CE. others and a 's? +2 =AC, are bes: pe and y= ba: ecewat at ar to or dane be rey's bers e operi tion 0 each avea fa ines, Woali opt 23 is prac ra sto of prot ne ide llected ES: Whence, The root of which is y=? pa ?), the breadth required. 2s Prob. 2. It is required to deter Having thou found the breadth, mine the side of a square inscribed the length may be obtained by dis in a given triangle. viding the area by the bread!h; or Let A B C (plate I. fig. 10,) re-otherwise we have, hy tinding the present the given triangle, and values of y instead of x, EFGH its inscribed square. Put bp the base A B=b, the perpendicu - pix y 6 And, because of the similar tri- bp— px whence b : a = x : b ba P b whence 2 of the square required. Prob. 4. Iu a right-angled trianCor. Hence it is obvious, that in sle, having given the lengths of two lines drawn from the acute all triangles whose bases and perpendiculars are constant, the side angles to the middle of the oppoof the inscribed square will be con- site sides, to find the sides of the stant also. triangle. Prob. 3. Giving the area or space Let ABC (Fig. 12) represent the of a rectangle, inscribed in a given proposed triangle, of which the triangle, to determine the sides of lines AD, CE, are given. Make the rectangle. AD=a, CE=b, AB=2x, CB=2y, Let ABC (fig. 11,) represent the AC=25. Then we have given triangle, and EFGH the inscribed rectangle, the area of x2 + 4y2= 12) again. 4.x? + y=al; 41202 15 CB. whence px=-by, And in the same manner we find and ... dya, the urc a. 62 2r 2 - AB. br 15 =AC. 15 the second., I= Prob. 5. In a riglit-angled trian gle, liaving given the hypotheninse bp -- by Whence again and side of the inscribed square, lo . find the base and perpendicular. P bpy-by2 = pa; Let ABC (Fig. 9) be the proposed triangle, AC the given hypothevo-pg- nuse, and BD the given square. 45 6 4x2 + y2=a?) Euc, 47. 1. , and se and pogled erally repro and consq. 2y = 2, (442) cle, of , and y, the en we 2 Make AC = h, FD=DE=S, AB Add and subtract double the last =x, BC-=y. equation from the tirst, and we Then we have have x2 + y2= 12 (Eucl. 47, 1.) x2 + 2xy + y2 = n2 + 2pn and (x - 5):s=s:(y- s) by - 2xy + y2 = n2 - 2pn, sim. trians. the roots of which are, Whence xy - 52 — sy +s?=s2; 2 +y= V(n2 + 2pn) or · • xy=s (x+y). and 2 — y= Vin? - 2pn). By adding double this to the first equation, we obtain Whence again, x=(ne + 2pn)+ x2 + 2xy + y2 = 12+2s (x+y), (12— 2pn=AB, and or (x + y)2 - 2s (x+y)=h2; y=v(n2+2pn)whence x+y=si Vina+s2). V (122 — 2pn) = BC. Now x +y being known, make it Wherefore the three sides AB, = n, then we have BC, AC, are determined. x2 + y2 = h2 In the second branch of the Apx + y =n plication of Algebra to Geometry, Square the second, and subtract or that which respects the higher it from double the first, and we geometry, or the pature and proobtain perties of curve lines, the nature 22 - 2xy + y2 = 222 - n?. of the curve is expressed, or deBy extracting, 2-y= V(212_n2). noted by an algebraical equation, Again • x+y=n; which is formed as follows: a line therefore, is conceived to be drawn to repren+1 (292_n) sent the diameter, or some other = AB; principal line of the curve; and n-v(2212) upon this line, at any indefinite and : BC, points, are erected perpendiculars, which are called ordinates; and as required. the parts of the first line cut off Prob. 6.-Having given the sum by ihem are terned abscisses. of the three sides of a right-angled Calling the abscis x, and its cortriangle, and the perpendicular, responding ordinate y, the known let fall from the right-angle upon nature of the curve, or the mutual the hypothenuse, to find the three relation of the other lines in it, sides of the triangle. will furnish an equation, involvLet ABC (fig. 12,) be the propos: ing x and y, with some other let. ed triangle, of which the sum of ter or letters which are known. the sides and perpendicular BD are And as x and y are common to given. Make the sum of the sides every point in the primary line, s, the perpendicular=P, AB=X, the equation, derived in this manBC=Y, AC=%. Then we have +y +% =S, ner, will belong to every position or value of the absciss and ordi. by the question 22 + y2=%? nate; and may be properly con(Euc). 