m ^Kl m iX. ...-. UN 4 JULIUS WMGENMUEIM 87 Mathematics Dept . AN , , , ELEMENTARY TREATISE ON THE GEOMETRICAL AND ALGEBRAICAL INVESTIGATION OF MAXIMA AND MINIMA, BEING THE SUBSTANCE OF a (zroum of awtum Delivered conformably to the Will of LADY SADLER: TO WHICH IS ADDED, <a gtlcttion of $toj)00itiott* DEDUCIBLE FROM EUCLID S ELEMENTS. BY D. CRESSWELL, A. M. FELLOW OF TRINITY COLLEGE, CAMBRIDGE. Second Edition, Corrected and Enlarged. CAMBRIDGE: Primed by J. Smith, Printer to the University; FOR J. DEIGHTON & SONS ; AND LAW & WHITTAKER, AVE-MARIA-LANE, LONDON. 1817 iv have been recorded, although, from my want of judgment or ability, they have not been fulfilled. I can, indeed, with great truth affirm, that I Was most desirous of being able to meet your Lordship s wishes, when I applied myself to this attempt. But, besides my own deficiencies, I had to encounter an obstacle of another kind. It was difficult to discover a subject important in itself, and claiming the early attention of the mathema tical student, which is not sufficiently illustrated in the Lectures constantly delivered by the Tutors of the College. This circumstance will, I trust, be accepted by your Lordship as my excuse, and it must, at the same time, be offered to the public as my apology, for not having accomplished more, nor attempted higher things. I gladly avail myself of this occasion publicly to acknowledge the several favours which your Lordship, with the utmost kindness, has been pleased to confer upon me. I am, my Lord, 9| jj with the sincerest gratitude and respect, your Lordship s v X. most .obliged (.wnifi and most obedient servant, D. CRESSWELL. APPENDIX I. CONTAINING A SELECTION OP PROPOSITIONS DEDUCIBLE FROM THE FIRST SIX BOOKS OF Censemus, autem, nihil utilius ad Geometriam penitius cog- noscendam haberi posse, quam hujusmodi contentio tyronis, in deducendis theorematis, vel solvendis problematis : qua fit, ut Geometria ipsa ejus animo inulto altius inside at, et investigationis fontes aperiantur. BOSCOVICH. a ^"DEDUCTIONS SiJSj " PROM THE FIRST SIX BOOKS OF GEOMETRICAL EXERCISES are peculiarly adapted to the improvement of the chief powers of the mind; and the sole motive which has prompted the publication of the following selection of Ques tions, is a desire to engage the Academical Student in that employment, from which he is likely to be most benefited in the beginning of his course. It is by no means intended to recommend to him the cultivation of this department of Science, to the neglect of all others ; but he is advised to make it the ground-work of his future acquirements in the Mathematics, and never to advance until he has laid that foundation well ; because it is the firmest upon which the superstructure of solid mathema tical knowledge can be built. No credit whatever is expected by the author to accrue to him from the execution of this design, #2 (*) APPENDCX I. unless it be of that very humble, but surely not dishonourable, kind, which belongs to an useful performance. Although some of the questions, which he has here published, are original, he is far from thinking it probable that even these have not occurred to others, at least as early as they did to himself. His materials are, for the most part, taken from works of established reputation, both ancient and modern. Well known, as they must be to the learned, they may, however, be useful, as a collection, to the student in Geometry. They have been chosen, either as exhibiting some remarkable property of lines or figures, omitted by Euclid, or as furnishing a mere exercise of ingenuity. Some Propositions, that are very obvious, and very easy of demonstration, are purposely inserted, as best suited to the ability of beginners ; and, perhaps, it may not be improper to add, that many others are the Geometrical Solutions of Problems belong ing to the several branches of Natural Philosophy. Such an arrangement has been given to the de ductions, as will, in many cases, lead to at least one method, it would be presumption to say the best method, of solution. They are distributed according to the same order as the several Books of Euclid, which are most studied; and further, they are placed, each under the last of the pro positions upon which it may be made to depend, or which need be quoted in its proof. They are not always, therefore, although they are often. APPENDIX I. (5) strictly speaking, corollaries of the proposition to which the} 7 are immediately subjoined. Sometimes, indeed, it is best, to make use of one of these de ductions as a step in the demonstration of another, which follows it; but this is not very often the case; and when it happens to be so, the reader is most commonly apprized of it. It is only, there fore, in those particular cases, in which some of the following theorems, or problems, are, properly speaking, corollaries of the proposition from Euclid, which is placed at the head of them, that the mode of solution, intended to be pursued, is clearly intimated by the arrangement which has been here adopted. In other cases, the difficulty of the deduction, such as it is, may mainly depend upon some other antecedent proposition in Euclid s book, although that, which is referred to, be also required in the course of the demonstration. To have dis tinctly pointed out all the elements upon which each demonstration is made to depend, would have been to leave too little for the ingenuity of the learner to perform. If, however, this book should happen to be used, under the direction of a tutor, it would be easy for him to supply as many of these purposely suppressed references, as he may judge to be necessary. To this second edition is annexed a specimen of propositions belonging to Natural Philosophy^ of which the solutions may very well be derived from the application of Geometry to that extensive and interesting subject. These examples, it is hoped, (6J APPENDIX i; may serve still further to illustrate the elegancy of geometrical constructions, and also to stimulate the curiosity of the student, by the diversified and perhaps more engaging forms, in which the ques tions thus involved are presented to his mind. ASJ in solving algebraical problems, his first step is to translate the conditions of the problem into the peculiar language of analytical calculation ; so here, it will be for him, in the first place, making use of the principles of Natural Philosophy, to reduce the question, under his consideration, to the sub stance of some geometrical proposition ; which being solved, the question itself may be regarded as solved also. If he has any taste for Plane Geometry, this will be far from being a disagreeable exercise ; and if he has acquired any skill in that, the most lucid of all the branches of mathematical learning, it will also be an easy task. DEDUCTIONS FROM THE FIRST SIX BOOKS <ft ol ^ OF ttcltt> . - . M //in!* <nFub.flin9qi*K{ .nil PROP IX - - - ~j tadi .bn A -: Jiji io eJniof -lalijij A GIVEN plane rectilineal angle being divided into any number of equal angles, to divide the half of it into the same number of angles, all equal to one another. PROP. X. (!7) (ll.) |> c,cf ; From the vertex of a given scalene triangle, to draw, to the base, a straight line which shall exceed the less of the two sides, as much as it is itself exceeded by the greater. (8) APPENDIX I. .XI. (HI.) In a straight line given in position, but inde finite in length, to find a point, which shall be equidistant from each of two given points, either on contrary sides, or both on the same side of the given line, and in the same plane with it ; but not- situated in a perpendicular to it. (iv.) If the three sides of a given triangle be bisected, the perpendiculars drawn to the sides, from the three several bisections, shall all meet in the same point : And that point is equidistant from the three angular points of the given triangle. (v.) Hence, to find a point, in a given plane, which shall be equidistant from three given points in the plane, that are not all in the same straight line. PROF. XVI. (VI.) ~ . There cannot be drawn more than two equal straight lines, to another straight line, from a given point without it. COR. A circle cannot cut a straight line in more points than two. APPENDIX I. (9) PROP. XVII The perpendicular let fall from the obtuse angle of an obtuse-angled triangle, or from any angle of an acute-angled triangle, upon the opposite side, falls within that side: But the perpendicular drawn to either of the sides containing the obtuse angle of an obtuse-angled triangle, from the angle op posite, falls without that side. o) ofwufl fcf JI I)ffiS ; . f (VIII.) If a straight line, meeting two other straight lines, makes the two interior angles on the same side of it not less than two right angles, these lines shall never meet on that side, if produced ever so far. COR. Two straight lines, which are both per pendicular to the same straight line, are parallel to each other. oov o/{| ,0* 3*iufv SCHOLIUM. Parallel straight lines being thus defined, " Two straight lines are parallel if they be in the same plane, and a straight line drawn from any point in the one, perpendicular to either of them, be also perpendicular to the other," the 35th Definition (10) APPENDIX I. of the First Book of Euclid becomes a corollary to the last article; and may be cited in the de monstration of Prop. 27. Book I.; also the 29th Proposition of this Book may be proved by a re- ductio ad absurdum, without the help of the 12th axiom, which itself becomes a corollary to that proposition. Nothing seems to be wanting, to render the definition here substituted unobjection able, but a proof, that if it be a property of two straight lines at any one point, it will also obtain at every other point of them. This proof has been given by Robert Simson in his Note upon E. 29. 1 . ; and it is made to depend only upon an axiom and upon the 4th and 8th Proposition of the First Book of Euclid. It has also been given, under a somewhat different form, by Borelli, in a single proposition, clearly and elegantly demonstrated; but the axiom borrowed from the Arabian mathe maticians, which he premises,, is, perhaps, less judi ciously chosen, than that which is the foundation of Simson s proof. The great object of this sub stitution is to avoid the necessity of having re course to the proposition, which Euclid has made his 12th axiom ; but neither is his 35th definition itself the best that might be given ; for, with the exception of the single supposition, that the two straight lines are in the same plane, it is wholly negative, and affords no practical test of the parallelism of two straight lines. A modern editor of the Elements, of great learn- ing and ability, has proposed the following axiom, APPENDIX I; (11) instead of the 1 2th : " Two straight lines cannot be drawn through the same point parallel to the same straight line, without coinciding with one another." But this pre-supposes the 35th Definition of Euclid, and is, therefore, objectionable, in asmuch as that definition is itself objectionable. Besides, if it be thus stated, according to its true meaning, " Two straight lines cannot be drawn through the same point, neither of which, when they are produced ever so far both ways, meets another straight line, given in position, but in definite in length," it does not appear to have a much better claim to the title of axiom, than the as sumption which it is intended to replace. Gamier has accordingly, in his Geometry, formally demon strated this very proposition : It is, indeed, most certain, that the mind cannot conceive two straight lines in a state of infinite extension ; which, how ever, it is led to attempt, by the 35th definition, and the axiom last considered. PROP. XX. (IX.) The three sides of a triangle taken together, exceed the double of any one side, and are less than the double of any two sides. APPENDIX I. Any side of a triangle is greater than the dif ference between the other two sides. Any one side of a rectilineal figure is less than the aggregate of the remaining sides. ^rLr^j^i/ ( XI1 -) ff^S^fiSS The two sides of a triangle are together, greater than the double of the straight line Which joins the vertex and the bisection of the base. PROP. XXI. ;i, Tlf ox&^ /; ni aaiiil (XIII.) If a trapezium and a triangle stand upon the same base, and on the same side of it, and the one figure fall within the other, that which has the greater surface shall have the greater perimeter. PROP. XXVI. (xiv.) If two right-angled triangles have the three angles of the one equal to the three angles of the APPENDIX I. (13) other, each to each, and if a side of the one be equal to the perpendicular let fall from the right angle upon the hypotenuse of the other, then shall a side of this latter triangle be equal to the hypo tenuse of the former. If the sides of any given equilateral and equi angular figure of more than four sides, be produced so as to meet, the straight lines, joining their several intersections, shall contain an equilateral and equi angular figure, of the same number of sides as the given figure. PROP. XXVII. (xvi.) If two opposite sides of a quadrilateral figure be equal to one another, and the two remaining sides be also equal to one another, the figure is a parallelogram, COR. 1. Hence may be deduced a practical method of drawing a straight line, through a given point, parallel to a given straight line. COR. 2. A rhombus is a parallelogram. (14) APPENDIX I. J*ROP. XXIX. , * (XVII.J Every parallelogram which has one angle a right angle, has all its angles right angles. (XVIII.) To trisect a right angle ; i. e. to divide it into three equal parts. (XIX.) Hence, to trisect a given rectilineal angle, which is the half, or the quarter, or the eighth part, and so on, of a right angle. PROP. XXXI. : a : -> V *H (XX.) r ,.- 1 .: M, P , ,J To find a point, in either of the equal sides of a given isosceles triangle, from which, if a straight line be drawn, perpendicular to that side, so as to meet the other side produced, it shaH be equal to the base of the triangle. (XXI.) In the hypotenuse of a right-angled triangle, to find a point, the perpendicular distance of which APPENDIX I., (15) from one of the sides, shall be equal to the seg ment of the hypotenuse between the point and the other side. (XXII.) In the base of a given acute-angled triangle, to find a point, through which if a straight line be drawn perpendicular to one of the sides, the segment of the base, between that side and the point, shall be equal to the segment of the per pendicular, between the point and the other side produced. (XXIII.) From a given isosceles triangle to cut off a trapezium, which shall have the same base as the triangle, and shall have its three remaining sides equal to each other. (XXIV.) To draw to a given straight line, from a given point without it, another straight line which shall make with it an angle equal to a given rectilineal angle. (XXV.) The two sides of a triangle are, together, greater than the double of the straight line drawn from the vertex to the base, bisecting the vertical angle. (16) APPENDIX I. cndl io PROP. XXXII. (XXVI.) If two triangles have two angles of the one equal to two angles of the other, the third angle of the one shall also be equal to the third angle of the other. (XXVII.) The angle at the base of an isosceles triangle is equal to, or is less, or greater, than the half of the vertical angle, accordingly as the triangle is a right-angled, an obtuse-angled, or an acute-angled triangle. (XXVIII.) If either of the equal sides of an isosceles tri angle be produced, towards the vertex, the straight line, which bisects the exterior angle, shall be pa/allel to the base. (XXIX.) The distance of the vertex of a triangle from the bisection of it s base, is equal to, greater than, or less than the half of the base, accordingly as APPENDIX I. (17) the vertical angle is a right, an acute, or an obtuse angle *. COR. 1. If any number of triangles have a right angle for their common vertical angle, and have equal hypotenuses, the locus of the bisections of the several hypotenuses is a quadrantal arch of a circle, having the common vertex for its center, and the half of any hypotenuse for its radius. COR. 2. A circle described from the bisection of the hypotenuse of a right-angled triangle as a center, at the distance of half the hypotenuse, will pass through the summit of the right angle. (XXX.) If either of the acute angles of a given right- angled triangle be divided into any number of equal angles, then, of the segments of the base, subtending those equal angles, the nearest to the right angle is the least ; and, of the rest, that which is nearer to the right angle is less than that which is more remote. * It is intended that this proposition should be demonstrated, ex absurdo, by the help of E. 32. 1. E. 5. 1. and E. 18. 1. But it is evidently deducible, with equal facility, from E. 31.3. It will, doubtless, often happen to the reader, in other instances, as well as in this, very readily to find out another mode of proof, when he does not, at the first attempt, discover the principle of solution, intimated by the arrangement which has been adopted in this Appendix. (18) APPENDIX r. o .viUDC rje .if fob s a 9(3 fie (xxxi.) If either angle at the base of a triangle be a right angle, and if the base be divided into any number of equal parts, that which is adjacent to the right angle shall subtend the greatest angle at the vertex ; and, of the rest, that which is nearer to the right angle shall subtend, at the vertex, a greater angle than that which is more remote. (XXXII.) To trisect a given finite straight line. COR. Hence, to inscribe a square in a given right-angled isosceles triangle. (xxxm.) To describe a triangle which shall have its three sides, taken together, equal to a given finite straight line, and its three angles equal to three given angles, each to each ; the three given angles being together equal to two right angles. (xxxiv.) If, in the sides of a given square, at equal dis tances from the four angular points, four other points be taken, one in each side, the figure con tained by the straight lines which join them, shall also be a square. APPENDIX I. (19) (xxxv.) >r// ,-tsrfJo rfafi* Jiu B5i!ll"3fciin24 iiioi ll If the opposite angles, of a quadrilateral figure be equal to each other, the figure shall be a parallelogram. (XXXVI.) In a given square to inscribe an equilateral triangle, having one of its angular points upon one of the angular points of the square, and its two remaining angular points one in each of two ad jacent sides of the square. (XXXVII.) If, at the extremities of the base of a given triangle, two straight lines be drawn, both above the base, and each of them equal to the adjacent side, and making with it an angle equal to the vertical angle of the triangle ; then, if two straight lines, let fall from the extremities of the two so drawn, make, with the base produced, two angles that are equal each of them to the vertical angle, they shall cut off equal segments from the base produced. (xxxvm.) To inscribe a square in a given rhombus. is (20) APPENDIX I. (XXXIX.) If four straight lines cut each other, without in cluding space, but so as to make three internal angles, towards the same parts, which together are less than four right angles, the two lines, which are not joined, shall meet, if produced far enough. U-ralirfia^ s wfi (XL.) ^.nVvig a al If the straight line, drawn from a point in the produced diameter of a circle to the convex cir cumference be equal to the half of the diameter, the angle, at the center, subtended by the concave circumference included between the diameter and the line so drawn, is the triple of the angle, at the center, subtended by the convex circumference in cluded between the same two lines. The converse of the proposition is also true. COR. Hence, if a straight line could be drawn from any point in the curve of a semi-circle to meet the diameter produced, so that the part of the line without the curve should be equal to the radius, any angle might be trisected. (XLI.) One of the two sides, which are about the right angle of a right-angled triangle, and the aggregate of the hypotenuse and the remaining side, being given, to construct the triangle. APPENDIX I, (81) PROP. XXXIV. (XLII.) The diameters of a parallelogram bisect each other. (XLIII.) The diameters of an equilateral four-sided plane rectilineal figure bisect one another at right angles. .:v!U]-fl Silj lo c bi? 701. /rfctlJ T5 )ot$ flcf (XLIV.) The diameters of a rectangle are equal to one another. "jfu Jurj.t O8 t B30lJ JM ; >. <J< ov/j ^ijijjl : ^Jf ;K ,- (XLV.) : ; ;;;^^ n; r :;; ^ If two opposite sides of a parallelogram be di vided each into the same number of equal parts, the straight lines, joining the opposite points of division, shall also divide the diameter of the parallelogram into the same number of equal parts, (XLVI.) To divide a given finite straight, line into any given number of equal parts. (22) APPENDIX I. , (XLVII.) Upon a given finite straight line, as a diameter, to describe a square. (XLVII i.) &>> > 9rfT Upon a given finite straight line to describe an equilateral and equiangular octagon. (XLIX.) If either diameter of a parallelogram be equal to a side of the figure, the other diameter shall be greater than any side of the figure. .... From a given point to draw a straight line cutting two parallel straight lines, so that the part of it, intercepted between them, shall be equal to a given finite straight line, not less than the per pendicular distance of the two parallels. If, from the summit of the right angle of a sca lene right-angled triangle, two straight lines be drawn, one perpendicular to the hypotenuse, and the other bisecting it, they shall contain an angle equal to the difference of the two acute angles of the triangle. APPENDIX I. (23) PROP. XXXVI. . "V .(ML) - To bisect a parallelogram by a straight drawn through a given point in one of its sides. (LIII.) A trapezium, which has two of its sides parallel, is the half of a rectangle between the same paral lels, and having its base equal to the aggregate of the two parallel sides of the trapezium. PROP. XXXVII. (LIV.) A plane rectilineal figure of any number of sides being given, to find an equal rectilineal figure, which shall have the number of its sides less, or greater, by one, than that of the given figure. Hence, first, to find a triangle, which shall be equal to any given plane rectilineal figure : secondly, to find a polygon of any given number of sides which shall be equal to a given triangle. (24) APPENDIX I. PROP. XXXVIII. . "^ : """* (LVI.) / . ~ , The diameters of any parallelogram divide it into four equal triangles. (LVII.) Of all triangles, which are between the same parallels, that which stands on the greatest base is the greatest. .cyfuisscjfiiJ odt b e*>bi3 fdfi&r&q owl sru (LVI ii.) The straight line, joining the vertex and the bisection of the base of any triangle, bisects every other straight line that is parallel to the base and is terminated by the two remaining sides of the triangle. (LIX.) ^ Hence, if two opposite sides of a trapezium be parallel to one another, the straight line, joining their bisections, bisects the trapezium. To bisect a given trapezium by a straight line drawn from any of its angles. APPENDIX I. (25) To bisect a given triangle, by a straight line drawn through a* given point in any one of its sides. PROP. XL. (LXII.) Equal triangles, which have their bases in the same straight line and which are between the same parallels, stand upon equal bases. PROP. XLI. (LXIII.) To describe a parallelogram, the area and peri meter of which shall be respectively equal to the area and perimeter of a given triangle. (LXIV.) The two triangles formed by drawing straight lines, from any point within a parallelogram, to the extremities of either pair of opposite sides, are, together, half of the parallelogram. (LXV.) If two sides of a trapezium be parallel, the tri angle contained by either of the other sides, and APPENDIX I. the two straight lines drawn from its extremities to the bisection of the opposite side,, is the half of the trapezium. *i 1o ono ^n in Jnioq fisvrg * ilgucmfi nwmfa (LXVI.) The triangle contained by the straight lines join ing the points of the bisection of the three sides of a given triangle, is one-fourth part of the given triangle, and is equiangular with it. jib ni 3Oei;J ibrf) 3vd dpaiw ^{gnsni fai/p^l (LXVII.) Hence, if the four sides of any given quadrila teral rectilineal figure be bisected, the figure con tained by the straight lines joining the several points of the bisection, shall be a parallelogram, which is the half of the given figure ; also the four sides of this parallelogram shall be, together, equal to the two diagonals of the given figure. COR. It is manifest that the straight lines, which join the opposite points of bisection of the sides of any trapezium, bisect each other. I- tffcJ PROP. XLIII. (LXVI 1 1.) To describe a parallelogram, which shall be of a given altitude, and equiangular with, and also equal to, a given parallelogram. APPENDIX I. COR. Hence, a rectangle may very readily be found, which shall be equal to a given square, and shall have one of its sides equal to a given straight line. PROP. XLV. (LXIX.) If there be any number of rectilineal figures, of which the firsj is greaj;e ( r than the second, the second than the third, and so on, the first of them shall be equal to the last together with the aggregate of all the differences of the figures,.