steps. EXAMPLES. Ex. 1. Given log sines of 3°4', 3°5', 3°6', 307', 3o8', to find that of 3°6'15". Angles. log. sines. | first diffs. second diffs. third diffs. 0023516 - 0000126 0023390 '0000001 36 8.7330272 - 0000127 ·0023263 37 + 0000004 8.7353535 0000123 ·0023140 38 8.7376675 Whence, u = 8:7283366, Au = .0023516, 4%ll, - '0000126 and A%= 9 - 0000001. Also x = 3°6'151 -34'=2'151 = the equidistant interval being 45 15 1. Consequently, by the formula, uş = , +42 + A^u + A3u= 32 128 8.733609993 log sin 3°6'15" nearly. Ex. 2. Given log sines of 1°, 1°1', 1°2', 1°3', to find that of 1°1'40". 1 1 1 1 1 Ex. 3. From the series find the term which lies 50' 51 52' 53' 54' 1 1 38552503 in the middle between and Ans. 52 53 2024006400 Ex. 4. Given the log tangents of 68°54', 68°55', 68°56', 68°57', to find that of 68°56'20". Ex. 5. Given the natural tangents of the same arcs, to find the natural tangent of 68°56'20": and compare its logarithm with the answer to the last example. Ex. 6. Given the sines of 7034', 7°35', 7°36', 7°37', 7°38', 7°39', to find the sine of 7°37'30". Ex. 7. Find also the same sine, supposing, first, that the sine of 7034' had not been given in the data ; and secondly, that sine 7°40' had been given in addition to the data. n .....=0. 3. When the first differences of a series of n equidistant terms are very small, any intermediate term may be interpolated by the following formula : m(m - 1) + U1 m(m - 1) (m - 2) Uz WA + 1.2 1.2.3 (m - 1) For (1 - 1)" = 1 n(7 –1) (0 – 2) + ....=0; and as Ui, U2, U3, are by hypothesis nearly equal, we have m(m - 1) non — 1) น U2 + и, 1.2 Uz ui = 0 nearly 1.2 n n n + EXAMPLES. Ex. 1. Given the square roots of 10, 11, 12, 13, 15, to find that of 14. Denote them by Un, U2, U3, U4, Ug, and us, of which ug is the term sought, and n = 5 the number of terms given. Hence we have U 5u2 + 10u; 104, + 5u: - Ug = 0, 10uz and hence u, = } {+ 101, — 104, + 51ky – }, or 46 V14 = 18.7083257 = 3:74166514 nearly. 5 Ex. 2. Given the square roots of 37, 38, 39, 41, and 42, to find that of 40. Ex. 3. Given the cube roots of 45, 46, 47, 48, 49, to find that of 50. Scholium. In Ex. 11, p. 285, the general expression for the sum of n terms of the moth order of figurate numbers is given. These numbers are so called, from the cir. cumstance of their capability of being arranged so as to form equilateral triangles, squares, regular pentagons, hexagons, ..; and they are accordingly called triangular, square, pentagonal, hexagonal, ... numbers. When the progression is 1 + 2 + 3 + 4 + ...n, a triangle of that number of balls n in each side may be formed of the sum of the series : when it is 1.+ 3 + 5 +7... (2n — 1), then a square of the side n may be formed : when 1 + 4 + 7 + 10 + (3n — 2) constitutes the series, it will form a pentagon of n balls in each side: when 1 + 5 + 9 + 13 + (4n 3), a hexagon: and generally, when the general term is {gn - (9 -1)}, then a (q + 2)-gonal figure will result qn for every integer value of q. If, now, in any one case q receive all possible integer values from 1 to any specified numbers, a series of polygons, of q + 2 sides each will be formed ; and if they were balls, laid stratum on stratum, they would altogether form a pyramid on a (q + 2)-gonal base ; and the integration of the general term of the (q + 2)-gonal polygonal series would give the number of balls in the pyramid. Amongst all these there are only two forms that are capable of being used for the piling of balls, on account of the balls pressing unequally on the contiguous ones, and strictly being incapable of a stable or permanent position : these are the pyramid on the triangular base, and the pyramid on the square base, p. 240. r(n + 1) First. Here (p. 283) 1 + 2 + 3 + = number in the base 1.2 m(n + 1) n(n + 1) (n + 2) of the triangular pile; and = number in the tria 1.2 1.2.3 angular pile; agreeing with pages 161 and 283. Second. For the square pile, we have nạ for the general term, or number in the base ; and (p. 283) the sum of all these strata, or courses, from 1 n(n + 1) (2n + 1) = number of balls in the square pile ; again, as in pages 162 1.2 3 and 283. n = on, is GEOMETRY. . DEFINITIONS AND PRINCIPLES. a 1. GEOMETRY treats of the forms, magnitudes, and positions of bodies produced according to any specified method of construction. All the other qualities of body are left out of consideration, and the attention confined exclusively to these, either singly or in combination. In this case the body is called a figure. 