47, 1.) x :x=p:y, by sim. sidered as expressing the nature trians.; or my=p. of the curve in all points of it, Now, add double this last equation and is usually called the equation to the second, then of the curve. Hence every partix2 + 2xy + y'= x2 + 2p% 22 + 2xy + y2 cular curve will appear to have 3? - 2- +2, by transposing in the first equa from that of every other; either an appropriate equation, differing tion and squaring. as to the number of the terms, the Whence, 2 + 2p%= 52--23% + m2, powers of the unknown quantities or, = AC. x and y, or the signs of the co2p + 2s efficients of the terms of the equaNow & being known, make it tion. =n; then the second and third APPLICATION of Geometry to equations become Algebra, is the converse of the 22 + y2 = n2 tirst of the two preceding cases. XY = pn. It relates principally to the find. ing the roots of an equation by as its point of regression, or vertex, geometrical construction. is uppermost, and the descending APPLICATION of Algebra and body must commence its motion Geometry to Mechanics. This is in it with a certain determinate found on the same principles as velocity. Varignon rendered the the Application of Algebra to Geo- question more general by investi. metry; and consists principally in gating the curve which' a body representing, by equations, the might describe in vacuo, so as to curves described by bodies in mo- approach through a given point tion; as in the theory of Projec- through equal spaces in equal tiles, &c. times, according to any law of gra. APPLICATION of Mechanics to Maupertuis also resolved the Geometry, consists chiefly in the same problem, in the case of a use that is sometimes made of the body descending in a medium, the centre of gravity of figures, for de- resistance of which is proportiontermining the contents of solids ate to the square of the velocity. described by those figures. Method of APPROACHES, a term APPLICATION of Geometry and used by Dr. Wallis, in his Álgebra, Astronomy to Geography, princi- to denote a method of resolving pally consists in the three followcertain problems, relating to square ing articles; viz. in determining numbers, &c.; which is done by by geometrical and astronomical tirst assigning certain limits to the operations the figure of the terres quantities required, and then aptrial globe; in finding the positions proaching nearer and nearer till a of places by their observed latio coincidence is obtained. This metude and longitude; and in deter-thod was invented by Dr. Pell, for mining, by geometrical operations, the solution of equations of the the positions of places that are not form x? — ay2 = 1; which pro. very remote from one another. blem was proposed by Fermat, as Astronomy and geography are a challenge to all the English maagain applicable to the theory of thematicians of his time; viz. to navigation. find rational and integral values APPLICATION of Geometry and of u and y, in the above equation, Algebra to Physics or Natural Phi- for every value of a, except when losophy. For this application we are it is a complete square.. indebted to Sir Isaac Newton, APPROXIMATION, in Algebra whose philosophy may, therefore, and Arithmetic, is the method of be called the geometrical or mathe approaching nearer and nearer to matical philosophy; and upon this the quantity sought, when there application are founded all the is no method of obtaining the ex. phyico-mathematical sciences.- act value. Hence a single observation or expe APPROXIMATION to the Roots riment will often produce a whole of Equations. As there is no di. science. Having ascertained, by rect method of determining the experience, that the rays of lighi, rools of equations beyond those of by reflection, make the angle of the fourth degree, and even in incidence equal to that of reflec- those of the third and fourth del tion, we hence deduce the whole gree. being very laborious by the science of Captoptics. The case direct rules, mathematicians have is also the same in many other endeavoured to find methods of sciences. approximating the roots : of these, APPROACH. The Curve of equa- Newlon's rule is the most popuble Approach, is of such