^ : ?sfa od t fe< (LXX.) fo] To find a rectangle, which shall have one of its sides equal to a given finite straight line, and which shall be equal to the excess of the greater of two given rectilineal figures above the less. SCHOLIUM. From the twenty-second, and the forty-fifth propositions, of this first book of Euclid, may be deduced a method of surveying, planning, and measuring irregular plots of ground, which have (28) APPENDIX I. rectilineal boundaries: and this method, which is purely geometrical, does not require the use of any instrument constructed for the purpose of estimating the magnitudes of angles ; its operations being performed, all of them, by means of a rule containing a scale of equal parts, a compass, and some standard measure of linear magnitude. It is manifest, that any rectilineal plot of ground may be divided into triangles ; and that the lengths of the sides of those several triangles may be found by the application of the standard measure, whether it be a foot, a yard, or any other standard measure of length, which has been chosen. Then, by the help of E. 22. 1, and of the scale of equal parts, an exact plan of the ground may be laid down on paper: and, lastly, a rectangle may (E.45.1 . Cor.) be described which shall be equal to the figure representing the plot of ground, and which shall have one of its sides equal to one, or to any given number, of the equal parts of the scale. If, there fore, one of its sides be made equal to one of the equal parts of the scale, it is plain that the number of such parts in the adjacent side, will shew the dimension of the plot of ground in square measure. For, it will indicate how many squares, each having one of the equal parts for its side, are contained in the rectangle that is equal to the plan of the ground ; and so many squares, it is evident, each having the standard measure, that was used, for its side, will there be in the plot of ground itself. If the side of the rectangle, constructed by APPENDIX I. (29) means of E. 45. 1, be taken any given multiple of one of the equal parts of the scale, then, if that multiple constitute any other standard measure of length, the dimension of the ground will still be found, as before, by finding how many of those multiples there are in the adjacent side; but it will be of a different denomination. But if the multiple, assumed for one side of the equal rectangle, be not any standard measure of length, the number of equal parts contained in the adjacent side must be counted ; and the product of this latter number, multiplied by the number of equal parts in the assumed side, will shew the di mension of the constructed rectangle, and of the plot of ground, also, which is required to be measured. For, although a rectangular surface can only be measured, in a direct manner, by the application of some lesser standard square, to its several parts in succession, yet, since it is evident, even from inspection, that any rectangle may be divided into a number of lesser squares, equal to the product of the numbers which shew how often a side of one of those lesser squares is contained in each of the two adjacent sides of the rectangle, that direct method of measurement is never employed. Hence, no such instrument as a standard square is wanted for the measurement of surfaces, and no such instrument is in use. By means of E. 35. 1, E. 41. 1, and E. 45. 1. the mensuration of parallel ogram^ triangles, and other rectilineal figures, is (30) APPENDIX I. reduced to the mensuration of a rectangle, which is effected, in the manner already described. Thus a parallelogram is denoted, in square measure, by the product of the number of standard equal parts in its base multiplied by the number of such parts in its altitude : and a triangle is also denoted, by the half of the product of the number of standard equal parts in its base multiplied by the number of such parts in its altitude. The plan of the piece of ground, required to be measured, having been previously drawn, if the deduction from E. 37. 1, set down in this book, be had recourse to, a triangle will be obtained, that is equal to the rectilineal figure so drawn : and the operations by which this result is arrived at, will be very easily and expeditiously performed, by the help of a parallel ruler. It will then only remain, to let fall, from the vertex of the triangle, a per pendicular on its base ; to measure both the perpen dicular and the base; and, lastly, to take the half of the product of the resulting numbers. The mode of planning and measuring which has here been described, is not, it is true, sufficient for all practical purposes. It contains, however, the first principles of the mensuration of plane surfaces. It has the advantage of being very simple and very easy to be understood : and it may, perhaps, afford some degree of satisfaction to the mathema tical student, to consider with how small a stock of Geometry he may be enabled to solve a problem of no small utility and importance. APPENDIX I. (31) PROP. XLVII. (LXXI.) If two triangles have two sides of the one equal to two sides of the other, each to each, and if the angles opposite to either pair of equal sides be each a right angle, the triangles shall be equal, and similar to each other. (LXXII.) To find a square which shall be equal to any number of given squares. (LXXIII.) Two unequal squares being given, to find a third square, which shall be equal to the excess of the greater of them above the less. (LXXIV.) If the side of a square be equal to the diameter of another square, the former square shall be the double of the latter. (LXXV.) In any right-angled triangle, the square which is described on the side subtending the right angle, as a diameter, is equal to the squares described upon the other two sides, as. diameters. DEDUCTIONS , * FROM THE FIRST SIX BOOKS OF BOOK II. PROP. I. IF two given straight lines be divided, each into any number of parts, the rectangle contained by the two straight lines, is equal to the rectangles contained by the several parts of the one and the several parts of the other. PROP. V. (II.) If a straight line be divided into two unequal parts, in two different points, the rectangle con- APPENDIX I. (33) tained by the two parts, which are the greatest and the least, is less than the rectangle contained by the other two parts ; the squares of the two former parts, together, are greater than the squares of the two latter, taken together ; and the differ ence between the squares of the former and the squares of the latter, is the double of the difference between the two rectangles. y,--- (in.) - nwil) , 7!l 1>w ^ In any isosceles triangle, if a straight line be drawn from the vertex to any point in the base, the square upon this line, together with the rect angle contained by the segments of the base, is equal to the square upon either of the equal sides. PROP. VI. l^m ^fl -f^-li (IV>) f- ^ The rectangle contained by the aggregate and the difference of two unequal straight lines is equal to the difference of their squares. COR. If there be three straight lines, the differ ence between the first and second of which is equal to the difference between the second and third, the rectangle contained by the first and third, is less than the square of the second, by the square of the common difference between the lines. (34) APPENDIX I PROP. VII The square of the excess of the greater of two given straight lines above the less, is less than the squares of the two lines, by twice the rectangle contained by them. *f fiiil )^irt< a (VI.) -,; , ,;^,,, ,,;,,/* The squares of any two unequal straight lines are, together, greater than twice the rectangle con tained by those lines. PROP. VIII. " / (VII.) If a straight line be divided into five equal parts, the square of the whole line is equal to the square of the straight line, which is made up of four of those parts, together with the square of the straight line which is made up of three of those parts. (VIII.) Upon a given straight line, as an hypotenuse, to describe a right-angled triangle, such that the APPENDIX I. (35) hypotenuse, together with the less of the two re maining sides, shall be the double of the greater of those sides. PROP. X. (IX.) In any triangle, the squares of the two sides are, together, the double of the squares of half the base, and of the straight line joining its bisection and the opposite angle. Hence, the squares of the sides of any paral lelogram are, together, equal to the squares of its diameters taken together.* If either diameter of a parallelogram be equal to one of the sides about the opposite angle of the figure, its square shall be less than the square of the other diameter, by twice the square of the other side about that opposite angle. * This proposition may also be readily deduced from E, 13, 2. (36) APPENDIX I. PROP. XII. (XII.) If two sides of a trapezium be parallel to each other, the squares of its diagonals are, together, equal to the aggregate of the squares of its two sides, which are not parallel, and of twice the rect angle of its parallel sides. PROP. XIII. (XIII.) The square of the base of an isosceles triangle is the double of the rectangle contained by either side, and by the straight line intercepted between the perpendicular, let fall upon it from the opposite angle, and the extremity of the base. (XIV.) If from any point, in the circumference of the greater of two given concentric circles, two straight lines be drawn to the extremities of any diameter of the less, their squares shall be, together, the double of the squares of the two semi-diameters of the two given circles. DEDUCTIONS FROM THE FIRST SIX BOOKS OF BOOK III. PROP. I. (1.) IF two circles cut each other, the straight line joining their two points of intersection is bisected, at right angles, by the straight line joining their centers. COR. Hence, if a trapezium have two of its adjacent sides equal to one another, and also its two remaining sides equal to one another, its diameters bisect each other at right angles. (38) APPENDIX I. SCHOLIUM. 81 -;r It is evident from the above deduction, that the double of the perpendicular,, let fall from the vertex of a triangle on its base, may very readily be found by the compass alone : And a reference to the Scholium in the twenty -seventh page, of,this Ap pendix, will shew that this expedient might be of great utility in Practical Geometry, for the men suration of the surfaces of triangles and other rectili neal figures. PROP. III. (II.) Through a given point within a circle, which is not the center, to draw a chord which shall be bisected in that point. PROP. XIV. ?J*d; i\iiifi^->i jl^ii.w Jii ^n is .*. (HI.) If two isosceles triangles be of equal altitudes, and the side of the one be equal to the side of the other, their bases shall be equal. APPENDIX I. (39) (IV.) Any two chords of a circle which cut a dia meter in the same point and at equal angles are equal to one another. Through a given point, within a given circle, to draw two equal chords, making with one another an angle equal to a given rectilineal angle. PROP. XVI. ns n r s x -r- * ii $ i| Jon elaioci owJ orfj, jjnioq nsvrg isrhans ni 9>b ?r:: , V If the diameters of two circles are in the same straight line, and have a common extremity, the two circles shall touch one another, O-U; L)lf f J(JH.K] U IK f)lf! jn^ "J.i ; ; : - " , ,.; ;/.:,. : (vn.) .;.*^ ^ To draw a tangent to a circle, which shall be parallel to a given finite straight line. COR. Hence, to draw a tangent to a circle, which shall make, with a given straight line, an angle equal to a given rectilineal angle. (40) APPENDIX I. (VIII.) . To describe a circle which shall have a given radius, and its center in a given straight line, and shall also touch another straight line, inclined at a given angle to the former. (IX.) To describe a circle, the circumference of which shall pass through a given point, and touch a given straight line in another given point. (x.) V To describe a circle, the circumference of which shall pass through a given point, and touch a given circle in another given point ; the two points not lying in a tangent to the circle. (XI.) To describe a circle, which shall touch a given straight line in a given point, and also touch a given circle. (XII.) Hence, if two circles touch each other exter nally, to describe another circle, which shall touch the one of them in a given point, and also touch the other. APPENDIX I. (41) ftoHKifp* ?x! Iforfe slow orb nlfitrv/ >i lo }7q orfi (XIII.) To describe two circles, each having a given radius, which shall touch the same given straight line, both on the same side of it, and shall also touch each other. (XIV.) To describe two equal circles, each having it s diameter equal to a given straight line, each touch ing a given circle, and each also passing through a given point without that circle : The given straight line being greater than the shortest distance, between the given point and the circumference of the given circle. o yd gviiw ilJod VTrTT , , PROP. XVII. sd . _ _ .JofliCioarto inioq od) m batoosrdf liv aivi, (xv ,,^.;v,-^.,, To find a point in the diameter, produced, of a given circle, from which, if a tangent be drawn to the circle, it shall be equal to a given straight m oniBa orlt fo no^yfoq /5 m/KfraoD Hi?/ , oriJ . t sd .Huda ^ XVI J "Mifrjsaijjoib 9i" Through a given point, either within, or with out a given circle, to draw a straight line, so that (42) APPENDIX I. i the part of it within the circle shall be equal to a given finite straight line, which is less than the diameter. folio 07/J oJmsob oT (xvn.) To draw a tangent to a given circle, such that it s segment,, contained between the point of contact, and an indefinite straight line, given in position, shall be equal to a given finite straight line. PROP. XVIII. : - - : - (XVIII.) .olonib If a straight line touch the interior of two con centric circles, and be terminated both ways by the circumference of the outer circle, it shall be bisected in the point of contact. (XIX.) ir i i j !_ j i i *u If a polygon be described about a circle, the straight lines joining the several points of contact will contain a polygon of the same number of angles as the former ; and any two adjacent angles of the circumscribed figure shall be, together, the double of that angle, of the inscribed figure, which lies between them. APPENDIX I. (43) PROP. XX. jfo-ib ydJ to T.J vc.iiib. -Mb < .,; (XX.) If two chords of a given circle intersect each other, the angle of their inclination is the half of the angle at the center standing upon the aggre gate, or the difference, of the arches intercepted between them, accordingly as they meet within, or without the circle. *jrf bia dJifcouui -_ N .o,qu ; (XXI.) To divide a given circular arch into two parts, so that the aggregate of their chords may be equal to a given straight line, greater than the chord of the whole arch, but not greater than the double of the chord of half the arch. (XXII.) To divide a given circular arch into two parts, so that the excess of the chord of the one above the chord of the other, may be equal to a given straight line, less than the chord of the whole arch. (xxin.) If from any given point, in the circumference of a circle, two straight lines be drawn to the ex- (44) APPENDIX I. tremities of a given chord, the angle which the one makes with any perpendicular to the chord, shall be equal to the angle which the other makes with the diameter of the circle that passes through the given point. -.vMiii olvn > : lo ,sfvK>i PROP. XXI. (XXIV.) The perpendiculars let fall from the three angles of any triangle upon the opposite sides, intersect each other in the same point. COR. This point is equidistant from the three straight lines joining the points in which the per pendiculars meet the three sides of the triangle. i>liffK)h silt rtfifto rj 63*r! Jon Jwd ^0*11; ekwlW 9*1* (XXV.) If from a given point within a circle, which is not the center, straight lines he drawn to the cir cumference, making with each other equal angles, the two, which are nearer to the diameter passing through the given point, shall cut off a greater circumference than the two, which are more re mote. (XXVI.) From either of the two given points, in which two given circles intersect each other, to draw a APPENDIX I. (45) chord cutting the one circumference, and meeting the other, such that the part of it, contained be tween the two circumferences, shall be equal to a given finite straight line. PROP. XXII. (XXVII.) If two opposite angles of a trapezium be to gether equal to two right angles, a circle may be described about it. (XXVIII.) A circle cannot be described about a rhombus, nor about any other parallelogram which is not rectangular. (xxix.) If from any point, in the circumference of a given circle, straight lines be drawn to the three angles of an inscribed equilateral triangle, the greatest of them shall be equal to the aggregate of the two less. (46) APPENDIX I. (xxx.) tl si!- " * ;v/t The first, third, fifth, &c. angles of any polygon, of an even number of sides, which is inscribed in a given circle, are together equal to the remaining angles of the figure ; any angle whatever being as sumed as the first.* (XXXI.) To make a trapezium, about which a circle may be described, having its four sides respec tively equal to four given straight lines, two of which are equal to each other, and any three to gether greater than the fourth ; the two equal sides of the trapezium, also, being opposite to each other. * The Proposition may be adapted to the case of a polygon of an odd number of sides inscribed in a circle, by dividing any one of its angles into two, by a straight line drawn from the center, and reckoning the two segments of that angle, each as one of the angles of the figure ; the number of angles is thus made even. The converse of this Theorem, and the second deduction from Prop. 18, may be applied to discover the relation which the angles of a polygon must have in order that it may admit of a circle being described about it, or inscribed in it. APPENDIX I* (47) PROP. XXIV. (xxxn.) If upon the two greater sides of an oblong, as diameters, two semi-circles be described, lying to ward the same parts, the figure contained by the two remaining sides of the oblong, and the two curve lines, shall be equal to the oblong. PROP. XXVI. (xxxm.) The straight lines joining the extremities of the chords of two equal arches of the same circle, toward the same parts, are parallel to each other. (xxxiv.) The arches of a circle that are intercepted between two parallel chords are equal to one another. (xxxv.) In equal circles the greater angle stands upon the greater circumference ; whether the angles com pared be at the centers or circumferences. (48) APPENDIX I. (XXXVI.) If from any given point, without a circle, there be drawn two straight lines cutting the cir cle, then of the circumferences which they inter cept, that which is the nearer to the given point is less than the other. , , PROP. XXVII. (xxxvn.) In equal circles, the greater of two circum ferences subtends the greater angle, whether the angles compared be at the centers or circum ferences. PROP. XXVIII. (XXXVIII.) In equal circles, the greater chord subtends the greater circumference. PROP. XXIX. (XXXIX.) In equal circles, the greater circumference has the greater chord. APPENDIX I. (49) If any equilateral and equiangular polygon be inscribed in a circle, a straight line drawn, from any of its angles, through the center of the circle, bisects the opposite side at right angles. (XL.) The two straight lines in a circle, which join the extremities of two parallel chords, are equal to each other. PROP. XXX. (XLI.) If from any point, in the diameter of a semi circle, there be drawn two straight lines to the circumference, one to the bisection of the circum ference, the other at right angles to the diameter, the squares upon these two lines are, together, the double of the square upon the semi-diameter. PROP. XXXI. (XLII.) If the chords of two arches of any the same circle cut each other at right angles, the squares of the four segments of the chords, are, together, equal to the square of the diameter. COR. Hence, if the diagonals of a quadrila teral rectilineal figure, inscribed in a circle, cut d (50) APPENDIX I. each other at right angles, the aggregate of the squares of the sides is the double of the square of the diameter of the circle. (XLIH.) To draw a straight line, cutting two concentric circles, so that the part of it which lies within the greater circle may be the double of the part which lies within the less. (XLIV.) To draw a straight line which shall touch two given circles. (XLV.) If the point, in which two straight lines that are perpendicular to each other meet, be applied to the circumference of a circle so that the straight lines themselves cut the circumference, the center of the circle is in the bisection of the straight line joining those two intersections. Thus the center of a given circle may readily be found by means of the instrument called a square. (XLVI.) If from the extremities of any diameter, of a given circle, perpendiculars be drawn to any chord APPENDIX I. (51) of the circle, that is not parallel to the diameter, the less perpendicular shall be equal to the segment of the greater contained between the circumfer ence and the chord. (XLVII.) If from the extremities of any diameter, $f a given circle, perpendiculars be drawn to any chord of the circle, they shall meet the chord, produced, in two points which are equidistant from the center. (XLVII i.) If upon either radius bounding a quadrantal cir cular arch as a diameter, a semi-circle be described, any chord of the semi-circle drawn from the center of the quadrant shall be equal to the perpendicular distance of the point, in which the chord produced meets the quadrantal arch, from the other radius. PROP. XXXII. (XLIX.) If the angle contained by two straight lines, one of which cuts a circle and the other meets it, be equal to the angle in the alternate segment of the circle, the straight line which meets, shall to uqh. the circle. (52) APPENDIX I* . A straight line touching a circular arch in the bisection of that arch, is parallel to its chord, (u.) If an equilateral triangle be described about a given circle, the straight lines joining the points of contact shall contain another equilateral tri angle ; and the side of the circumscribed triangle is the double of the side of the inscribed triangle so contained. PROP. XXXIII. (LII.) Upon a given finite straight line to describe a segment of a circle, which shall be similar to a given segment of another circle. (LIII.) If any equilateral rectilineal figure, of an even number of sides, be inscribed in a given circle, a curvilineal figure may be found that is equal to it, and that is bounded by arches of circles, each of which circles is equal to the given circle. V V ^ ; ,-VV M^-V (LIY.) n^ jVH^^, The base, the vertical angle, and the altitude of a triangle being given, to construct the triangle. APPENDIX I. (53) To find a point in a given straight line, from which if straight lines be drawn to two given points, on the same side of the given line, they shall contain an angle equal to a given rectilineal angle. The vertical angle, the base, and the aggregate of the three sides of a triangle being given, to construct the triangle. (LVII.) The perimeter and the three angles of a triangle being given, to construct the triangle. (LVIII.) From two given points, in the circumference of a circle, to draw two equal chords of that