2. The object of geometrical inquiry is twofold :-first, theoretical or speculative; and secondly, practical or operative. In one case it is a science ; in the other it is an art founded upon science. 3. As a body of knowledge it is divided into sections, each of which is called a proposition ; and every proposition is either a theorem, a problem, or a porism. 4. A theorem is a proposition in which some statement is made concerning a specified figure, in addition to the conditions of its construction, but inevitably flowing from those conditions. In order to its obtaining reception as a truth, it must be proved or demonstrated. 5. A problem is a proposition in which when certain figures are given, or already exhibited in construction, some other figure dependent upon these is to be found which shall fulfil some assigned conditions. It requires for its completion a discovery of the method of constructing the figure sought, and a demonstration that the method does effect the proposed object *. 6. The words in which a proposition is stated constitute the enunciation ; it is called the general enunciation when there is no reference made to an exhibited figure; and the particular enunciation when such a figure is referred to. 7. The general term solution is often applied to the whole series of constructions and reasonings that are necessary to complete a proposition after the general enunciation is given. ; * The peculiar shade which distinguishes a porism from a problem and a theorem (of the nature of each of which it in some degree partakes) cannot here be explained to the student intelligibly. See, however, a note to Prob. I. of this work; and for a history of the methods devised for investigating them, the article Porisms in the Penny Cyclopædia, by J. O. Halliwell, Esq. F.R.S. To these may be added, that a lemma is a proposition which is premised, or solved beforehand, in order to render what follows more simple and perspicuous. A corollary is a consequent truth, gained immediately from some preceding truth, or demonstration. A scholium is a remark or observation made upon something going before it: generally of a collateral rather than of a direct application. 8. Every kind of body that can be supposed actually to exist has three dimensions, length, breadth, and thickness; and the branch of geometry which takes all these dimensions simultaneously into consideration is hence called the geometry of three dimensions. It is also very frequently called solid geometry; and often again, though not in strict propriety, the geometry of planes and solids. 9. The boundary of a body is the superficies or surface. It includes the dimensions, length and breadth, but not that of thickness. We may speak and think, and reason about the superficies, without taking note of the thickness of the body, whose superficies it is. Hence, for all the purposes of geometry, the figure which is called a superficies has no thickness. 10. The boundaries of a surface are lines. The line has therefore no breadth nor thickness, but length only. 11. The extremities of lines, and their mutual intersections, are called points. A point, therefore, has no dimensions, but position only. 12. A straight line, or a right line, is that which preserves, at all its points, the same direction *. 13. A curve line is one which, at its successive points, changes its direction. 14. A plane surface is one in which any two points being taken, the straight line passing through them shall lie wholly in that plane. If this be not the case, the superficies is said to be a curved surface. 15. A circle is a figure lying wholly upon a plane superficies, and is composed of one curved line (called the circumference) such that all straight lines drawn from certain point in that surface to the circumference are equal to one another. This point is called the centre of the circle. The circumference itself is often called a circle, and also sometimes the periphery. 16. The radius of a circle is a line drawn from the centre to the circumference. 17. The diameter of a circle is a line drawn through the centre, and terminating at the circumference on both sides. ODO 18. An arc of a circle is any part of the circumference. * When the term line is used in the following treatise, it designates a straight line : and the curved line is called simply a curve. In more general reasoning, the generic term line is applied to all kinds of lines, and another mode of classification of them adopted, which will be explained in a future stage of the work. |