a nature, lar, and is founded on the followthat a body descending by the sole ing principles : power of its own gravity approach It any two numbers, being subes the horizon equally in equal stituted for the unknown quantity times. This curve has been found, in an equation, give results with by Bernoulli, Varignon, Mauper- opposite signs, an odd number of tuis, and otbers, to be the second roots must be between these numcubical parabola, so placed that) bers. sa v3+ :)= ; This appears from the property .248 =8+12F+612+13 of the absolute term, and from this -2x=-4-2f -5--5 from the nature of our substitution, it may be omitted as inconsider. 25 36 -5+ V31 6 of x. therefore r= 2:1 nearly. let f=:1+g, fs = 0.001 + 0.03g + 0.3g? + g ed to the substitution of negative 6/?=0.06 + 1:24 + 6g? quantities and the negative roots. 105 =1. + 10g -1=1 0.061 + 11.23g + 6 3g? + g3=0. And neglecting gs and gʻ, for the Here we easily see, that one root reasons above stated, this equation is between 2 and 3; for these num. becomes bers being substituted for x, will 0.061 + 11.23g = 0, 0.061 give, the one a positive, and the org = .0054. 11.23 worked out, will give X = 2:09455147. ing this value for x in the propos Another method of approximated equation, we have, ing towards the roots of equations 14 3.8'2 = ar'2 V in general, is by the following for-froot, we are rather under than mula, first published by Barlow: over the real state of convergency; Let 2" tax-17.xn- +241-3+ for generally, if the first supposi. &c.=w, by any equation ; tion be a simple integer, the first and zin+ax'n-+bxin–2+czn3+ approximation may be obtained lo &c. =v, at being an approximate three places, which, though it be value of 2; then not always correct in the last (10-0)2x1 ==x+ place, yet by employing it as a (1 - 1)w + (n + 1) xnt new approxiniation, the next value (1-1)ux-+(1-3).xn-+ &c.; of x will, in most cases, be found (W-V)22! true to six or seven fignres, which or, x=x+ is a degree of convergency that (1-1) v+(n+1) xin + cannot be obtained by any other (n-1)axın-I+(1-3)bxın-2+ &c. rule that we are acquainted with, The first formula being applica- at the same time that the operation ble to the cases in which w is is much more simple than by any greater than unity, and the second other rule whatever. when a' is less than unity. An example will sufficiently ilThese formnlæ are general for lustrate the preceding remarks. equations of all dimensions, but Example. Find the value of x when reduced to particular cases, in the cubic equation they become much more simple : x3 + 3x2 + 3x = 130; thus, reducing them both into one, Assume a = 4, then by formula (1) X'3 we have for 64 128 48 (1.) Equations of the third Degree, 3x 12 130 (w--V)21 x=x+ w or v+2013+ axl2 = 124 306 deno. = 130 minator (2.) Equations of the fourth Degree, (w/v)2X r='+ 6 306) 24 (4:08 nearly. (W-Vr' I=x' + Hence, adopting this new value, 2w or 20 + 3x6 + 2ax'+ + we have X'3 67.9173 135.83416 b.xs_dar 49.9392 And in the same way we might 3x = ar'2 = 49.9392 19.24 find the particular formula answer. E 130 ing to an equation of any other = 130.0965 315.7738 degree. = 130 Application of the preceding For-w-v= 0965 mula.-In order to this it may be 4:08 observed, that æe is an approximate value of x, found first by 315-77) •39372 (-001247. trial, as near the true root as pos- Whence, 4:08 —:001247 = 4:078753 sible, which in all cases ought to Answer. be true to the nearest integer. This root is true in the seventh Then substituting this value of a', figure ; and, in the same manner, a nearer approximate value of x the root of any other equation may will be obtained, which will in all be determined. cases double the number of figures. APSES, in Astronomy, are the Then, considering this as a new two points in the orbits of the plavalue of x', another still nearer nets where they are at their greatvalue of x will be determined; and est and less distance from the sun so on. or earth; the former being called In stating the degree of approx. the higher apsis, and the latter the imation to be that of doubling the lower apsis : but the higher apsis number of figures in the assumed is more commonly called the 4 2.x'3 3x2 = W |