circle, which, produced if necessary, shall make with one another an angle equal to a given rectilineal angle. PROP. XXXV. To produce a given straight line so that the rectangle, under the given straight line, and the part of it produced, shall be equal to a given square. APPENDIX I. . . (LX.) If through any point in the common chord of two circles, which intersect one another, there be drawn any two other chords, one in each circle, their four extremities shall all lie in the circum ference of a circle. (LXI.) If through the given extremity of any diameter of a circle straight lines be drawn to meet an inde finite straight line without the circle, which is perpendicular to the diameter produced, the rect angles contained by the segments of these lines lying between the given point, the point in which each of them cuts the circumference again, and the indefinite line, shall be equal to each other. (LXI i.) From the obtuse angle of an obtuse-angled tri angle, to draw a straight line to the base, the square of which shall be equal to the rectangle contained by the segments, into which it divides the base. (LXIU.) To make a rectangle which shall be equal to a given square, and shall have its two adjacent sides, together, equal to a given straight line 5 the side APPENDIX I. (55) of the given square being less than the half of the given straight line. PROP. XXX VL (LXIV.) If two tangents be drawn to a given circle, from any the same point without it, they shall be equal to each other; only two tangents can be drawn to a circle from the same given point without it ; and if from a given point without a circle, two equal straight lines be drawn to the convex cir cumference, one of which touches the circle, the other shall also touch it. . ;iU 10 tii Vir. teiif -lto (LXV.) If a quadrilateral rectilineal figure be described about a circle, the angles subtended, at the center of the circle, by any two opposite sides of the figure, are, together, equal to two right angles. (LXVI.) If two given straight lines touch a circle, and if any number of other tangents be drawn, all on the same side of the center, and all terminated by the two given tangents, the angles which they sub tend, at the center of the circle, shall be equal to one another. (56) APPENDIX I. COR. The two segments^ which any two tan gents, so drawn, cut off from the two given tan gents, also subtend equal angles, at the center of the circle. (LXVII.) To produce a given straight line, so that the rectangle contained by the whole line thus pro duced, and the part of it produced, shall be equal to a given square. (LXVIII.) If, from the bisection of any given arch of a circle, a straight line be drawn cutting the chord of that arch, or the chord produced, and the cir cumference also of the circle, the rectangle con tained by the two parts of the straight line so drawn, the one lying between the point of bisection and the circumference, the other between the point of bisection and the chord, shall be equal to the square of the chord of half the arch. ^ (LXIX.) Hence, from the bisection of a given arch of a circle, to draw a straight line, such that the part of it intercepted between the chord, or the chord pro duced, of the given arch and the circumference, shall be equal to a given straight line. APPENDIX I. (57) (LXX.) Hence, through any given angle of a given equilateral four-sided figure, to draw a straight line terminated by the sides produced, containing the angle opposite to the given angle, which shall be equal to a given straight line. SCHOLIUM. In the application of Algebra to Geometry, the equation upon which the construction of a Pro blem depends may, sometimes, be of more di mensions than two, and yet the solution may be obtained by means of Plane Geometry. If, there fore, the resulting equation be of an higher order than a quadratic, a trial must be made whether it be not divisible by some binomial, one of the terms of which is a divisor of the last term of the equation, and the other a similar power of the un known quantity ; it may thus, perhaps, be reduced to a quadratic equation. If it be a biquadratic, DES CARTES method (WOOD S Alg. Art. 329.) should be had recourse to. Thus the last question, if the figure be a square, may be solved by a simple and elegant Geome trical construction. But, if it be treated algebrai cally, the resulting equation is a biquadratic. For, let a denote the side of the square, b the given straight line, and x and y the two segments of (58) APPENDIX I. the sides produced, cut off by the line which is to be drawn through the angular point equal to b. Then, (I.) x*+y z = b* (E. 47. 1.), and (II.) (x-a) .y = ax (E. 4. 6) Whence, x 4 - 2ax* + (2a z - b^x 1 + 2ab*x - tftf = ; which equation is reduced, by Des Cartes method, to this cubic ; / + (a 2 - 2&% 4 - (a 4 - b^)f - (a 6 + 2a*b* -f a*V} = 0; and a*+& 2 being one of the divisors of the last term, y* 1 a 2 b* is found, upon trial, to divide this equation without a remainder; the quotient is i/ 4 4-(2a 2 -^)3/ a +( 4 4-a 2 6 3 )=0; which is an equation of the quadratic form, and, therefore, the question may be solved by means of Plane Geometry. But, now, the straight line, which is to be in vestigated, being supposed to have been actually drawn, let x be put for the segment between either of its extremities, and the angular point of the square through which it passes; from its other extremity let a perpendicular be drawn to it, so as to meet the opposite side of the square produced ; and let y denote the segment of the produced side of the square, lying between that angular point and the perpendicular: Then, the side of the square being denoted, as before, by a, and the given straight line by b, it follows from Ded. I. Prop. 26. B. 1, that the perpendicular is equal to x, and by the APPENDIX I. (59) help of E. 47. 1, and E. 4. 6, the following equa tions are easily deduced : = b* 2 (b- jc) . x i.e. ? 2 .= ^- 2az; And, thus, the straight line, required to be drawn, is determined by the solution of an equation, which presents itself, in the very first instance, in the com mon form of an adtected quadratic. If, in the analytical investigation of this ques tion, whether the figure be a square or a rhom bus, Trigonometrical forms be introduced, the re sulting equation will be a quadratic; and, as the sine of any angle is also the sine of its supple ment, that equation gives four different positions of the line to be placed between the two sides of the figure produced. So that when the problem is made general, the sides must be produced both ways. (LXXI.) If two circles cut each other, and from any point, in the straight line produced, which joins their intersections, two tangents be drawn, one to each circle, they shall be equal to one another. (LXXII.) If two circles cut each other, and if two tan gents drawn, one to each circle, from any point (60) APPENDIX I. without them, be equal, the straight line, joining the intersections of the circles, shall, if it be pro duced, pass through the common extremity of the equal tangents. (LXXIII.) The straight line which passes through the intersections of two circles, that cut one another, bisects the straight line which touches both the circles. (LXXIV.) iiQfJiKJp Two circles being given, neither of which lies within the other, to draw a straight line, such that the tangents to the two circles, drawn from any point of the line, shall be equal to one another. (LXXV.) To divide a given straight line in,to two parts, so that the square of the one shall be equal to the rectangle contained by the other and a given straight line. (LXXVI.) If a given circle be cut by any number of circles,, which all pass through the same two given APPENDIX I. (61) points without the given circle, the straight lines, joining the points of each of these intersections, are either all parallel, or all meet when produced in the same point. (LXXVII.) If a perpendicular be let fall from the right angle, of a right-angled triangle, on the hypote nuse, the rectangle contained by the hypotenuse and either of the segments, into which it is divided by the perpendicular, is equal to the square of the side adjacent to that segment. (LXXVIII.) To draw a tangent to a circle, such, that the part of it intercepted between two straight lines, given in position, but of indefinite length, shall be equal to a given finite straight line: 1st, When the indefinite straight lines both pass through the center of the circle : 2dly, When they are parallel to one another : 3dly, When they are not parallel, but are equi distant from the center. (62) APPENDIX I. (LXXIX.) If two straight lines, which touch two given circles, the one touching both the circles on the one side of them, the other on the other, be cut by a third tangent, which touches the two circles on contrary sides of them, then, of the segments into which the two first tangents are thus divided, those which are alternate are equal to one another. (LXXX.) The perimeter, the vertical angle, and the alti tude of a triangle being given, to construct the triangle. (LXXXI.) If from the intersection of any two tangents to a circle, any straight line be drawn cutting the chord which joins the two points of contact and again meeting the circumference, it shall be divided by the circumference and the chord into three seg ments, such, that the rectangle contained by the whole line and the middle part, shall be equal to the rectangle contained by the extreme parts. (LXXXII.) To make a rectangle which shall be equal to a <riven square, and have the difference between its two adjacent sides equal to a given straight line. APPENDIX I. (63) (LXXXIII.) From a given point without a circle, to draw a straight line cutting the circle, so that the rect angle contained by the part of it without, and the part within, the circle, shall be equal to a given square.* PROP. XXXV 1 1. (LXXXIV.) To describe a circle which shall touch a given straight line, and pass through two given points, both on the same side of the given line. (LXXXV.) To describe a circle which shall touch two given straight lines, and pass through a given point between them. (LXXXVI.) To describe a circle which shall touch two given straight lines, and also touch a given circle. * The limitation to this Problem is evident from the con struction by which it is solved: The same remark may also be applied to several other problems, in this Appendix, in the enunciations of which the necessary limitation is not specified. (64) APPENDIX I. (LXXXVII.) To describe a circle which shall pass through a given point, and touch both a given circle, and a given straight line. (LXXXVIII.) In a straight line of indefinite length, but given in position, which cuts a given circle, to find a point, from which if a straight line be drawn to touch the circle, it shall be equal to a given finite straight line. ( LXXXIX.) To describe a circle that shall touch a given straight line, and that shall also touch two given circles, neither of which lies within the other. To describe a circle which shall touch a given circle, and pass through two given points, either both without the circle, or both within it. (xci.) To find a point in a straight line given in posi tion from which if two straight lines be drawn to two given points, without the given line, they shall have, first, their difference, and, secondly, their aggregate, equal to a given finite straight line. APPENDIX I. (65) 8M( (XCH.) M The base and the altitude of a triangle being given, together with the aggregate or the dif ference, of the two remaining sides, to construct the triangle. 4$tft$- v >I?S> Bftfi Iwi^ ; u (xcin.) Three points being given, to find a fourth, from which if straight lines be drawn to the other three, two of them shall be equal, and the difference between either of these and the third shall be equal to a given straight line. (xeiv.) To describe a circle that shall touch three given circles, of which two are equal to one another. (xcv.) To find a point, in the circumference of a given circle, from which if two straight lines be drawn to two given points, without the circle, the chord joining the intersections of the lines so drawn and the circumference, shall be parallel to the straight line joining the two given points. DEDUCTIONS FROM THE FIRST SIX BOOKS OF BOOK IV. PROP. IV. _ THREE straight lines being given, which, when produced, do not all three meet in the same point, and of which the middle line is not parallel to either of the others, to describe a circle which shall touch each of them. The three straight lines, which bisect the three angles of a triangle, meet in the same point. If a circle be inscribed in a right-angled tri angle, the excess of the two sides, containing the right angle, above the third side, is equal to the diameter of the inscribed circle. APPENDIX I (67) The straight line bisecting any angle of a tri angle, inscribed in a given circle, cuts the circum ference in a point which is equidistant from the extremities of the side opposite to the bisected angle, and from the center of a circle inscribed in the triangle. In a given circle, to inscribe three equal circles, touching each other and the given circle. To inscribe three circles in an isosceles triangle, touching each other, and each of them touching two of the three sides of the triangle. PROP. VI. V " j* . " (vii.) ."v" , The square, inscribed in a circle, is equal to the half of the square upon its diameter. (yin.) In a given circle, to inscribe a rectangle equal to a given rectilineal figure, not exceeding the half of the square upon the diameter. 3fb (68) APPENDIX I. (ix.) -> . , i^i"*- If from any point, in the circumference of a given circle, straight lines be drawn to the four an gular points of an inscribed square, the aggregate of the squares of the four lines, so drawn, shall be the double of the square of the diameter. PROP. VII. In a given circle, to inscribe four circles equal to each other, and in mutual contact with each other and the given circle. COR. In the same manner, four equal circles may be inscribed in a given square, touching each other and the sides of the square. PROP. VIII. To inscribe a circle in a given rhombus. 8 s, .- -, (X"-) - rs? . ^ To inscribe a circle in a given trapezium, of which two opposite sides are, together, equal to the other two sides taken together. APPENDIX I. (69) PROP. X. (XIII.) Upon a given finite straight line, to describe an equilateral and equiangular decagon. PROP. XL (xiv.) Upon a given finite straight line, to describe an equilateral and equiangular pentagon. ^rS Jb liiJ i! (xv-) . ." : " :); The angle of a regular pentagon exceeds a right angle by one-fifth part of a right angle ; and is three times as great as the angle contained by any two sides of the figure, which are not adjacent to each other, produced so as to meet. (xvi.) If isosceles triangles could be described, having their angles at the base, any required multiple of the angle at the vertex, any regular rectilineal figures whatever of an uneven number of sides, might be inscribed in a circle ; and if isosceles tri angles could be described, having each of the angles (70) APPENDIX I. at the base to the angle at the vertex, in the ratio of any odd numbers to two, any regular figures whatever of an even number of sides might be in scribed in a given circle ; the bases of these several triangles being the sides of the several inscribed figures. SCHOLIUM. It is manifest, that, by the help of E. 9. 1, E. 2. 4, E. 6. 4, E. 11. 4, and E. l6. 4, the cir cumference of a circle may be divided into three, six, twelve, &c. equal parts; into four, eight, sixteen, &c. equal parts ; into five, ten, twenty, &c. equal parts ; and into fifteen, thirty, sixty, &c. equal parts. But the problem, announced in the last deduction, has not yet been solved geometrically : Nor has any other general method, depending upon plane geometry, been discovered, by which the circum ference of a circle may be divided into any given number of equal parts. PROP. XV. (xvn.) The square of the side of a regular pentagon, inscribed in a given circle, is equal to the square of the side of a regular decagon, together with the APPENDIX I. (71) square of the side of the regular hexagon, both in scribed in that given circle. (xvill.) Upon a given finite straight line/ to describe an equilateral and equiangular hexagon. DEDUCTIONS PROM THE FIRST SIX BOOKS BOOK V. PROP. XII. (I.) . IF any number of equal ratios be each greater than a given ratio, the ratio of the sum of their antecedents, to the sum of their consequents, shall be greater than that given ratio. PROP. XIII. . (II.) If the first of four magnitudes have a greater ratio to the second than the third has to the fourth,, APPENDIX I. (73) the second shall have to the first a less ratio the fourth has to the third. PROP. XVI. *b ^ihbfvd fcj (HI.) IlS^to ^b f !)ru;. * If the first of four magnitudes have a greater ratio to the second than the third has to the fourth, the first shall have to the third a greater ratio than the second has to the fourth. PROP. XVII. (IV.) If the first, together with the second, of four magnitudes, have a greater ratio to the second, than the third, together with the fourth, has to the fourth, the first shall have a greater ratio to the second than the third has to the fourth. PROP. XVIII. If the first of four magnitudes have a greater ratio to the second than the third has to the fourth, (74) APPENDIX I. the first, together with the second, shall have to the second a greater ratio than the third, together with the fourth, has to the fourth. If the first term of a ratio be less than the second, the ratio shall be increased by adding the same quantity to both terms ; but if the first term be greater than the second, the ratio shall be dimi nished by adding the same quantity to both, . (vn.) ;i/ - ; "" -. "-" <"* If the first of four magnitudes have a greater ratio to the second than the third has to the fourth, the first, together with the third, shall have to the second, together with the fourth, a greater ratio than the third has to the fourth, and a less ratio than the first has to the second. (VIII.) If the first, together with the second, have to the second, a greater ratio than the third, together with the fourth, has to the fourth, then shall the first, together with the second, have to the first, a less ratio than the third, together with the fourth, has to the third. (IX.) If the first, together with the second, have to the third, together with the fourth, a greater ratio APPENDIX I. (75) than the first has to the third, then shall the second have to the fourth a greater ratio, than the first, together with the second, has to the third, together with the fourth. PROP. XIX. (x.) If any number of magnitudes be proportionals, their differences shall, also, be proportionals. PROP. XXII. If there be three magnitudes, and other three, and if the first have a greater ratio to the second, in the former set, than the first has to the second, in the latter; and if, also, the second have to the third, in the former set, a greater ratio than the second has to the third, in the latter ; then shall the first have a greater ratio to the third, in ihe former set, than the first has to the third, in the latter. PROP. XXIII. t If there be three magnitudes, and other three, and if the first have to the second, in the former (16) APPENDIX I. set,, a greater ratio than the second has to the third, in the latter ; and if, also, the second have to the third, in the former set, a greater ratio than the first has to the second, in the latter; then shall the first have to the third, in the former set, a greater ratio, than the first has to the third, in the latter. PROP. XXV. (inn.) If three magnitudes be proportionals, the two extremes are, together, greater than the double of the mean. COR. An arithmetic mean proportional, be tween two given magnitudes, is greater than a geometric mean proportional between the same two magnitudes. (XIV.) If three magnitudes be proportionals, the excess of the greatest above the mean, is greater than the excess of the mean above the least of them. -< ; [-. : - (XV.) " - . r ^ If there be two sets of magnitudes, the one geometric, and the other arithmetic, proportionals, and if the two first magnitudes be the same in both, APPENDIX I. (77) any other magnitude in the former set, shall be greater than the corresponding magnitude in the latter. COR. The two first magnitudes, in both the sets, being the same, if the second of the geometric pro portionals be greater than the second of the arith metic proportionals, then, much more, will every other magnitude, in the former set, be greater than the corresponding magnitude in the latter. (XVI.) If there be two series of magnitudes, the one arithmetically proportional, the other geometrically proportional, but each having the same magnitude for its first term, and if the last term of the arith metic series be not less than the last term of the geometric series, any other term of the former series shall be greater than the corresponding term in the latter. DEDUCTIONS PROM THE FIRST SIX BOOKS OF BOOK VI. PROP. I. (I.) IF the bases of four rectangles be proportionals,, and their altitudes be also proportionals, the rect* angles themselves shall likewise be proportionals. COR. 1. If four straight lines be proportionals, their squares shall also be proportionals. COR. 2. Conversely, if four squares be propor^ tionals, their sides shall likewise be proportionals. PROP. III. and A. Simsorfs Euclid. portion (n.) To cut a given straight line in harmonic pro- tion. APPENDIX I. (79) PROP. IV. (HI-) If the base of an isosceles triangle be produced to meet a straight line drawn from the opposite angle perpendicular to either of the equal sides, that side shall be a mean proportional between the base and the half of the line which is made up of the base and of the part produced. The diameter of a circle is a mean proportional between the sides of an equilateral triangle and hexagon described about the circle. (v.) Through a given point, within a triangle, to draw a straight line cutting the sides, either of them being produced, if necessary, so that the rect angle contained by the segments between the vertex and the cutting line, may be equal to a given square. _ :u. tfl* >;,,,.-:. (VI.) V ,,.,: - .?,, n If from a given point, without a circle, two straight lines be drawn to the concave circum- (80) APPENDIX I. ference, they shall be reciprocally proportional to the parts of them between the given point and the convex circumference. The straight lines, drawn from the bisections of the three sides of a triangle to the opposite angles, meet in the same point. (VIII.) >rij io cm sham #i ill dflJ to TIBII >ra bafi To find, within a given rectilineal angle, the Z0CM of all the points, from each of which, if two straight lines be drawn, so as to meet the lines con taining the given angle, and so as always to be parallel to two straight lines given in position, they shall be to one another in a given ratio. ohTO "(ix.) ^"V**-** ^ If a circle be touched, in the same point, both externally and internally, by two other circles, and through the point of contact two straight lines be drawn, the parts of them intercepted between the circumference of the given circle, and that of the circle which touches it internally, shall have to one another the same ratio as the parts which are chords of the other circle. From the center of a given circle, to draw a straight line to meet a given tangent to the circle, APPENDIX I. (81) so that the segment of the line between the circle and the tangent shall be any required part of the tangent. From a given triangle to cut off a rhombus ; the base of the rhombus being part of the base of the triangle, and /having its extremity in a given point of that base. If two given circles touch each other internally, and a chord of the greater, which is perpendicular to the straight line joining their centers, also touch the less, to describe a circle which shall touch the two given circles, and also touch the chord on the same side as the less circle touches it. (XI 1 1.) If two triangles have one angle of the one, equal to one angle of the other, and also another angle of the one, together with another angle of the other, equal to two right angles, the sides about the two remaining angles shall be proportionals. (XIV.) If, from the extremities of the base of a given triangle, there be drawn two straight lines, both on the same side of the base, and each equal to the adjacent side, and making with that side an angle (82) APPENDIX I. equal to the vertical angle of the triangle, then the straight lines which join the extremities of the lines so drawn, and the further extremities of the base, shall cut off, from the sides, equal segments towards the vertex ; and each of those segments shall be a mean proportional between the other segments, that are towards the base. (xv.) lam " [Jio If at the extremities of the hypotenuse of a right-angled triangle two straight lines be drawn, on the same side of the hypotenuse as the right angle, each equal to, and each perpendicular to, the adjacent side, the two straight lines joining each of their extremities and the further extremity of the hypotenuse, shall cut each other in the same point of the perpendicular drawn to the hypote nuse from the right angle.* fen nrtovisnu ,; (XVI } * win ownr " The semi-circumference of a circle having been divided into any number of equal parts, and chords having been drawn, from either extremity of the diameter, to the several points of division, the first chord has to the second, the same ratio which the second has to the aggregate of the first and third ; * This proposition explains a circumstance belonging to the figure of E. 47. 1 ; and it may very easily be proved, ex absurdo, by the help of the third deduction from E. 32. x. xn. set dowa in this Appendix* APPENDIX I. (83) or the same ratio which any other chord has to the aggregate of the two chords that are next to it. PROP. VI, . (XVII.) If two trapeziums have an angle of the one equal to an angle of the other, and if, also, the sides of the two figures, about each of their angles, be proportionals, the remaining angles of the one shall be equal to the remaining angles of the other. PROP. VIII. (xvni.) To divide a given finite straight line into two parts, such, that another given straight line, not greater than the half of the former, shall be a mean proportional between them. (XIX.) If two straight lines touch a circle at opposite extremities of its diameter, any other tangent of the circle, terminated by them, is so divided in its point of contact, that the radius of the circle is a mean proportional between its segments. (84) APPftNDIX t. (XX.) ; ^ ! !* !^"!; If two given circles touch each other, and also touch a given straight line, the part of the line between the points of contact, is a mean propor tional between the diameters of the circles. (XXI.) Two straight lines being given, which are the two first of a series of proportionals, to find the rest ; and, if the series decrease, to find a line which shall be greater than the aggregate of any number, whatever, of its terms, but to which the aggregate may approximate indefinitely. COR. The first term of a decreasing series of proportionals is a mean between the excess of the first term above the second, and the line which is the limit of all the terms. PROP. X. (XXII.) To describe a square which shall have a given ratio to a given rectilineal figure. (XXIH.) To divide a given finite straight line into two parts, the squares of which shall be to one another in a given ratio. APPENDIX 1. (85) (XXIV.) To cut off from a rectangle a similar rectangle which shall be any required part of it. (XXV.) To find two points, situated in two adjacent sides of a given oblong, at equal distances from two opposite angles, from which, if two straight lines be drawn parallel to the sides of the figure, they shall cut off from it any part required. (XXVI.) Hence, within a given rectangle, to describe another rectangle which shall be any required part of it, and shall have its four sides all equally distant from the four sides of the given rectangle. (XXVII.) To divide a given straight line into two parts, such, that the rectangle contained by the whole line and one of its parts, shall have a given ratio to the square of the other part. (XXVIII.) The base, the vertical angle, and the ratio of the two sides of a triangle being given, to con struct it. (86) APPENDIX I. (XXIX.) One given circle lying within another, to find a point from which, if two tangents be drawn, one to each of the given circles, they shall be to each other in a given ratio. (xxx.) A given finite straight line being divided into any two given parts, to divide it again, so that the rectangle contained by the two former given parts shall have a given ratio to the rectangle contained by the two latter parts. .^ (XXXI.) To draw a straight line to touch a given arch of a circle, so that being terminated by the semi- diameters, produced, which bound the arch, it shall be divided by the point of contact, into two parts that are to one another in a given ratio. (XXXII.) Ol * i> i - -> ( i O I ) 1 ; I. J V : . ! Two points being given, one in each of two parallel straight lines, and a third point being also given, without them, to draw, from that third point, a straight line so to cut the parallels, as that the segments of the parallels, between it and the two first points, shall be to one another in a given ratio. APPENDIX 1. (87) (XXXIII.) To find a point within a given triangle, from which if three straight lines be drawn to the three angles of the triangle, it shall thereby be divided into three parts that are each to each in given ratios. (xxxiv.) The base, the perpendicular distance of the vertex from the base, and the ratio of the two sides of a triangle being given, to construct it. (xxxv.) To divide a given circular arch into two parts, so that the chords of those parts shall be to each other in a given ratio. (XXXVI.) To inscribe a square in a given trapezium, which has the two sides about any angle equal to one another, and the two sides about the opposite angle also equal to one another. (XXXVII.) To inscribe a square in a given trapezium. PROP. XL (xxxvni.) To determine the locus of the vertices of all the triangles which can be described on a given base, (88) APPENDIX I. so that each of them shall have its two sides in a given ratio. (xxxix.) Hence, to find a point, from which if three straight lines be drawn to three given points, they shall be each to each in given ratios. a jj > .axwtoci. < (XL.) l; >,! ,K Hence, a straight line being divided into three given parts, to find a point without it, at which the three parts shall subtend equal angles. (XLI.) To find a point in a straight line, given in posi tion, from which, if two straight lines be drawn to two given points, both on the same side of the given line, they shall be to each other in a given ratio. (XLII.) In a given parallelogram to inscribe a parallel ogram that shall have it s two adjacent sides in a given ratio to one another, and that shall be the half of the given parallelogram. PROP. XII. (XLI ii.) Through the bisection of the base of a given triangle, to draw a straight line cutting the sides. APPENDIX I. (89) of which one is produced, so that the segments of the line, between the bisection of the b.ase and the two sides, shall be to one another in a given ratio. (XLIV.) From a given point, without a given rectilineal angle, to draw a straight line cutting the two lines which contain the angle, so that the distances of the two intersections from the given point, shall be to one another in a given ratio. (XLV.) From a given point, without a given rectilineal angle, to draw a straight line cutting off from the lines which contain the angles, segments, towards the summit of the angle, which shall be to one another in a given ratio. iri -OS (XLVI.) .d-MJ-j From a given point, to draw a straight line to cut a given circle, so that the distances of the two intersections from the given point, shall be to each other in a given ratio. (XLVII.) To find, between two given parallel straight lines, the locus of all the points, from each of which if two straight lines be drawn to the two given parallels, so as always to make with them, towards (90) APPENDIX I, the same parts, given angles, they shall be to one another in a given ratio. (XLVIII.) Two given circles lying wholly without one another, through a given point, which is between the two circles, and which is posited in the straight line joining their centers, to draw a straight line that shall be terminated by the convex circum ferences, and divided, by the given point, into two parts, that are to one another in a given ratio. (XLIX.) To find a point, from which if three straight lines be drawn to meet as many given straight lines, so as to make, each with the line on which it falls, an angle equal to a given angle, the lines so drawn shall be, each to each, in given ratios. PROP. XV, (L.) To make an isosceles triangle, which shall be equal to a scalene triangle, and shall also have an equal vertical angle with it. APPENDIX I. (91) PROP. XVI. If a straight line, drawn from the vertex of an isosceles triangle cutting the base, be produced to meet the circumference of a circle described about the triangle, the rectangle contained by the whole line so produced, and the part of it between the vertex and the base, shall be equal to the square of either of the equal sides of the triangle. (Lll.) *! Of four straight lines which are continual pro portionals, the two extremes being given, and also a line which is equal to the difference of the other two, to find those two lines. (LIII.) The semi-aggregate of two straight lines, and also another straight line, which is a mean pro portional between them, being given, to find the two lines. !>TC ^*>nt ifoirw ot , //I MI} lo sDiTftfarfo $d3 (LIV.) To make a triangle, which shall have its two sides equal to two given straight lines, each to each, and shall have its base equal to the perpen dicular distance of the vertex from the base. (92) APPENDIX I. - -^.v : (tv.) ; _ " I " If from any point in the diameter, or the dia meter produced, of a given parallelogram, perpen diculars be let fall on the two adjacent sides, pro duced, if necessary, which meet the diameter, the perpendiculars shall be reciprocally proportional to the sides on which they fall. >v,* ^yny\l M (LVI.) , ;{^, .-;.--" ,! Through a given point, within a triangle, to draw a straight line to meet the sides, either of them being produced, if necessary, so that the rect angle contained by the segments, into which the line is divided by the given point, may be equal to a given square. (LVII.) To find a point, from which if three straight lines be drawn to three given points, their dif ferences shall be severally equal to three given straight lines; the difference of any two of the straight lines to be drawn, not being greater than the distance of the two points to which they are to be drawn. (LVIII.) To describe a circle, which shall pass through a given point, and touch two given circles. APPENDIX I. (93) To describe a circle that shall touch three given circles. PROP. XVIII. Upon a given finite straight line, to describe an equilateral and equiangular polygon, having the number of its sides equal to four, eight, sixteen, &c.; or to three, six, twelve, &c.; or to five, ten, twenty, &c. ; or to fifteen, thirty, sixty, &c. sides. PROP. XIX. (LXI.) To cut off from a given triangle any part required, by a straight line drawn parallel to a given straight line. PROP. XX. (LXH.) To describe a polygon, similar to a given poly gon, and having a given ratio to it. (94) APPENDIX I (LXIII.) Any regular polygon, inscribed in a circle, is a mean proportional between the inscribed and cir cumscribed regular polygons of half the number of sides. (LXIV.) If from two points similarly situated, one in each of any two homologous sides of two similar polygons, two straight lines be drawn making equal angles with those sides, they shall cut off from the polygons two similar figures; and the one shall be the same part of the one polygon, that the other is of the other. PROP. XXI T. (LXV.) If any two chords of a circle intersect each other, the straight lines joining their extremities shall cut off equal segments from the chord which passes through the common intersection of the two former chords and is there bisected. (LXVI.) Two similar rectilineal figures being given, to find a third figure also similar to them and a mean proportional between them. APPENDIX I. (95) >-)( OWJ PROP. XXIII. (LXVII.) Equiangular parallelograms have to one another the same ratio as the rectangles contained by the sides about equal angles in each. COR. Triangles, having equal vertical angles, are to one another as the rectangles contained by the sides about those equal angles. (LXVIII.) Through a given point, either without or within a given triangle, to draw a straight line which shall cut off from the triangle any part required. COR. Hence, and by the help of Trigonometry, any given rectilineal figure may be divided into two parts, which are to each other in any given ratio, by a straight line drawn from a given point, situate without the given figure. (LXIX.) If two sides of a trapezium be parallel, and a straight line be drawn cutting them, and meeting also the other two sides, (any of the sides being produced, if necessary) the two rectangles con tained by the respective segments of the parallel sides, have to each other the same ratio, as the (96) APPENDIX I. two rectangles contained by the segments into which the line, so drawn, is severally divided by each of the two parallels. PROP. XXX. (LXX.) A given straight line being cut in extreme and mean ratio, if from the greater segment the less be taken, the greater segment also will thus be cut in extreme and mean ratio ; and if a straight line, equal to the greater segment, be added to the given line, the line which is made up of the given line and this segment, is also cut in ex treme and mean ratio. (LXXI.) Upon a given straight line as an hypotenuse, to describe a right-angled triangle, which shall have its three sides continual proportionals. (LXXH.) The perimeter being given of a right-angled triangle, having its three sides proportionals, to construct the triangle. (LXXIII.) The radius of a given circle having been divided in extreme and mean ratio, the greater segment APPENDIX I. (97) shall be equal to the side of an equilateral and equiangular decagon inscribed in the circle. PROP. XXXI. f V (LXXIV.) Any rectangle is the half of the rectangle con tained by the diameters of the squares of its two sides. PROP. XXXIII. (LXXV.) In different circles the semi-diameters which bound equal sectors contain angles reciprocally proportional to their circles ; and conversely. (LXXVI.) To trisect a given circle, by dividing it into three equal sectors. (LXXVI i.) If, from the greater of two unequal sides, of a given triangle, be cut off a part equal to the less, that segment shall have to the remaining seg ment, a ratio greater than the ratio which the angle adjacent to the remaining segment, has to the angle adjacent to the segment first cut off. (98) APPENDIX I. (LXXVill.) The greater of any two unequal arches, of a given circle, has a greater ratio to the less arch, than the chord of the greater has to the chord of the less. COR. The greater angle, at the base of a scalene triangle, has a greater ratio to the less angle, than the greater side has to the less side. OV/j PROP. D. Simsorfs Euclid. (LXXIX.) If, from the center of the circle, described about a given triangle, perpendiculars be drawn to the three sides, their aggregate shall be equal to the radius of the circumscribed circle, together with the radius of the circle inscribed in the given triangle. In the investigation of the six next following deductions, it is necessary to quote the theorem, which is the Second Proposition of the Twelfth Book of Euclid s Elements. 1; ;<r> Ji:{ & fto )!> -ni .ofjanuhl i tdvitj # (txxx.) To divide a given circle into any required num ber of equal parts, by circles described within it, about its center. APPENDIX I. (99) (LXXXI.) To find a circle, which shall be equal to the excess of the greater of two given circles above the less. (LXXXI i.) If, in any given circle, two chords cut each other at right angles, the four circles described upon their segments, as diameters, shall, together, be equal to the given circle. (LXXXI ii.) A circle is a mean proportional between any regular polygon, described about it, and a similar polygon, the perimeter of which is equal to the circumference of the circle. (LXXXIV.) If a figure be bounded by two circular arches^ subtending at their respective centers angles reci procally proportional to the circles to which they belong, a square may be found, that shall be equal to it. (LXXXV.) A circle is equal to the half of the rectangle contained by its circumference and its semi- diameter. COR. The circumferences of circles are to one another as their semi-diameters. (100) APPENDIX I. The following Propositions were omitted in their M\: proper places : o *f ^ii BOOK I. > VWIxr PROP. XXXIV. (XLII. A.) ^ . . -?J CJ IbJfp J (XI If any number of parallelograms be inscribed in a given parallelogram, the diameters of all the figures shall cut one another in the same point. PROP, xxxvm. ; ; ; (LVI. A.} If two triangles have the two adjacent sides of a parallelogram for their bases, and have their common vertex situated in the diameter, or in the diameter produced, they shall be equal to one another. BOOK III. PROP. XVI. r? (vii. A.) The diameter of a circle having been produced to a given point, to find in the part produced, a APPENDIX I. (101) point from which if a tangent be drawn to the circle, it shall be equal to the segment of the part produced, that is between the given point and the, point found. PROP. XXXVI. (LXXIV. A .) To find a point from which if straight lines be drawn to touch three given circles, none of which lies within another, the tangents so drawn shall be equal to one another. PROP. XXXVII. (LXXXV. A.) To describe a circle which shall have its center in a given straight line, which shall pass through a given point, and shall, also, touch another given straight line. > tid Jlwis ii <i jiii!^ jbso .bnuoi d r ait oT ZX-d) APPENDIX II. CONTAINING A SERIES OF PROPOSITIONS WHICH MAY BE SOLVED AND DEMONSTRATED BY THE PRINCIPLES OF Natural AND .1} to ^ T av:A f!3vja< :JH v *lmif iiwia * P* fif p 1 * APPENDIX II. PROPOSITIONS IN ri NATURAL PHILOSOPHY. ( ) To determine the directions in which two forces, represented, in quantity, each by a given straight line, must act, so that the equivalent com pound force may be represented by another given straight line ; the two former straight lines being, together, greater than the latter. nv,, ,b i nif *feu (II.) ;!" o>- .aieiq tain s.o- To resolve a given force into two forces, of which the directions shall make with one another an angle equal to a given angle, and the quantities shall, together, be equal to the quantity of another given force. To resolve a given force into two forces, of which the directions shall make with one another (106) APPENDIX II. an angle equal to a given angle, and which shall have the difference of their quantities equal to another given force. (IV.) To resolve a given force into two forces, the quantities of which shall be to one another in a given ratio, and the directions shall make with each other an angle equal to a given angle. There being given the quantity and the direc tion of one of the two forces which act upon a body, the direction of the other force, and also the force which is equivalent to them both, to find the quantity of the other force. Three points A> B, C 9 being given upon any plane, it is required to determine a point, P 9 on the same plane, to which if straight lines be drawn from A* B, and C, they shall be the directions in which three forces, each given in quantity, must act upon a body at P, so as to keep it at rest. The quantity and the direction being given of a force, which acts upon a body at rest, to deter mine the quantities and the directions of three equal forces, which also acting upon the body, APPENDIX II. (107) each in a direction at right angles to the directions of the other two, shall keep it at rest. (VIII.) The quantities and the directions being given, of two forces which act upon a given straight lever, at the extremities of its arms, and thereby keep the lever at rest on a fulcrum, to determine, geo metrically, the place of the fulcrum. The place of the fulcrum in a straight lever being given, and the quantities of two forces which are to act at its extremities, to determine the direc tions in which the two forces must act, so that these forces and the pressure on the fulcrum may be to each other as three given straight lines, of which any two are greater than the third. Two forces, each given in quantity, are to act upon a straight lever, of indefinite length, at two given points, both on the same side of the ful crum : It is required to determine the place of tho fulcrum, and the directions of the given forces, so that the pressure on the fulcrum may be to each of the given forces, as each of two given straight lines to a third given straight line, and may take place (108) APPENDIX II. . in a direction which shall make with the lever an angle equal to a given angle.* The force of gravity being supposed to tend to the earth s center, but to be a constant force, the pressure on the fulcrum of a straight lever, at the ends of which two given weights are appended, is the less, the nearer the lever is to the earth s center. To determine the position of a given straight lever, loaded at a given point with a given weight, so that it may rest when placed between two planes, each of them inclined to the horizon at a given angle. (XIII.) The arms of a bent lever being given, in length, and the angle being given, which they make at the fulcrum, to determine the position of the lever, when two given weights, appended at its extremi ties, balance one another. (XIV.) A flexible string, loaded at one of its ends with a given weight, and passing over a fixt pulley, has * Any two, of the given straight lines, are supposed to be greater than the third. APPENDIX II. (109) its other end fastened to a given point in the same vertical plane with the pulley, and it is further loaded, at a given point between the pulley and the fastened end of the string, with another given weight : It is required to determine any number of positions of this latter weight, in each of which it will balance the former weight. Three given weights being affixed to the same string, and the two extreme weights being made to act upon the middle weight, by means of two fixt pullies, which are in the same vertical plane, to determine the position of the string, when the whole system of weights is at rest. (XVI.) The position being given of a string, loaded with two weights, at two given points, and having its ends fastened at two given points in its vertical plane, it is required to find two straight lines, which shall be to one another as the two weights. (XVI I.) A string of given length being supposed to be fastened at two given points in a vertical plane ; to find the point in the string at which a given ring will rest. (xvm.) A system of given spheres, of equal weights and magnitudes, being so placed, as that the two (116) APPENDIX II. extreme spheres resting on two inclined planes, the whole system is supported in the form of an arch, it is required to determine the places of the centers of the spheres, and the position of the two inclined planes. (XIX.) To find the center of gravity of a trapezium, which has two of its sides parallel to one another. , v-fix i owl 9fte ban If the sides of a given triangle, taken in order, be cut proportionally, and the points of section be joined; the center of gravity of the triangle so formed will be the same as that of the given triangle. (XXI.) The centers of gravity of all the parallelograms which can be inscribed in a given parallelogram, will be in the same point as the center of gravity of the given parallelogram : And the center of a circle is the common center of gravity of all the rectangles that can be inscribed in it. (XXII.) If a given body remain at rest either within or without a given circle, or in its circumference, and another given body describe the circumference; to find the path of their center of gravity. APPENDIX II. (HI) (XXIII.) If any number of equal bodies be placed in the circumference of a given circle, to determine the locus of the several centers of gravity of any one of them, and each of the rest. (XXIV.) To find the path of the common center of gravity of two bodies, which move uniformly through two sides of a given triangle, in the same time, and are to one another inversely as those two sides. (XXV.) The paths of two bodies, which move with known uniform velocities, being given, so that their places at any given instant are known, it is required to determine the relative path, and the relative velocity, of the one, as seen from the other. (XXVI.) The position being given of two perfectly elastic balls on a table, the sides of which constitute a given polygon, to find the direction in which the one ball must be sent, so that after impinging successively on each of the sides, it may at last strike the other ball. (XXVII.) The positions being given of two perfectly elastic balls (A) and (B) 9 on an horizontal table, whichis (112) APPENDIX II. in the form of a rectangular parallelogram, to find the direction in which (A) must be sent, so that impinging (l) on one side, (2) on two of the sides, (3) on three of the sides, or (4) on four of the sides, it may at last strike the other ball. (XXVIII.) Given the position of a perfectly elastic ball (A), on an horizontal plane, to find the locus of the points in which another given elastic ball (B) can be placed, so that after (A) impinges upon (J5), (A) and (B) may strike two given points respectively. (xxix.) Two planes, which rest upon the level ground, so as to have their common section perpendicular to the horizon, are inclined to one another at a given angle, and a perfectly elastic ball, sent along the ground, so as always to strike a given point in the one, is reflected towards the other : It is required to determine the locus of the points, on the ground, from which if the ball be sent, it shall, in each case, after the two reflections, return to its point of projection. (xxx.) If the secant of a circular arch be in a vertical position, the times in which a heavy body, falling from a state of rest, would describe the secant, the APPENDIX II. (US) tangent, and the radius drawn through the extremity of the arch, are all equal. (xxxi.) If any number of heavy bodies be let fall, at the same instant, the one from the upper extremity of the vertical diameter of a given circle, the rest from the several upper extremities of chords drawn to the lower end of that diameter, so as to describe the diameter and the chords respectively, it is re quired to find the locus of the positions of the bodies, at any given time of their descent. (xxxn.) To determine the position of a straight line to be drawn from a given point to a given inclined plane, so that the time in which a heavy body falls from rest, down that line, may be equal to the time in which it would fall down the given plane. (xxxni.) From a given point in an horizontal plane, to draw a straight line to a given perpendicular to the plane, which shall be described by a heavy body, falling from rest, in the same time as the given perpendicular would be described, i (xxxiv.) From the right angle of a given, right-angled triangle, having one of the sides containing the APPENDIX II. right angle parallel to the horizon, to draw to the hypotenuse a straight line, such, that the times of descent down the line so drawn, and down the hypotenuse, shall be equal. (xxxv.) To place between an horizontal straight line of indefinite length, and a given perpendicular to it, a straight line of given length, less than the given perpendicular, which shall be described by a heavy body, falling down it from a state of rest, in the same time as the perpendicular. (XXXVI.) To place between an indefinite horizontal straight line and a given perpendicular to it, a straight line which shall make with the horizon an angle equal to a given angle, and shall be described by a heavy body in the same time as the given perpendicular. (xxxvu.) In a given circle, the plane of which is vertical, to draw a diameter which shall be described by a heavy body, in any given time, greater than the time in which the vertical diameter is described. (XXXVIII.) If, from any point in the circumference of a vertical circle, straight lines be drawn to the extre- APPENDIX II. mities of the vertical diameter, and if these lines be cut by a parallel to that diameter, the times of descent down the two segments, which are towards the diameter, shall be equal. (xxxix.) If two straight lines, drawn from a given point of an horizontal plane, to a given vertical straight line, be equidistant from the straight line which makes with the horizon, at the given point, an angle equal to half a right angle, they shall be described, by falling bodies in equal times. Kvi.KVj-oj, (XL.) r^flfHtaMt/uJj The time in which a heavy body, moving with the velocity acquired in falling down a given inclined plane, describes a given horizontal space, is to the time of its falling down the plane, as the half of the given space so described, is to the length of the inclined plane. (XLI.) If the plane of a scalene triangle be vertical, and the greater side exceed the less by the half of the base, the time of falling down the greater side is equal to the time of falling down the less, together with the time of describing the base, with the velocity thence acquired. (116) APPENDIX II. (XLII.) The base and one side being given in length, of a scalene triangle, the plane of which is vertical, to construct the triangle, so that the time of falling down the greater side may be equal to the time of falling down the less, together with the time of describing the base, with the velocity thence ac quired. (XLIII.) The plane of a scalene triangle being vertical, and one side and the angle which it makes with the base being given, to construct the triangle, so that it shall have the same property as that described in the last problem. (XLIV.) If the plane of a scalene triangle be vertical, the time of falling down the greater side shall be equal to the time of falling down the less, together with the time of describing, with the velocity thence acquired, a part of the base, that is equal to the double of the excess of the greater side above the less. (XLV.) If two bodies, after falling from the summit of a vertical scalene triangle, down the sides, begin APPENDIX II. (H7) to move along the base, each with its acquired velocity, to determine the point in which they will meet. (XLVI.) To find a point, in an horizontal plane, such, that if two straight lines, of given unequal lengths, be drawn from it to a given vertical straight line, the times of falling down them shall be equal. (XLVII.) The tangent of a circular arch being horizontal, if a body, after falling down the vertical radius to the center, move along the secant with its acquired velocity, the space which it will describe, in the same time as that of its fall, shall be equal to the radius, together with the perpendicular distance of the point, in which the secant cuts the circumfer ence, from the tangent. (XLVI 1 1.) If two heavy bodies let fall, at the same time, from two given points in a vertical straight line, move, after reaching the ground, with their acquired velocities, along the same horizontal straight line, to determine the point at which t"he one will over take the other. (XLIX.) Two given parallel straight lines, of unequal lengths, being equally inclined to the ground plane (118) APPENDIX II. in which they terminate, and being at a given distance from each other, if two heavy bodies, let fall at the same instant from their summits, move after reaching the ground, with their acquired velo cities, along the same horizontal straight line, to determine the point in which the one will overtake the other. The data being the same as in the last problem, excepting that the two given straight lines, al though equally inclined to the horizon, are not parallel to one another, to find where the two bodies will meet. tjrij (LI.) To place, on the ground, two given unequal vertical straight lines, at such a distance asunder, as that the time of falling down the greater^ and then moving, with the velocity acquired, to the less, shall be equal to the time of falling down the less, and then moving, with the velocity thence acquired, to the greater. vn t 94 K-iir,"..- . :-n (i, vr (HI.) The plane of a given right-angled triangle being vertical, and one of the sides containing the right angle being parallel to the horizon, and being taken as the base, to find a point in the perpendicular, produced, if necessary, such that the time of falling from it to the base, and afterwards describing the APPENDIX II. base, with the acquired velocity, shall be equal to the time of falling down the perpendicular and then describing the base, with the velocity so acquired. (LIII.) If the vertical diameters, of two vertical circles, be in the same straight line, and have a common lower extremity, from which several straight lines are drawn cutting both the circumferences, the times in which heavy bodies fall down the segments of these lines, between the two circumferences, arc all equal. (LIV.) If the plane of a circle be vertical, and a tangent be drawn to it at the upper extremity of its vertical diameter, the time in which a heavy body falls, from any point in the tangent, to the convex cir cumference, and afterwards with its acquired velocity, describes a chord drawn parallel to the tangent, from the point in which the body strikes the convex circumference, is equal to the time of falling from the same point to the concave circumference, and of afterwards describing, in like manner, the chord drawn parallel to the tangent, from the point in which the concave circumference is struck. Of all inclined planes having a common base, to determine that which shall be described by a falling body in the least time. APPENDIX II. (LVI.) - 1 Two sides of a triangle being given, the plane of which is vertical, to construct it, so that the time of falling down the third side may be a minimum. (LVII.) Two points, equidistant from an horizontal plane, being given, to find a point in the plane, such, that if straight lines be drawn from it to the two given points, the time in which a heavy body falls down the one and then ascends the other, may be a minimum. (LVIII.) vvSd ft /fetiiw m !knh. adi* ;,ife*TOi& To find the straight line of quickest descent to a given plane, from a given point above it. (LIX.) uioii vaifKtlo sm. irsrfj *>J ferjp-, : - nh To find the straight line of quickest descent from a given plane, to a given point below it. To find the straight lines of quickest descent to the circumference of a given vertical circle, from a given point, either within or without the circle, but in the same plane with it. APPENDIX II. (121) (LXI.) To find the straight lines of slowest descent, (l) to the convex, (12) to the concave circumference of a given circle, from a given point above, and in the same plane with, the circle. (LXII.) Two given circles being in the same vertical plane, to find the straight lines of quickest and of slowest descent, drawn from the circumference of the one to the circumference of the other. (tXIII.) If a body descend from the highest point of a vertical cycloid of which the base is parallel to the horizon, and the vertex downwards, the time of falling down any arch of the cycloid varies as the arch of the generating circle, intercepted between the base of the Cycloid, and a parallel to the base drawn from the place of the body at the end of that time. . (LXIV.) If any number of bodies fall down different vertical cycloids, that have their bases in the same parallel to the horizon, and terminated at one extremity in the same point, from which the bodies begin their motion, at the same instant, each of them will arrive, sooner than the rest, at that per- APPENDIX II. pendicular to the horizon, which is the axis of its curve. (LXV.) A given point on the ground, and two given points in the opposite sides of a vertical rectangle being all in the same vertical plane, it is required to find the direction and the velocity of projection, with which a ball, sent from the given point on the ground, shall pass through the other two given points. (LXVI.) The velocity of projection being given, to find the direction in which a body must be thrown, so that the aggregate of the altitude and amplitude shall be a maximum. ;q ?i 9 t x (LXVII.) orb If a body be projected from a given point on the ground, with a given velocity, in a direction making with the horizon an angle equal to half a right angle, and if another body be thrown hori zontally, with an equal velocity, from a point directly above the given point, and at a distance from it equal to the space due to the common velo city of projection, the two bodies so thrown will strike the ground in the same point. (LXVIII.) The velocity with which a ball is shot from a cannon at a mark on the ground, and also the velo- APPENDIX II. (123) city of sound, being given, to find a point in the perpendicular drawn, from the mark, to the straight line joining the place of the cannon and the mark, at which the explosion of the cannon, and the blow upon the mark, will be heard at the same time. (LXIX.) ,Iq If any number of bodies be projected from the same given point, with .equal and given velocities, the focus of the parabola described by any one of them shall be in the surface of a sphere, having the point of projection for its center, and the space due to the common velocity for its radius : And the vertex of the parabola described by any of the bodies shall be in the surface of a spheroid, of which the major-axis is horizontal, and is the double of the minor-axis. . 3ih til If any number of projectiles be thrown from a given point, at the same instant, and with equal velocities, at any given time, they shall all of them be found in the circumference of some circle, the center of which is in the vertical straight line drawn through the given point of projection. . (LXXI.) To find a point in a given horizontal plane, and the direction in which a body must be projected from that point, with a given velocity, so that a (124) APPENDIX II. given point above the plane may be the highest point of the trajectory described. -irlt LfiB floor ^lifq.erb^alrriotsnii (LXXII.) To find the velocity with which a perfectly elastic ball must be projected up a given inclined plane, so that, being reflected at the top by a ver tical plane, it may come to the hand again. (LXXIII.) itt> yufc vd b j { Of!.). To 81.:- To find the locus of all the points from which, if a perfectly elastic ball be successively thrown, with a given velocity, so as always to strike a given vertical plane in the same given point, it shall in each case return to the point from which it was projected. (LXXIV.) To find the least velocity with which a body, projected from the top of a given inclined plane, so as to describe a parabola, shall strike the bottom of the plane. (LXXV.) To determine the furthest point, in a given horizontal plane, which can be hit by a body pro jected with a given velocity, from a given point without the plane. (LXXVI.) To find the locus of the foci of all the parabolas described by perfectly elastic balls, let fall from a given horizontal line, upon a given inclined plane. APPENDIX II. (125) (LXXVII.) The same supposition being made as in the preceding problem, to find the point in the horizon tal line, from which, if a perfectly elastic ball be dropped upon the inclined plane, the range shall be a maximum. (LXXVII i.) To determine the highest and lowest points in a given vertical plane, which can be hit by a body projected, with a given velocity, from a given point without the plane. (LXXIX.) To determine the highest and lowest points in a given inclined plane, which can be hit by a body projected, with a given velocity, from a given point without the plane. (LXXX.) The base of an inclined plane being given, to find its altitude, so that a ball being projected up the plane with a given velocity, may, after leaving the plane, strike the base produced in a given point. (LXXXI.) If a perfectly elastic ball projected from an horizontal plane, perpendicularly upwards, with a given velocity, impinge upon a perfectly hard and immoveable plane, inclined to the horizon, at a (126) APPENDIX II. given angle, to find where the ball will again strike the horizontal plane. (LXXXII.) A perfectly elastic ball being supposed to be let fall from a given point, above an horizontal plane, to find where a given inclined plane must be placed, so that the ball, after impinging upon it, may strike the horizontal plane in a given point. (LXXXII i.) If any number of given inclined planes have a common section which is parallel to the horizon, to determine the locus of the furthest points of the planes, which can be hit, by a body projected from a given point in the common section, with a given velocity. (LXXXIV.) If any number of bodies, projected with equal velocities, from the same given point, in different horizontal directions, all strike a given plane, which is situated below the given point, to determine the locus of the points of the plane that are hit; (l) if the given plane be horizontal; (2) if the given plane be vertical; and (3) if the given plane be inclined to the horizon. (LXXXV.) To determine the graduation of a given glass cylinder, so that the several numbers of degrees may APPENDIX II. (127) indicate the specific gravities of different fluids, in which the cylinder floats, the specific gravity of pure water being represented by unity. (LXXXVI.) A cylinder filled with water being given, the altitude of which is (a) inches, to divide it into (n) parts, on which the lateral pressures shall be equal. (LXXXVI i.) The place of the lateral orifice, in a close cylin drical vessel, containing water, being given, and also the point of the ground struck by the issuing fluid, to find the altitude of the water in the vessel. (LXXXVIII.) The direction of a tube, inserted at a given point in the side of a close cylindrical vessel, con taining water, being given, and also the point of the ground struck by the spouting fluid, to determine the altitude of the water in the vessel, and the track of the spouting fluid. (LXXXIX.) To find the place of an orifice to be made in the side of a given cylinder, filled with water, and inclined to the horizon at a given angle, so that the effluent fluid may just strike the base of the cylinder. (128) APPENDIX II. To find the place of an orifice to be made in the side of a given cylinder filled with water, and inclined to the horizon at a given angle, so that the fluid may spout the farthest from its base on an horizontal plane. (XCI.) If a hollow sphere elevated above the ground and filled with the water, be supposed to be bored through, in every point of its surface, to find a con cave surface, which shall be touched by the several streams of the spouting fluid. (XCH.) A prismatic vessel has, at a given point in its side, a circular orifice of given dimensions, and is kept full by a certain supply of water : It is required to find the dimensions of another such an orifice to be made at another given lower point in the side, so that the same supply of water may keep the vessel full, when this lower orifice is open, and the upper orifice closed. (xcin.) A vessel, from which the water issues, at the bottom, is supplied at a given rate with water, poured into its top, and the surface is thus kept at a certain known altitude : It is required to find the APPENDIX II. (129) proportionate additional supply of water, with which the vessel must he fed, so that the surface of the fluid may be kept at any greater given altitude. (xciv.) In the straight line joining the extremities of two given finite straight lines, to find a point from which the two given lines being seen; they shall be of the same apparent magnitude. (xcv.) In a straight line, parallel to the straight line which joins the extremities of two given finite straight lines toward the same parts, to find a point, from which the two given lines being seen, they shall be of the same apparent magnitude. (xcvi.) To find a point, from which two sides of a given triangle being seen, they shall be of the same given apparent magnitude. (XCVII.) The vertical angle of a triangle being less than the exterior angle of an equilateral triangle, to find within it a point, from which if the three sides be seen they shall be of the same apparent magnitude. (xcvm.) In a given indefinite straight line, to find a point, seen from which a given finite straight line, (130) APPENDIX II. lying wholly without the indefinite line, shall be of the greatest apparent magnitude. (xcix.) To find the locus of all the points, in which an eye can be placed, so that two unequal straight lines, both lying in the same straight line, may always appear of the same magnitude. To find the point, in which an eye must be placed, so that three given straight lines, all situated in the same straight line, may appear of the same magnitude. Two points being given, one on each side of a given indefinite straight line, to find a portion of the indefinite line, such that its apparent magni tude, when seen from one of the given points, shall be the double of its apparent magnitude, when seen from the other. b-rtrf-9lJ1 (en.) rf niffJiw- To find a point in the circumference of a circle, seen from which two given unequal portions of a given straight line, without the circle, shall be of equal apparent magnitudes. APPENDIX II. (131) (cm.) If a plane mirror revolve about an axis, the angular motion of the image of any object is the double of the angular motion of the mirror. (civ.) A given straight line being inclined to the horizon at a given angle, to find the position of a plane mirror, from which the image of the given straight line shall be inclined to the horizon at any other given angle. A given vertical object being supposed to be at a given height above an horizontal plane, to find the point in the plane, where the apparent mag nitude of the object is greatest. (CVI.) The eye being supposed to be placed in any point of a given parallelogram, to determine the least length and breadth of a parallel plane mirror, in which the whole of the image of the given figure shall be visible. (cvn.) A circle which is inclined at a given angle to the horizon, is placed opposite to a plane vertical i 2 (132) APPENDIX II. mirror: it is required to determine the greatest length and breadth of the least portion of the mirror, in which the whole image of the circle shall be visible to an eye placed in its center. . ! (CVIII.) The eye being in the bisection of a given straight line, which is parallel to a vertical mirror, to draw through the place of the eye another straight line, of a given length, so that its image shall be of the same apparent magnitude as the image of the first-mentioned given straight line. (CIX.) The eye and a given luminous point being situated between two plane mirrors, inclined to one another, the eye s distance from any one of the images of the given point is equal to the aggregate of the incident ray and the reflected rays, belonging to that image. (ex.) A radiant point, and the position of the eye, being given, both on the same side of a plane mirror, and the mirror being supposed to move in a direction perpendicular to its own plane, to find the locus of the several points of the mirror, from which rays are reflected to the eye. APPENDIX n. (133) (cxi.) To find an incident ray, parallel to the axis of a given reflecting circular arch, which shall be reflected so as to pass through a given point in the axis. (cxii.) To find an incident ray, parallel to the axis of a given reflecting circular arch, that shall be re flected so as to pass through a point in the circum ference, which is at the distance of half a quadrant from the intersection of the axis and the mirror. (CXIII.) If two rays fall on the convex side of a re flecting circular arch, parallel to its axis, the arch of the mirror intercepted between the two incident rays, is one-third of the arch contained between the directions of the reflected rays, on the contrary side of the point in which they cross one another. (cxiv.) A reflecting circular arch, and two points in the same plane with it, being given, to find a ray pro ceeding from the one, supposed to be a luminous point, which shall be reflected in a direction passing through the other; (l) when both points are in the circumference ; (2) when only one of the points is in the circumference; (3) when both points are equidistant from the center, but are not (134) APPENtUX II. in the circumference ; (4) when both points are in the axis of the reflector. (CXV.) Two points, one of which is a radiant point, being given, both situated in the same perpendi cular to a refracting surface, and both on the same side of the surface, to find a ray proceeding from the one, which shall be refracted in a direction passing through the other ; the ratio of the sine of incidence to the sine of refraction being also given. (CXVI .) If a luminous point describe a circle, having its center in the axis of a double convex, or a double concave lens, and its plane perpendicular to the axis, the image shall describe another circle, having its plane also perpendicular to the axis. (CXVII.) To find a point in the axis of a given glass lens, such that when it is the focus of incident rays, the focus of emergent rays shall be at an equal distance from the center of the lens. (CXVIII.) To find a sphere, such that if a luminous point, placed before a double convex lens, move in any track over part of its surface, the image shall APPENDIX II. (135) describe a similar track, and shall move over an other part of its surface. (CXIX.) One of the surfaces of a given glass lens being coated with a reflecting substance,* to find a point in the axis of the lens, such that all rays which fall upon the lens diverging from it, or converging to it, shall pass through it again, after being once reflected, and twice refracted, by the lens. (CXX.) To find the focus of a small pencil of parallel rays, falling upon a given glass lens-mirror, of in considerable thickness. (CXXI.) If one side of a meniscus, bounded by con centric spherical surfaces, be coated with a reflecting substance, the distance of the principal focus of rays, will be the same as in the case of simple reflexion from the spherical mirror. (CXXII.) The radius of the refracting surface of a glass lens-mirror, of inconsiderable thickness, being * A lens so prepared, is called, in two of the following propositions, a lens-mirror. (136) APPENDIX II. given, to find the radius of the reflecting surface, so that when a very small pencil rays proceeds from a given point in the axis, the directions of the rays, after one reflection, and two refractions, shall again pass through the given point. . (CXXIII.) I If a ray parallel to the axis of a glass sphere, one half of which is covered with a reflecting substance, fall upon the refracting half of the sphere, after refraction it shall be reflected to a point in the axis, at a distance from the reflecting surface nearly equal to one-sixth part of the sphere s diameter. (cxxiv.) The orbits of the earth and a planet being supposed to be concentric circles, and to be in the same plane, to find the point in which the planet, seen from the earth, appears to be stationary. i<i us orn/ifc 9ili ad Hiw ; (CXXV.) To determine the angle which, at any given time, the straight line joining the moon s cusps will make with the horizon ; the moon s orbit being supposed to coincide with the ecliptic. (cxxvi.) According to the Simple Elliptic Hypothesis, the earth being -situated in one focus of the moon s APPENDIX II. (137) orbit, the plane of the same lunar meridian will continually pass through the other focus. ; (CXXVII.) bo The earth being supposed to be a sphere, and the angles being observed, which the straight line, join ing the tops of two distant mountains, makes with the plumb-line, at each of the tops, to determine the dimensions of the earth. (cxxvm.) The time of the beginning of morning twi light, or of the end of evening twilight, being given, to find the altitude of the atmosphere, which reflects the sun s light. (cxxix.) The annual parallax of a fixt star being sup posed to be sensible, to determine the points of the earth s orbit in which the annual parallax in lati tude, and the annual parallax in longitude, are the greatest. (cxxx.) The earth being supposed to move with an uniform velocity, in a circular orbit, and the sun being supposed to be placed in a given point, which k (138) APPENDIX Itv is not the center of the orbit, to determine in what points of the orbit the difference of the angles, de scribed about the center of the circular orbit and the center of the sun, is a maximum ; the angles being measured from the apogee, off* bun ?mrlq: R , Ji-fB > ^ IT -iiioj ..oriftjdgiijTte //.birr: ;f 3ofgn/5 riff , - ! ; "; -> - , , isri -; dilJ on . i MAXIMA AND MINIMA. INTRODUCTION. GEOMETRY, like every other system of useful knowledge, is said to have had its rise in the actual necessity of inventing it. What is wanting in au thentic history on this point, is compensated by the highest degree of probability. Not only does the name itself indicate the occasion upon which it was given, but the natural features of the par ticular region, which has been assigned as the nursery of this science, are such as to give credi bility to the traditionary account of it. In a country where the quantity of cultivated soil is necessarily small, where the population has always been numerous, where fertility is placed in imme diate contrast with the extreme of barrenness, and where the boundaries of property are liable to be A 2 INTRODUCTION. swept away by an annual inundation, the practice and the theory of the correct mensuration of plane surfaces could hardly fail to originate. But if such be the origin of Geometry, that particular branch of it, which is the subject of the first of the follow ing Sections, was not, perhaps, unknown to the ancient Egyptians. Where there was a choice of figure, some consideration is likely to have been bestowed on the most oeconomical plan of division, or enclosure: and many of the mathe matical truths which belong to this subject are so obvious, that they could not well escape even a cursory survey. If a field, in the form of a rect angular parallelogram, were to be divided equally between two persons, it might be done either by drawing the diagonal, or by drawing a straight line perpendicular to one of the sides through the point of its bisection. Which of these two methods of bipartition would require the least fencing is suffi ciently evident. The advantage of the square, in this respect, compared with an equal triangle or parallelogram, and some other elementary truths of the same kind, may also be supposed to have been early known. The error arising, in common cases, from considering the earth s surface as plane, instead of treating it as a convex surface, is al together inconsiderable : Not to mention, that some of the leading propositions, relating to maxima of enclosed space, and to minima of boun dary, are found to be the same in Plane and in Spherical Geometry. INTRODUCTION. 3 It must, however, be allowed, that the full investiga tion ofsuch maxima and minima would not be called for by the wants of men. The case which would prompt it, the transfer of a learned nation into a country wholly unenclosed, and not abounding, in any part of it, with materials for fencing, has never yet occurred. The mythology of the ancients, in deed, exhibits an instance, which might have afforded full scope for speculations of this kind. But this is fabulous. In long established communities the divi sions of land have taken place gradually, and upon no regular plan ; nor is there often an opportunity of making any extensive application, in this way, of the principles of science. It is, however, worth knowing, how a given quantity of surface may be most ceconomically fenced, in a given manner, and which is the best of all figures in this respect, where there is no restrictions, or the best figure of its kind, if the number of sides be prescribed : it is not an uninteresting problem to determine, what form a given length of outline must be made to assume, in order that it may contain the greatest area ; or, what is the best method of subdividing a given space into a given number of equal and similar parts, so as to leave no interstices. We cannot wonder, then, that the immutable relation which exists between the space enclosed and the species of its boundary, should have engaged the attention of the ancient mathematicians. Apollonius of Perga, who flourished under Ptolemy Euergetes, treated largely of Maxima A2 * INTRODUCTION. and Minima, in the fifth book of his work on Conic Sections. This work of the " Great Geo metrician," consisting of eight books, remained entire until the fourth century, when it was in the hands of Pappus. It appears, indeed, to have been known, in its perfect state, to Eutocius, at the latter end of the fifth century. But, from that time to the year l65S, the first four books only were known to be extant. The discovery of a Compendium of the three following books in Ara bic, by Borelli, and the publication of the very ingenious attempt of Viviani to supply from con jecture the matter of the fifth book, bear nearly the same date : but it is very certain that that emi nent scholar of Galileo could not have previously seen the Arabic manuscript, which was found by accident in the Medicean Library at Rome. The Mathematical Collections of Pappus have reached us in a mutilated state. They are now hardly accessible at all, excepting through a Latin translation which is full of errors. This author appears to have been led to the considera tion of Geometrical Maxima and Minima by ob serving the cells of bees : and it is to be regretted that we have not his complete investigation of that most astonishing structure. What he has deli vered, in his fifth book, has found its way into the writings of many modern mathematicians. In the literary correspondence which took place between Torricelli, Fermat, and Roberval, toward the beginning, and the middle* of the seventeenth INTRODUCTION. 5 century, geometrical questions of this kind seem to have been proposed : there is one, in particular, sent from Fermat to Torricelli, the solution of which will be given in the course of this work. Dr. Barrow, in the Addenda to his Lectiones Geometricae, a work of great originality and eru dition, has shewn how theorems relating to maxima and minima may be deduced from the considera tion of tangents. His two fundamental theorems are demonstrated with admirable perspicuity and elegance. Amongst the works of later mathematicians, not to mention those of L Huillier and other foreigners, the Elements of Thomas Simpson con tain a series of propositions, " on the Maxima and Minima of Geometrical Quantities," in which there is not much that is original. Waring, in his Treatise on Curve Lines, has determined several maxima and minima, of which it may be said, that they are curious rather than useful. But the two last-mentioned authors, as well as Pappus among the ancients, and most of the modern writers upon this subject, have admitted into their reasonings what appears to be a sophism. It is easily shewn, for example,, that of all triangles standing upon the same base and of equal perimeter, that which is isosceles is the greatest : hence they have concluded, at once, that whatever the rectili neal figure be, of a given perimeter, when it is greatest, its sides are ail equal. Thus they have 6 INTRODUCTION. taken for granted that the quantity under con sideration has a maximum value: which is not al lowable in any geometrical proposition. Such an assumption may be properly made in the analytical investigation of maxima and minima ; because if no maximum nor minimum exists, the process itself will shew that to be the case, by the nature of its result : and if no absurdity appear in the result, and it still be doubtful whether the quantity determined give a maximum or a minimum value, there are means of ascertaining to which of the two it be longs. But,, in this application of Geometry, it is necessary to prove that the variable magnitude is greater, or less, than any other of the same kind with itself, before the conclusion can be fairly drawn. The first division of the following publication is purely geometrical, and an easy application for the most part of the Elements of Euclid. Wherever any theorem or problem is wanted, which is not contained in that book, it has been supplied : in those cases, in which a proposition relating to maxima and minima appearsLto depend principally upon some more simple geometrical truth, this latter has been separately premised ; in order that it may be distinctly seen upon what elements each main proposition is founded. From a wish to accom modate this work, as far as it could well be done, to those who have studied only the first four books of Euclid, the doctrine of proportion has been, as much as possible, avoided : although the use of it might have shortened some of the demonstrations. V INTRODUCTION. 7 The propositions of the first and second Sec tions, of this first Part, form a distinct and im portant subject: they lead to results which have, most of them, been long known, but which are, perhaps, no where to be found collected, arranged, and strictly demonstrated. The maxima of the first Section are, each of them, connected with a minimum: that is, the same species of figure which renders the surface greatest when the peri meter is given, renders the perimeter least when the surface is given. This remarkable property is shewn, in a general theorem, necessarily to obtain. In the questions of the second Section, on the contrary, the area is a maximum when the peri meter is a maximum ; and it is a minimum when the perimeter is a minimum. In one description of. them, whilst the perimeter remains the same in length, the area also remains the same, whatever be the number of sides of the figure. The third Section consists of miscellaneous propositions ; classed, however, according to a division, which refers them to lines, angles, and surfaces. It could not escape observation, if the mention of it were suppressed here, that it is part of the plan of this work to invite a comparison between Geometry and Algebra, and to illustrate the ad vantages peculiar to each. The relative advantages of these two great branches of science, in the in vestigation of mathematical truths, are now, indeed, well understood. But it may not be improper to 8 INTRODUCTION. offer some remarks on the great difference which there is between them in producing those collateral effects, which have been ascribed to the mathe matics, considered as a discipline of the mind. In the very entrance upon our discussion of this topic, in order to avoid all misconstruction, it may not be wholly needless to state expressly, that what follows is intended to refer solely to the case of academical students, who apply themselves to the mathematics, not so much on account of the in trinsic value of that science itself, as for the sake of those indirect advantages, which are supposed to flow from the cultivation of it ; such as the habits of close attention, of weighing the validity of proofs, of searching into the connexion of related truths, and of methodizing the materials of thought. With this particular view, then, let it be remem bered, our enquiry is to be conducted : and it will turn principally on the comparative merits of the analytic and the synthetic modes of reasoning, so considered. Both these modes of reasoning may, indeed, be used in every department of the mathe matics. But throughout the whole province of arithmetic, numerical as well as algebraical, ele mentary as well as infinitesimal, both in the inves tigation of theorems and in the solution of problems, the analytic method is, almost exclusively, em ployed : whilst the truths of Geometry are, for the most part, demonstrated synthetically ; and the student in acquiring them becomes habituated to the use of that method of teaching. Now, there INTRODUCTION. 9 exists, in the first place, this manifest distinction between a synthetic proof in Geometry, and an analytic process in Algebra, that in order to com prehend the former, the whole chain of reasoning must be kept in view, as it is continued from the beginning of the proposition to the end : whilst in pursuing the latter method, the attention is fixt only upon each single step, as each of them suc cessively offers itself; and the conclusion is to be admitted independently of all but the last of them, whenever it is arrived at. Stronger and more un ceasing attention,, therefore, is required in the former case, than in the latter, and the judgment, as well as the memory, is called more urgently into action. There is, however, analysis, as well as synthesis, in Geometry. All those propositions, the truth of which Euclid has deduced ex absurdo, are, in reality, demonstrated analytically : and, in the same manner, a series of conclusions legitimately drawn from a certain supposition, may so terminate as to shew that supposition to be true. But, in both these cases, it is evident that the connexion of the several steps with the original hypothesis must be closely attended to, in order that the force of the proof may be clearly seen. Arithmetical specu lations, on the contrary, most commonly hinge upon the solution of an equation, or upon the finding of a fluent : and whatever obstacles the authour of such propositions may have had to encounter, there is seldom any serious difficulty in following him along the path which he has traced out. 10 INTRODUCTION. There is, besides, a more absolute precision in all the forms and in the language of geometrical disquisition. Pure Geometry is always precise and logical; it carries on its demonstrations by the exact comparison of ideas, adhering to the constant use of terms, the meanings of which are always verified by a reference to accurate definitions. Its reasonings proceed by means of syllogisms, in which, for the sake of brevity, the minor proposition is suppressed. But even in the proofs of those theorems of Algebra, in which little depends upon the employment of its peculiar symbols, the reasoning is seldom close and exact. The absurdities which have been published with a view of explaining the rule for Algebraic multi plication ; the common method of shewing that the numerator and denominator of a fraction in its lowest terms are measures of the numerator and denominator, respectively, of every other equal fraction; the imperfect state in which the proof of the rule for finding the greatest common measure of two complex algebraic quantities, h,as been left by most elementary writers, as if they could only be accurate as far as Euclid is accurate, from whom they have copied, but who did not contemplate the nature of quantities expressed algebraically ; the de fects in the demonstration of the binomial theorem ; and many more examples might be adduced in sup port of the assertion made above. Nay, it is well known, that some propositions of the greatest importance in Algebra have never yet received a INTRODUCTION. 1 1 satisfactory proof : and although mere metaphysical objections ought * not to stop the progress of any science, it is time that these faults were remedied : the most eminent writers in this department, how ever, always appear to be in haste to quit the pro vince of severe reasoning, and to exhibit their skill in the management of symbols. Thus it appears, that even when the same method is used in both, Geometry affords a better exercise, than Algebra, for the mental powers. That it exercises, without oppressing and fatiguing them, will scarcely be denied by any man, of even middling abilities, from his own experience. Different individuals may, indeed, find it more or less difficult to retain, and to recollect, the proofs of a long series of geome trical propositions ; but fully to comprehend these proofs, at the time when they are considered, to * " Quoique les veritcs mathematiques soient toutes d une certitude parfaite, elles n ont pas toutes le meme degre d* evi dence : ce sont sur-tout les notions premieres qu il est difficile de porter au point de clarte desirable ; mais on seroit arrete dans la carriere, des les premier s pas, si parce que certains principes fondamentaux restent enveloppes de quelque obscuritc, on refu- soit d aller plus avant. Ainsi les anciens, parce qu ils n avoient pu parvenir & eclaircir entierement la theorie des paralleles, n ont point ete pour cela retardes dans leurs recherches. Ainsi, quoique la notion de V infiiii presentat aux modernes des difficultes, Us n ont pas laisse de donner & 1* analyse infinite- sirnale tout le developpement qu on pouvoit attend re de leur sagacite. Ainsi, enfin, Pobscurite dans laquelle est restee la notion des quantites negatives n a nullement entrave la rnarche des algebristes." Carnot^ Geom, deposit, p. 48 L 12 INTRODUCTION. perceive the concatenation which binds the parts of the series together, and thus indirectly to gain all the essence of a system of logic, without the tediousness of its technical terms and rules, requires nothing more than common sense and sedulous application. What has been hitherto said, upon this subject of comparison, relates chiefly to the progress of the student in making himself master of the discoveries of other men. But it is not only in reading and digesting what has been written upon the mathematics, that his mind is disciplined: another most important employment of his faculties consists in the application of knowledge so acquired to cases which are new to him. Now the questions proposed to a learner to be answered algebraically, as a trial of his skill and talents, are usually of such a kind as not to demand any extraordinary exertion of his reason. He is not called upon to attempt the investigation of new and recondite theorems. The difficulty presented to him, is seldom more than the mere translation of the conditions of the question, into a language, the peculiarity of which is, that it is so concise as to exhibit several propositions in a small compass. This having once been effected, and it is seldom an arduous task to perform, the attention is then withdrawn from the things signified, and confined to the signs : and from performing the mere opera tions of Algebra, it will scarcely be contended that any improvement of the reasoning faculties is to be derived. But the exercise of the understanding INTRODUCTION. IS is of a very different kind, when it is occupied in the solution of a geometrical problem. Whether it proceeds, strictly speaking, analytically, or whether it makes a particular construction, in the way of trial and conjecture, and then pursues the consequences of it, until they either end in the attainment of that which was proposed, or else in dicate that some other method must be had re course to, its faculties of judging, recollecting and inventing are continually exercised. Algebra is, doubtless, the more powerful and convenient instrument for use. " Idem omnino mihi," says EULER, " cum Newtoni Principia et Hermann i Phoronomia perlustare ccepissem, usu venit, ut quamvis plurium problematum solutiones satis percepisse mihi viderer, tamen parum tantum discrepantia problemata resolvere non potuerim." But the same causes, which give analytics their superiority in that respect, prevent them from being so valuable, considered as a mental discipline. The great praise, it may be further remarked, which has been bestowed upon the Mathematics as conducing to strengthen the mind, has proceeded from men, who lived when Geometry constituted the principal part of them : and those who have lately denied them this merit, seem to have been biassed in their estimate by a partiality for extended analytics. If the view which has here been taken of this subject be just, it should seem to be no disservice to our established system of education, to afford INTRODUCTION. scope for the efforts of our junior students in an easy extension of those rudiments of knowledge which they learn from Euclid. It is impossible for them to enter upon a more fertile field than that of Geometry, which really seems to admit of the exercise of as much genius and invention as poetry itself: and after having thus strengthened their faculties, and accustomed themselves to the comparison of clear ideas, they will proceed with better success to the remaining part of their aca demical course. A permanent taste for the Mathematics will thus be formed, and a study, which is now too frequently thrown aside as soon as it has answered a temporary purpose, will be come a valuable resource to amuse and adorn their future leisure. ON MAXIMA AND MINIMA. PART I. ON THE APPLICATION OF EUCLID* S ELEMENTS TO aUESTIONS CONCERNING MAXIMA AND MINIMA. SECTION I. 1. DEF. A VARIABLE magnitude is said to be a " maximum " when it is the greatest of its kind, or the greatest under certain conditions : and it is called a " minimum " when it is the least of its kind, or the least under certain conditions. PROP. I. 2. Problem. Through a given point, situate between two given straight lines, which are not parallel, to draw a straight line terminated by the two given lines, and bisected in the given point. Let AX and AY be two given straight lines which are not parallel, and D a given point between them ; it is required to draw a straight line through D, terminated by AX and AY, and bisected in D. 16 THE GEOMETRICAL INVESTIGATION Through D draw (*E. 31. 1.) DE parallel to AY, and DH parallel to AX, from HY cut off (E.3. l. ) HG equal to DE or AH; join G, Z),and produce GD to meet .4X in F. The straight line GF is bisected in Z). For the two triangles FDE, DGH have the side DE equal to the side GH, by the construc tion ; and the angles adjacent to these sides are equal, each to each, because (E. 29. 1.) DE is parallel to AG ; wherefore (E. 26. 1.) FD is equal to DG; and FG is bisected in D. PROP. II. 3. Theorem. Of all triangles having the same vertical angle, and the bases of which pass through the same given point, the least is that which has its base bisected by the given point. * The letter E refers to the Elements of Euclid, the former of the two numbers to the Proposition, and the latter to the Book. OF MAXIMA AND MINIMA. 17 Let BAG (Fig. Art. 2.) be the common vertical angle, and D the point through which the bases of all the triangles pass. Draw (Art. 2.) the base FG such that it shall be bisected in Z>, and let BDC be the base of any other of the triangles ; the tri angle AFG is less than the triangle ABC. Through G, the extremity of FG which is not further than the other extremity, from A, draw (E. 31. 1.) GI parallel to AB and meeting BC in /; then, because FD is equal to DGr, and the angle BI)F to the angle GDI (E. 15. 1.) and the angle BFQ to the angle FGI (E. 29. 1.), the triangle GDI is equal (E. 26. 1.) to the triangle BDF; but the triangle GDI is less than the triangle GDC; therefore the triangle BDFis less than the triangle GDC ; if, therefore, the trapezium ABDG be added to both, the tri angle AFG is less than the triangle ABC. . PROI>. III. 4. Theorem. The greatest parallelogram which can be inscribed in a given triangle, so as to have the vertical angle of the triangle for one of its angles, is that which is formed by drawing two straight lines from the bisection of the base, each parallel to a side of the triangle. Let ABC (Fig. Art. 2.) be the given triangle ; let its base BC be bisected in K, and let KL be drawn parallel to AC, and KM parallel to AB. Also let D be any other point in BC, and let DHznd DEbe drawn parallel to AB and AC respectively ; B 18 THE GEOMETRICAL INVESTIGATION the parallelogram AK is greater than the paral lelogram AD. Through the point D draw (Art. 2.) the straight line FDG so that it may be bisected in D ; the tri angles BKL, A CMmay be shewn to be equal in the same manner as the triangles BDF 9 GDI were shewn to be equal in Art. 3. ; therefore LK is equal to MC; and LK is also (E. 34. 1.) equal to AM; therefore AM is equal to MC; and (E. 41. 1.) the parallelogram AK is the double of the triangle MKC- 9 it is, therefore, equal to the two triangles MKC, LBK, and is the half of the whole tri angle ABC. In the same manner the parallelo gram AD may be shewn to be half of the triangle AFG: but (Art. 3.) the triangle ABC is greater than the triangle AFG ; therefore, also, the paral lelogram AK is greater than the parallelogram AD. PROP. IV. 5. Theorem. Of all equiangular parallelograms of equal perimeters, that which f is equilateral is the greatest. a OF MAXIMA AND MINIMA. HT 19 Let ADKE be any equilateral parallelogram, and AP any other parallelogram equiangular with it, and of equal perimeter; AK is greater than AP*. Join K 9 P, and produce KP both ways to meet AD and AE, produced, in B and C. And since ND is a parallelogram, RN is equal to DK, and RD to /L/V (E. 34. 1.) ; but ^D together with DK is equal to AR together with RP, each being the half (E. 34. 1.) of equal perimeters ; from each of these equals take AR together with RN or DK, and there will remain RD equal to NP ; there fore, also NK is equal to NP ; therefore the angle NKP is equal to the angle NPK (E. 5. 1.); but the angle NKP is equal to the angle ABC, and angle NPK to the angle ACS (E. 29. l), and also to the angle DKB ; wherefore the four angles DBK, DKB, EKC, ECKwe equal to each other; and the side DK of the triangle BDK is, by the hypothesis, equal to the side KE of the triangle KEC-, therefore (E. 26. 1.) B K is equal to KC; and (Art. 4.) AK is greater than A P. 6. COR. 1. The square is the greatest of all * If it be required to make an equilateral parallelogranl equiangular with a given parallelogram AP, and having also an equal perimeter ; produce A& to C, and make QC equal to QP. Join C, P ; and let CP produced meet AR produced in B ; bisect (E. 10. 1.) CB in K -, and through K draw (E. 31. 1.) KD and KE parallel to A 2 and AR respectively; then it is manifest, from the demonstration of Art. 5., that AK is equi lateral and that its perimeter is equal to that of the equiangular parallelogram AP. B 2 30 THE GEOMETRICAL INVESTIGATION rectangles of equal perimeters ; which may also be deduced from E. 5. 2. 7. COR. 2. The space which can be enclosed by a straight line of given length, and an indefinite straight line, the given finite line being divided into two segments, which are to be inclined to each other at a given angle, is greatest when the segments are equal. For (Art. 5.) the double of that space will be a maximum when the segments are equal. PROP. V. 8. Theorem. Of all triangles having two sides of the one equal to two sides of the other, each to each, that which has the two sides perpendicular to each other is the greatest. Let the triangle ACB have the two sides AC, CB at right angles to each other, and let A CD It . be any other triangle standing upon AC 9 and having the other side CD equal to CB. The triangle ACB is greater than the triangle ACD. Draw DE (E. 12. 1.) perpendicular to EC and OF MAXIMA AND MINIMA. join A, E. Then, because the angle DEC is a right angle, the angle EDC is less than a right angle (E. 17. 1.); and, therefore, (E. 19. I.) .DC is greater than EC; but DC is equal to BC; therefore EC is greater than EC-, again, be cause each of the angles DEC 9 ECA is a right angle, DE (E. 28. 1.) is parallel to AC, and the triangle ADC is equal (E. 37. 1.) to the triangle AEC. But the triangle ABC is greater than the triangle AEC 9 because BC has been shewn to be greater than EC; therefore the triangle ABC is greater than the triangle ADC. 9. COR. 1. A square is greater than any other given parallelogram of equal perimeter: And the perimeter of a square is less than that of any other equal parallelogram. If the given parallelogram be not equilateral, find (Note, Art. 5.) an equilateral parallelogram equiangular with it, and of equal perimeter. This is (Art. 5.) greater than the given paral lelogram ; and if the given parallelogram be rect angular, it will be a square : if not, it will be a rhombus : but (Art. 8.) the half of a square is greater than the half of a rhombus of equal peri meter ; wherefore the whole square is greater than the whole rhombus ; much more then is the square greater than the given parallelogram of equal perimeter. Conversely, let A be a square, and B any other equal parallelogram ; the perimeter of A is less than that of B. For if it be not less, it is either equal to it or greater; but it cannot be equal ; for THE GEOMETRICAL INVESTIGATION then, as hath been shewn, A would be greater than B ; which is contrary to the supposition : Neither can the perimeter of A be greater than that of B ; for then A would manifestly be greater* than a square of equal perimeter with B> the which square, as hath been proved, is itself greater than B ; much more, then, would A be greater than B ; which is contrary to the supposition. Where fore, the perimeter of the square A is less than that of any other equal parallelogram B, since it can neither be equal to it, nor greater than it. 10. COR. 2. The space which can be enclosed by a straight line of a given length, and an inde finite straight, line, the given finite line being divided into two segments, is greatest when the segments are equal, and perpendicular to each other. 11. COR. 3. If space be to be enclosed by any number of given finite straight lines together with an indefinite straight line, and if the semicircle described upon the assumed portion of the inde finite line as a diameter do not pass* through all the angular points of the figure, a greater space may be enclosed under the same conditions. Let AB, BC, CD, DE be the given finite straight lines, placed so as to enclose a space with the indefinite straight line AX\ and let the semi circle ABCE, described upon AE as a diameter, * It is self-evident, that of two squares, that which has the greater side is the greater, and conversely : That this is true of all regular polygons of the same number of sides, is shewa in Art. 23. OF MAXIMA AND MINIMA pass through the angular points B and C, but not through D ; join A, D and E, F; draw (E. 1 1. 1.) DG perpendicular to AD, and make DG equal to DE ; join A, G ; the angle EFD is a right angle (E. 31. 3.) ; therefore (E. 17. 1.) the angle ADE is not a right angle; therefore (Art. 8.) the triangle ADG is greater than the triangle ADE ; if, there fore, the figure ABCD be added to both, ABCDG is greater than ABODE. But, secondly, if the semicircle ABDE, de scribed upon AE as a diameter, pass through the angular points B and D, but not through C, which point is not adjacent either to the first or the last of the given finite straight lines, which bound the 2* THE GEOMETRICAL INVESTIGATION polygon, join A, Cand E, C; and since the point Cis not in the semicircle, the angle ACE (E. 31.3. and l6. 1.) is not a right angle: Draw, therefore, (E. 11.1.) CG perpendicular to AC, and make CG equal to CE ; through A and G draw the indefinite straight line AGX ; and upon CG, which is equal to CE, make the figure CHG (E. 18. 6.) similar, and therefore (E. 2O. 6.) equal to CDE, i. e. to that part of the polygon which stands on CE : Then, since (Art. 8.) the triangle ACG is greater than the triangle ACE, if to the former be added the figures ABC and CHG, and to the latter the figures ABC and CDE, which has been shewn to be equal to CHG, it is manifest that the whole figure ABCHG is greater than the whole figure ABCDE, which is enclosed under the same conditions. PROP. VJ. 12. Problem. In a given indefinite straight line, to find a point, from which if two straight lines be drawn to two given points on the same side of the indefinite line, and in the same plane with it, their aggregate shall be a minimum. Let XYbe the indefinite straight line, and A, B the two given points, both on the same side of it ; it is required to find a point in XY, from which, if two straight lines be drawn to A and B, their aggregate shall be a minimum. From A draw the straight line ACD perpen dicular to XY, and make CD equal to AC; join OF MAXIMA AND MINIMA. 25 D, B, and let DB meet XY in E ; also join E y A ; AE, together with EB, is a minimum. For, let P be any other point in XY 9 and join P with" A, B, D. Because AC is equal to CD, and CE common to the two triangles ACE, DCE, and that the angle ACE is equal to the angle ECD (E. Def. 10. l.)AE is equal (E. 4. 1.) to DE ; and in the same manner AP may be shewn to be equal to DP ; therefore DB is equal to AE together with EB, and DP together with BP is equal to AP together with PB ; but DP together -with PB is greater (E. 20. 1.) than Z>5; therefore, also, AP together with PB is greater than AE together with EB*. * If XY be the circumference of a great circle in a sphere, and if A^ J5, be two given points, in the sphere s surface, both of them on the same side of XY, the same kind of con struction may be made as in the above proposition, and a point in XFmay be thereby determined, such that the aggregate of the arches of two great circles, drawn to it from A and B, shall be 26 THE GEOMETRICAL INVESTIGATION 13. COR. If several points be taken in the indefinite straight line XY y the aggregate of the two straight lines drawn from A and B to that which is nearer to the point E, is less than the aggre gate of the two straight lines drawn from A and B to that which is more remote from E. This is manifest from E. 21. 1. PROP. VII. 14. Theorem. The perimeter of an isosceles triangle is less than that of any other equal tri angle, standing upon the same base. Let ABC be an isosceles triangle, and AQC any other equal triangle standing upon the same M be a minimum. See a Treatise on Spherics, by the Authour of this Treatise (Art. 70. 92. 66. 97. 74.). It may, perhaps, in many instances, furnish amusement to the reader, whilst he is engaged with the problems in this book, to endeavour to solve the analogous problems, on the surface of a sphere, by an application of the principles of Spherical Geometry. OF MAXIMA AND MINIMA. 27 base AC-, the perimeter of ABC is less than that of AQC. Join B, Q and produce BQ both ways to X and Y; BQ is parallel (E. 39. 1.) to AC, and, therefore, the angle ABX is equal to the alternate angle BAG, and the angle CBYto the angle EC A (E. 29. 1.) ; but the angle BAG is equal (E. 5. 1.) to the angle BCA ; therefore the angle ABX is equal to the angle CBY ; it follows, therefore, from the demonstration of Art. 12., that AB together with BC is less than AQ together with QC; and, be cause the base AC is common to the two triangles, the perimeter of the triangle ABC, is less than that of the triangle AQC. Otherwise : Let AB be produced to D so that BD may be equal to Aft, or BC, and let D, Q be joined ; the angle DBQ (E. 29. 1.) is equal to BAG, and QBC to BCA; but BAG is equal (E. 5. 1.) to BCA, wherefore DBQ is equal to QBC; and BD is equal to BC, and, therefore, (E. 4. 1.) DQ is equal to QC; but AQ together with QD, is greater (E. 20. 1.) than AD; i. e. than AB, BC; there fore AQ, together with QC, is greater than AB, together with BC; and the base AC is common to the two triangles ; therefore the perimeter of the triangle ABC is less than the perimeter of the tri angle AQC. 15. COR. 1. If any polygon be not equi lateral, another equal polygon may be found, of the same number of sides, which has a less perimeter. Let the two sides CD, DE of the polygon 28 THE GEOMETRICAL INVESTIGATION ABODE be unequal; join C, E and (E. 10. 1.) r K bisect CE in L; through D draw (E. 31. 1.) XY parallel to CE, and through L (E. 31. 1.) LK per pendicular to Cl ; and join C, A" and j, K. By the construction, and E. 4. 1., the two triangles CLK, ELK are equal ; so that CK is equal to KJE, and therefore, (Art. 14.) CK, together with KE, is less than CD, together with DE ; add to both the rest of the perimeter EABC, and the whole perimeter EABCK is less than the whole EABCD. \6. COR. 2. An isosceles triangle is greater than any scalene triangle of equal perimeter, and standing upon the same base. If not, it is either equal to, or less than the sca lene triangle ; but it cannot be equal to the scalene triangle; because it would then (Art. 14.) have a less perimeter, which is contrary to the hypothesis ; neither can it be less ; because then, also, its peri meter must be still less than before; for if two unequal isosceles triangles stand upon the same base, and on the same side of it, the less must lie wholly within the greater; and, therefore, Hi O / TO * v .vv\. f v\vJ 3D* OW j iJ J XJt OF MAXIMA AND MINIMA, lip (E. 21. 1.) have a less perimeter; otherwise, an angle at the base, belonging to one of the tri angles, would, at the same time, be greater and less than an angle at the base belonging to the other. Wherefore the isosceles triangle is greater than the scalene triangle, since it can neither be equal to it, nor less than it. 17. COR. 3. Hence, if any polygon be not equilateral, a greater polygon may be found of the same number of sides, and of equal perimeter. Let the sides AB, J5Cof the polygon ABCDE (Fig. Art. 15.) be unequal ; join A, C; from the centre A, at a distance equal to the semi-aggre gate of the two sides AB, BC, describe a circle, and let it cut another equal circle, described from the centre C, at an equal distance, in H. Join A, H and C, H; the isosceles triangle AHC is, therefore, of the same perimeter as ABC, and (Art. 16.) is greater than ABC; add to each the polygon ACDE, and AHCDE is greater than ABCDE, a polygon of the same number of sides, and of equal perimeter. ; : ^ ol" twp:.*^$y>L (.1 M,.$] fad ,v*Lr. PROP. VIII. 18. Theorem. If the bases of two equal iso sceles triangles be equal, the side, also, of the one is equal to the side of the other. Let ABC, DCE be two equal isosceles tri angles, having the base EC of the former equal to 30 THE GEOMETRICAL INVESTIGATION the base CE of the latter; the side AB is equal to Df> < &d ad> Jfi * i> n r9h _ the side DC. Let the bases be placed contiguously at the extremities and in the same straight line ; join A, D and from A and D draw (E. 12. 1.) AF and DH perpendicular to BE. Then (E. 40. 1.) AD is parallel to BE, and (E. 28,. and 29. 1.) AH is a parallelogram; also, because AF is common to the two right-angled triangles AFC, AFB, which have the angles at B and C equal, BF (E. 26. 1.) is equal to FC\ in the same manner, CII is equal to HE \ therefore FC is equal to CH, each being the half of two equal lines; and FH is equal to BC; but (E. 34. 1.) FH is equal to AD ; therefore, also, SC is equal to AD, and (E.33. 1.) AB is equal to DC. 19 COR. In the same manner it may be shewn that if the bases of two equal right-angled tri angles be equal, the remaining sides of the one are equal to the remaining sides of the other, each to each. OF MAXIMA AND MINIMA. 31 PROP. IX. 20. If the base of an isosceles triangle be less than the base of an equal equilateral triangle, its side shall be greater than the side of the equi lateral triangle. Let a b c be an isosceles triangle, and ABC an equilateral triangle equal to it ; and let b c be less than BC-, then is a b greater than AB. From 0, and A, draw a d perpendicular to b c, and AD perpendicular to BC. It may be shewn, as in the preceding proposition, that b c and BC are bisected in the points d and Z); therefore, the triangle ABD is equal to the triangle abd, each (E. 38. 1.) being the half of the two equal tri angles. From DB cut off (E. 3.1.) DE equal todb-, join^, E ; through B draw (E. 31. 1.) EG parallel to EA, and let it meet DA produced in G ; also, through A draw HAI parallel to BC, and join G, E and G, C. Then, (E. 4. 1 ) the angle 32 THE GEOMETRICAL INVESTIGATION GBC is equal to the angle GCB ; and, therefore, because the two triangles GAH, GAI have the angles at the bases HA and I A equal (E. 29. 1.) and have the side GA common, HA is equal (E. 26. 1.) to AI\ but HA is also equal to BE (E. 34. 1.), because HE is a parallelogram ; therefore Al is equal to BE, and (E. 33. I.) IE is equal and parallel to AB. Again, be- because Ae two triangles ABE, EGA (E. 37. 1.) are equal , add to each the triangle AED, and the whole triangle GED is equal to the whole triangle ABD, i. e. to the triangle aid-, and DE was made equal to db ; therefore (Art. 19.) GE is equal to ab. But the angle ElCis necessarily an acute angle; for it is less (E. 16. 1.) than the angle EXC> which (E. 29. 1.) is equal to the angle of the equilateral triangle BAC-, therefore the angle E1G is neces sarily (E. 13. 1.) an obtuse angle, and EG is greater than (E. 17. and 19. l.) El. But EG was shewn to be equal to a b, and El to AB. There fore ab is .greater than AB. ! PROP. X. . 21. Theorem. The perimeter of an equilateral triangle is less than the perimeter of any other equal triangle. Let ABC be an equilateral triangle, and p b c any other equal triangle; the perimeter of the tri angle ABC is less than that of the triangle p b c. If the triangle pbc be not isosceles, find (Art. OF MAXIMA AND MINIMA. S3 15.) an equal isosceles triangle a b c, which will have a less perimeter; draw (E. 12. 1.) AD perpendicular to BC, and a d perpendicular to be; and from DB, produced if necessary, cut off (E. 3. 1.) DE equal to db; join A, E ; through B draw (E. 31. 1.) BG parallel to EA, and through A draw ^/parallel to BC, meeting BG produced in /; and join E, G. It may be shewn, as in Art. 20., that the triangle EGD is equal to the triangle ABD, and that, therefore, (Art. 19.) EG is equal to ab. And, first, if ED be not less than DA, the angle DAE is not less (E. 5. and 18. 1.) than the angle AED, and, therefore, the angle AGI is not less than the angle AIG (E. 29. and 34. 1.) ; therefore (E. 6. and 19. 1.) AI or EB is not less than AG. But EG and GA are together (E. 20. 1.) greater than EA, and EA is greater (E. 19. 1.) than AB, be cause the angle ABE is necessarily obtuse ; much more then are GE and EB together greater than 34 THE GEOiMETRICAL INVESTIGATION AB ; add BD to both ; therefore GE and ED are together greater than AB and BD. But QE an<J ED are equal to the semi-perimeter of the triangle abCy and AB and I?Z) are equal to the semi-peri meter of the triangle ABC; wherefore the whole perimeter of the former is~ greater than the whole perimeter of the latter ; much more then is the peri meter of the triangle p b c greater than that of the triangle ABC. But if ED be less than DA, and greater than DB, at the point A make (E. 23. 1 .) the angle EAR equal to the angle AED. The angle AED is greater (E. 18. 1.) than the angle DAE; therefore the angle EAHis greater than the angle EAD, and AH and AE lie on contrary sides of AD ; again, the angle AED is less than the angle EAC, for AC, which is equal to BC, is, by the hypothesis, less than CE ; therefore the angle EAH is less than the angle EAC; and AH falls between AD and AC ; therefore AH meets BI between AD and AC; join E, H. The angle AHI is equal (E. 29. 1.) to the angle EAH; i. e. to the angle AEB, by the con struction. And the angle AEB is equal (E. 34. 1 .) to the angle AIH; therefore (E. 6. 1.) A I is equal to AH. But the exterior angle BGD is greater (E. 16. 1.) than the angle BAG; and the angle BGD is equal (E. 15. 1.) to the angle AGH 3 and the angle BAG is equal to the angle GAC, because the perpendicular AD bisects the triangle BAC-, but the angle GAC has been shewn to be greater than the angle GAH;>m\ich more, then, is the OF MAXIMA AND MINIMA. 35 angle AGH greater than the angle GAH\ there fore (E. 19. 1.) AH, or AI, or (E. 34. 1.) EB is greater than GH ; but the two EG, GH are to gether (E. 20. 1.) greater than EH ; much more, then, are GE, EB together greater than EH. And since AH is equal to EB, and HB common to the two triangles HAB, HEB, and that the angle AHB is equal to the angle EBH, the side EH is equal (E. 4. 1.) to the side AB ; wherefore GE and EB are together greater than AB, and it may be shewn, as in the first case, that the peri meter of the triangle ABC is less than the peri meter of the triangle abc-, much more, then, is it less than that of the triangle p b c. Lastly, if the base of the isosceles triangle be the less of the two, its side is greater (Art. 20.) than the side of the equilateral triangle. And if on one of the sides of this isosceles triangle, another isosceles triangle be constructed, equal to it, as in Art. 15., its perimeter shall (Art. 14.) be less than that of the former isosceles triangle ; and the peri meter of this latter triangle is greater than that of the equal equilateral triangle, by the preceding cases. Therefore, in this case, also, the perimeter 1 of the equilateral triangle is less than that of the given triangle. 22. DEF. A regular polygon is a plane recti lineal figure which is equilateral and equiangular. : PROP. XI. 23. Theorem. Of all regular polygons, con- C2 56 THE GEOMETRICAL INVESTIGATION tained by the same number of sides, that which has the greatest perimeter is the greatest ; and that which is the greatest has the greatest perimeter. Let AD, a d be two equilateral and equiangular polygons, and let AD be greater than ad; the- perimeter also of AD is greater than that of a d. For (E. Cor. 1. 32. 1.) the angles of the one figure are equal to the angles of the other, each to each ; and since, the figures are both of them equi lateral, they are (E. Def. 1. 6.) similar to each other. Wherefore (E. 2O. 6.) AD is to ad as the square on CD is to the square on c d ; but AD is greater than ad\ therefore (E. l6. and 14. 5.) the square on CD is greater than the square on c d y and CD is greater than c d ; but the perimeter of AD is the same multiple of CD as the perimeter of ad is of c c?; therefore (E. 15. 5.) the perimeter ef AD is greater than that of a d. The converse of the proposition is proved in the same manner. OF MAXIMA AND MINIMA. 37 PROP. XII. 24. Theorem. If the perimeter of a regular polygon be less than the perimeter of any other equal rectilineal figure of the same number of sides, the regular polygon is greater than any other -rectilineal figure of the same number of sides, and of equal perimeter: And, conversely, if a regular polygon be greater than any other polygon of the same number of sides and of equal perimeter, then is its perimeter less than that of any other equal polygon of the same number of sides. Let A be a regular polygon, and B a rectilineal figure of the same number of sides and of equal perimeter ; then if the perimeter of A be less than that of any other equal rectilineal figure of the same number of sides, A is greater than B. For if it be not greater, it is either equal to it or less ; but it cannot be equal ; for then, by the sup position, its perimeter would be less than, and not equal to, that of B; neither can it be less, for then its perimeter must be less (Art. 23,) than if it equalled JB, and, therefore, less than the perimeter of B ; therefore A is greater than B 9 since it can neither be equal to it, nor less than it. Conversely ; let A be a regular polygon, and B an equal rectilinear figure ; then, if A be greater than any other rectilineal figure of the same number of sides and of equal perijneter, the perimeter of A, k less than that of B. For if it be not less, it is either equal to it, or 38 THE GEOMETRICAL INVESTIGATION greater ; but it cannot be equal ; for then, by the hypo thesis, A would be greater than B ; but it is also equal to B -, which is absurd : neither can the perimeter of A be greater than that of B ; for then, if C be an other regular polygon, of the same number of sides, having its perimeter equal to that of B, and there fore less than that of A, it follows from Art. 23. that A is greater than C; and C, by the hypothesis, is greater than B ; much more, then, is A greater than B-, but it is also equal to J5; which is absurd. Since, therefore, the perimeter of A can neither be equal to that of B, nor greater than it, the peri meter of A is less than that of B. 25. COR. An equilateral triangle is greater than any other triangle of equal perimeter*. [j norft ; * If any point be taken in the arch of a parabola, included between the vertex and focal ordinate, the distance of that point from the focus, together with its perpendicular distance from the focal ordinate, is equal to the distance between the focus and the vertex. From this property Art. 25. may be thus de duced. Take the straight line SB equal to the semi-perimeter, OF MAXIMA AND MINIMA. 39 tor its perimeter was shewn (Art. 21.) to be less than the perimeter of any other equal triangle. PROP. XIII. 26. Theorem. The perimeter of a square is less than that of any other quadrilateral rectilineal figure which is equal to the square. Let A BCD be any quadrilateral rectilineal figure which is not a square ; the perimeter of an equal square is less than that of ABCD. Join A, C; and if AB be not equal to BC, nor AD to CD, bisect (E. 10. 1.) AC in E ; through of any isosceles triangle; draw &4 perpendicular to SB, and niake it equal to the half of SB ; let the parabola CAB be de scribed, having its focus in S, and its vertex in A, and let XY be its directrix. Let SA, produced, meet XY in L: trisect LS in the points D and JE; through E draw EF parallel to SB, and FG perpendicular to SB. Join S 9 F: then, by the property of the curve, SF is the side of an equilateral triangle of the given perimeter; and if any point P be taken in AfBj and S, P be joined, SP is the side of an isosceles tri angle of the same given perimeter. Draw PN perpendicular to SB, and PO parallel to SB. The rectangle EG is equal to the equilateral triangle, and the rectangle OJV to the isosceles tri angle. And since AL is equal to AS, and LD to ES, DA is equal to EA, and, therefore, the straight line joining D, and F, will touch the curve in F; join D, F; and let DF meet SB produced in Q; because DE is equal to ES, and ,DP(E. 2. 6.) is equal to FQ ; therefore (Art. 4.) the rectangle EG is greater than the rectangle ON ; i. e. the equilateral triangle is greater than any isosceles triangle of equal perimeter, and is, therefore, (Art. 16.)- greater than any other triangle of equal perimeter. 40 THE GEOMETRICAL INVESTIGATION B and D draw (E. 31. 1.) BFand DG parallel to AC, and through E draw (E. 11. 1.) PEG per pendicular to AC, and let it meet BF and DG in F and G ; join A, F, and C, F, and y/, G, and C, G; then (Art. 14, 15.) the perimeter of AFCG is less than that of ABCD. Again, if AF be equal to AG, the figure AFCG is equilateral ; but if AF be not equal to AG, bisect FG in JW; through A and C draw AK, and C/, parallel to FG, and through jfiT draw KHI perpendicular to FG ; join F, K, and JIT, G, and G,, /, and /, F. The figure F#G/ is (Art. 14, 15.) equal to FAGC and has a less perimeter; therefore, also, it is equal to ABCD, and has a less perimeter than ABCD; but KG is equal to G7, (E. 4. 1.) for KH is equal to HI, and //G is common to the two triangles KHG, IHG, which are right-angled at //; therefore the figure FKGI is equilateral, and is either a rhombus or a square ; if it be a rhombus its perimeter is greater (Art. 9.) than that of an OF MAXIMA AND MINIMA. 41 equal square ; wherefore the perimeter of the square is less than that of the equal quadrilateral figure ABCD. 27. COR. The square is greater than any other quadrilateral rectilineal figure of an equal perimeter (Art. 26. and 24.). SCHOLIUM. 28. What has been legitimately proved of the trapezium and the triangle (Art. 27. and 25.) namely, that the perimeter being given, the area is a maximum when the figure is equilateral, is usually inferred from Art. 17. to be true of all polygons whatever. It has been shewn, in the Introduction, that this mode of reasoning is de fective. Before such a conclusion had been drawn, it ought, at least, to have been shewn, that, by re peating the process described in Art. 17., the sides of the polygon approximated to a ratio of equality. This may be readily proved to obtain in the case of the triangle, either by geometry, or algebraically. Let CAB be the isosceles triangle resulting from -* F JE JS Z> the first operation (Art. 17.) upon the given tri angle. Produce AB to D 9 and make BD equal to BC. Bisect AD in E; then is AE, or ED, equal to the side of the next isosceles triangle to be 42 THE GEOMETRICAL INVESTIGATION described, in a similar manned, upon AC, having its perimeter equal to that of ACE. From AB, produced if necessary, cut off AF equal to AC; then, since AE is equal to ED, and AF to BD 9 FE is equal to EB ; and, therefore, FB is the double of FE; but FB is the difference between AB and AC; and FE is the difference between AC and the side of the next isosceles triangle described on AC as a base. Thus the difference between the base and the side of the resulting isosceles triangles is halved at each step; and, therefore, the three sides may be made to approach inde finitely near to a state of equality. This proof is equally applicable to the case of any three mag nitudes whatever, whether any two of them be greater than the third or not ; which may also be shewn algebraically. For, let a, b,c represent any series of three mag nitudes ; let a second series be found from it, by taking the semi-sum of a and b for the two first terms, and c for the third ; in the same manner let a third series be formed from tfre second, by taking the semi-sum of its two unequal terms, for the two first terms of this new series, and so on. The several series thus derived will be, a-\-b a + b , a+b I. , -,c; whence c -- ft -f b + 2c a -f ft -f 2 c a+b II. - - , _ , -_ . t a whence 2 4 OF MAXIMA AND MINIMA* 5 f- 43 a + b + 2c TTT wh ence, , 00 jtii nocin .oidfiftoiJ!)i -fa >j &c. &c ........................... . ...... ... r i r fj^s yet ? -. . 1. i.\ _ ft) + - -C - - and the differ ence between the two unequal terms is C ^ . Df Let w be considered as indefinitely great ; then the limit of the terms of the n th series is evidently a + b 4- c * iij *~F~ v Thus,, by continuing the process^ the terms of the resulting series may be made to approximate indefinitely to a state of equality. The same kind of reasoning may be extended to shew, that if any number of magnitudes be treated in the same manner, the aggregate of their differences will be halved at each step ; it follows, therefore, that, being so treated, they will approximate indefinitely to a state of equality ; and, since it has been proved that, if the sides of any polygon be so treated, the 44 THE GEOMETRICAL INVESTIGATION figure is at each step enlarged, whilst its perimeter remains the same, it may be thence inferred that the greatest polygon of a given perimeter, and a given number of sides, is equilateral. The common mode of proving that this greatest polygon is also equiangular, as well as equilateral, is still more objectionable, upon the same grounds. The method of PAPPUS, which may be seen in the propositions immediately following, is very different ; it is strict to the extent, to which it is here carried, and ingenious, although it be not concise. PROP. XIV. 29. Theorem. If the angles of the two right- angled triangles be equal, each to each, the square described upon the aggregate of the two hypote nuses is equal to the square described upon the aggregate of the bases, together with the square described upon the aggregate of the two remain ing sides. Let ABC, DCE be two triangles, having the 7? OF MAXIMA AND MINIMA. 45 angles ABC, DCE right angles, the angle ECA equal to the angle CED, and the angle at A equal to the angle at D. The square described on the aggregate of AC and DE is equal to the square described on the aggregate of BC and CE, to gether with the square described on the aggregate of B A and CD. Let the bases BC, CE be placed in the same straight line. It may be shewn, as in E. 4. 6., that BA and ED meet when produced, and that if they be produced to meet in F, the figure FC is a parallelogram; therefore (E.34. 1.) FD is equal to AC, and FA to Z>C; but (E. 47. 1.) the square described on FE, is equal to the squares described on FB, BC ; that is, to the square described on the aggregate of AB, DC, together with that de scribed on BC, CE. PROP. XV. 30. Problem. Two isosceles triangles being given, which stand upon unequal bases, and are not similar to each other, to describe, upon the same bases, two other isosceles triangles similar to each other ; and having their perimeters, taken together, equal to the perimeters of the two given triangles. Let AEB, CFD be two dissimilar isosceles tri angles, standing upon the unequal bases AB, CD, of which AB is the greater ; it is required to de scribe upon AB and CD two isosceles triangles, which shall be similar to each other, and shall 4b THE GEOMETRICAL INVESTIGATION have their perimeters, taken together, equal to the perimeters of the two given triangles. Take the straight line OH, and make it equal to AE, EE, CF, and FD, taken together ; divide (E. 10. 6.) GHm K 9 so that HK is to KG as H CD to AE ; bisect GK in L, and HK in M; since, by the construction, HK is to KG as CD to AE, GH is to GK, (E. 18. 5.) as CD together with AE is to AB-, but (Constr. and E. 20. 1 .) GH is greater than CD together with AB; therefore (E. 14. 5.) GA^is greater than AE\ and since GK is to KH as AE to CD, and GK has been shewn to be greater than AB, KH (E. 14. 5.) is greater than CD. Wherefore, of the three lines AB, GL, and LK, any two are greater than the third ; for GL is equal to LK, and AE together with either of them is greater than the other ; and GK, i. e. GL to gether with LK, has been proved to be greater than AE. In the same manner it may be shewn, that of the three lines CD, KM, and MH any Si, i OF MAXIMA AND MINIMA. 47 two are greater than the third ; therefore upon AB (E. 22. 1 .) describe the isosceles, triangle ANB, having its sides AN, BNe&ch equal to GL or LK; and upon CD describe the isosceles triangle COD, having its sides CO, OD each equal to KM or MH. The two triangles ANB, COD are similar to each other, and have their perimeters, taken together, equal to the perimeters of the triangles AEB, and CM). For, AN, BN, CO, and OD, are together equal to GH, which was made equal to AE, EB, CF, and FD taken together ; therefore the perimeters of the triangles ANB, COD are together equal to those of the triangles AEB, CFD ; and because CD is to AB as KH to GK, (E. 15. 5.) CD is to AB as KM to LK; i. e. CD is to AB as CO to AN; wherefore (E. l6. 5.) CD is to CO as AB to AN, and (E. 5. 6.) the two triangles ANB, and COD are similar to each other. PROP. XVI. 31. Theorem. The two isosceles triangles, standing upon unequal bases, which are similar to each other, are greater, taken together, than the two isosceles triangles together, which stand upon the same bases and have equal perimeters with the two former, but are not similar to each other. I^et GBC, FDB be two isosceles triangles, not similar to each other, standing upon the bases BC 9 BD ; of which let jBObe the greater. Upon the same bases describe (Art, 30.) the two isosceles triangles 48 THE GEOMETRICAL INVESTIGATION ABC, EDB of equal perimeter with the former, F and similar to each other ; wherefore (E, 1 9. 6.) the triangle A EC is greater than the triangle EDB. The two triangles ABC, EDB are together greater than the two triangles GBC, FDB. Let the bases DB, BC be placed in the same straight line ; join F, E and A, G ; and produce FE and AG to meet DC in L and M\ produce FL to H, and make LH equal to LF; join H 3 G and H,B;FD is equal to FB, and ED to E B, and FE is common to the two triangles DFE, and UF.E; therefore (E. 8. 1.) the angle DFE is equal to the angle BFE ; also the angle FDL is equal (E. 5. 1.) to & the angle FBL, and FL is common to the two triangles FLD, FLB ; therefore (E. 26.1.) the triangle FLD is equal to the triangle FLB, and the angle FLD to the angle FLB; and (E. 15. and 4. 1.) FB is equal to HB. In the same man- ner EL may be shewn to bisect the triangle EDB, and AGMto bisect the triangles ABC, GBC. OF MAXIMA AND MINIMA. 49 But, by the hypothesis, FB together with EG is equal to EB together with BA ; but FB has been proved equal to HB ; therefore HB together with BG is equal to EB together with BA. But (E. 20. I.) HB together with BG exceeds 7/G; wherefore, also, EB together with BA exceeds HG\ and (Art. 29.) the square on the aggregate of EB and I$A is equal to the square on the aggre r gate of EL, AM, together with the square on the aggregate of LB, BM, that is, to the square on the aggregate of EL, AM together with the square on LM; therefore these two last squares are together greater than the square on HG, i. e. (Art. 29.) greater than the square on the aggre gate of FL, GM, together with the square on LM- 7 take away the common square of LM, and there remains the square on the aggregate of EL 9 AM greater than the square on the aggregate of FL 9 GM; wherefore EL together with AM is greater than FL together with GM; take from both these the common parts EL, GM? and there remains QA greater than EF. And, because the triangle jBM exceeds, by the hypothesis, the similar triangle EBL, (E. 10. 6 f and Art. 5.) BM exceeds BL ; wherefore the rect angle contained by QA and BM, which (E. 41. 1.) is the double of the triapgle AGB, exceeds the rectangle contained by JZF, and BL, which is the double of the triangle FEE ; therefore the tri angle AGB exceeds the triangle FEB ; and if to each of these (triangles be added the two triangles 50 THE GEOMETRICAL INVESTIGATION BGM, BEL, the triangle AMB together with the triangle BEL will exceed the triangle FLB to gether with the triangle BGM; wherefore; also, the doubles of the two former triangles will to gether exceed the doubles of the two latter taken together ; i. e. the triangle ABC together with the triangle EDB is greater than the triangle GBC together with the triangle FDB *. PROP. XVIL v 32. Theorem. If a polygon be not equilateral and equiangular, a greater polygon may be found which has an equal perimeter, and the same num ber of sides. Let ABCDE be any given polygon ; if it be not equilateral a greater polygon may be found of JS J> equal perimeter, and the same number of sides, by- Art. 9.; let, therefore, ABCDE be equilateral * The sides of the two dissimilar isosceles triangles are, in the figure, taken equal, each to each; that being the case which is used in Art. 32 ; and the triangle DEB then necessarily falls within DFB, and BAG without BGC. OF MAXIMA AND MINIMA. 51 but not equiangular ; join A, C and C, E ; then, be cause the angle at B is not equal to the angle at D, the two isosceles triangles ABC, CDE are not (E. 32. l.) similar, and (E. 24. 1.) AC and CE are unequal , upon AC and CE describe (Art. 30.) the two similar isosceles triangles AFC, CGE having their perimeters, when taken together, equal to the perimeters of the triangles ABC, CDE, taken together; then (Art. 31.) the two triangles AFC, CGE, are together greater than the two ABC, CDE together ; add to both the remaining part of the figure, and the polygon AFCGE is greater than the given polygon ABCDE, and it has the same number of sides, and an equal peri meter. 33. COR. 1. Of all polygons of the same num ber of sides, and of equal perimeter, if any one be greatest it is that which is equilateral and equi angular. For, if there be such a maximum, the figure is either equilateral and equiangular or not; but it cannot be a maximum, if its sides be not equal to each other, and also its angles equal to each other ; because, in that case, (Art. 32.) a greater figure might be found. If, therefore, the figure be a maximum, it must be equilateral and equiangular. 34. COR. 2. Of all equal polygons, of the same number of sides, if the perimeter of one be a minimum, it is that of the regular polygon, (Art. 24. and 33.) 52 THE GEOMETRICAL INVESTIGATION SCHOLIUM. It is usually asserted, in the place of Art. 32, that an equilateral and equiangular polygon is the greatest of all isoperi metrical polygons of the same number of sides - 9 and the steps by which this property is commonly demonstrated are these: 1. Of rectilineal plane figures, in which all the sides, but one, are given, the greatest is that which may be inscribed in a semi-circle, having the unde termined side for its diameter. 2. Hence, of all rectilineal plane figures, contained by sides which are given both in number and length, the greatest is that which may be inscribed in a circle. 3. The greatest of all rectilineal plane figures of the same perimeter and number of sides, is that which is equilateral. 4. Therefore the greatest of such figures is that which is equilateral, and which, at the same time, is capable of being inscribed in q. circle ; i. e. which is equilateral and equiangular. But from the proof given of the first step nothing more can be strictly concluded, than what is ex pressed in Art. 11. ; therefore, the deduction con tained in the second step is not fairly drawn. Again, in the third step the same unauthorized extension is made of Art. 17. that is made of Art. 11. in the first step. No more, therefore, is thus proved than what is stated in Art. 32. The advantage of the method of proof here adopted is, that it requires only one hypothesis, namely, that OF MAXIMA AND MINIMA. 53 the greatest of all isoperimetrical polygons, of the same number of sides, is necessarily equilateral; whereas the more common method of proof rests upon two distinct suppositions. It may also be remarked, that if two polygons, contained by straight lines given both in length and number, be inscribed in the same, or equal, circles, they shall be equal to each other, what ever be the order of their sides. For the polygon inscribed in a circle is equal to the difference be tween the circle, and the segments of the circle cut off by its sides ; and the aggregate of the seg ments cut off will manifestly (E. 28.3. E.21.3. and E. 24. 3.) be the same, whatever be the order of the sides of the inscribed figure. Further, it may be shewn, as is done by LEGENDRE, that, if the sides of a polygon be given both in length and number, the radius of the circle in which a polygon, so bounded, may be inscribed can only be of one certain length. For, if it be possible, let two such polygons be inscribed in two circles having unequal radii ; then, if the centers of the circles and each angular point in both figures be joined, the angles at the center will, in both cases, be equal to four right angles; but the angles at the center of the greater circle are together less than the angles at the center of the other ; for the straight lines by which they are subtended are equal, each to each ; and the same straight line subtends a greater angle (E, 21. 1.) at a less perpendicular distance from its bisection 5 54 THE GEOMETRICAL INVESTIGATION wherefore the angles at the centers of the two circles are at the same time equal and unequal, which is absurd. PROP. XVIII. 35. Problem. To find the center of a circle which may be described about any given regular polygon. The construction and the proof are the same as in E. 14. 4. PROP. XIX. 36. Theorem. If a straight line be drawn from either of the acute angles of a right-angled tri angle to cut the opposite side, that side shall have to the segment cut off from it, a greater ratio than the whole acute angle has to the part of it cut of by the straight line. Let BCA be a right-angled triangle, and let the straight line BD be drawn from the acute OF MAXIMA AND MINIMA. 55 angle B, cutting the opposite side AC in D ; AC has a greater ratio to DC than the angle ABC has to the angle DEC. For DB is (E. 17. and 19. 1.) less than BA, and greater than BC ; if, therefore, a circle EDF be described, from J5 as a center, at the distance BD, it will cut BA between A and B in jE, and BC produced in F. The triangle ABD is to the tri angle DEC as AD to DC (E. 1.6.); but the tri angle ABD is greater than the sector EED; and the triangle DEC is less than the sector DBF; therefore (E. 13. 5.) AD has to DC a greater ratio than the sector EBD has to the sector DBF, that is, (E. 33. 6.) a greater ratio than the angle ABD has to the angle DBF-, and, componendo *, AC has to DC a greater ratio than the angle ABC has to the angle DEC. PROP. XX. 37. Theorem. Of regular polygons, which have equal perimeters, that which has the greatest number of angles is the greatest. * EUCLID cannot be quoted for this step ; it may thus be proved. Let A have to B a greater ratio than C has to Z>. Find (E. 12. 6.) JE a fourth proportional to D, C, and JB, so that E i B :: C : D; therefore (E. 10. 5.) A is greater than E; add B to both ; and A together with B is greater than E together with B ; therefore (E. 8. 5.) A together with B has a greater ratio to B than E together with B has to B , i. e,. (E. 18. 5.) than C together with D has to P. 56 THE GEOMETRICAL INVESTIGATION Let the two regular polygons ABC, DEF, have equal perimeters; but let the polygon DEF have the greater number of angles; then is it greater than the polygon ABC. Find (Art. 35.) the centers, G and it, of the circles circumscribing the polygons ABC, from G and H draw (E. 12. 1.) &K per- jpendicular tb AC, and tiL perpendicular to and (E. 3. 3.) they will bisect the angles AGC 9 DHF, and the bases AC, DF; also join A, G and G, C, and D, H and H> F. Then since the figure DEF has (hyp.) more angles than the figure ABC, it has also more sides ; and since the two peri- tneters (hyp.) are equal, DFis less than AC-, and; therefore, DL, which is the half of DF, is less than AK, which is the half of AC. From AK cut offMK (E. 3. 1.). equal to DL y and join M, G. AC is to the perimeter of the figure ABC as the angle AGC is to four right angles; because^ all the angles at the center G, subtended by the 1 equal sides of the polygon^ are equal to each other$ OF MAXIMA AND MINIMA. 57 and are together equal to four right angles ; also, the perimeter of the figure DEFis to Z)jPas four right angles are to the angle DHF; therefore (E. 22. 5.) ex cequali, AC is to DFas the angle AGC to the angle DHF-, therefore, also, (E. 15. 5.) AK is to DL 3 or MK, as the angle AGK to the angle DHL. But (Art. 36.) the ratio of AK to MK is greater than the ratio of the angle AGK to the angle MGK ; wherefore the ratio of the angle AGK to the angle DHL, is greater than that of the angle AGK to the angle MGK> and (E. 10. 5.) the angle MGK is greater than the angle DHL; therefore (E. 3&. 1.) the angle GMK is less than the angle HDL. Make (E. 23. 1.) the angle KMN equal to the angle HDL ; then since MK is equal to DL, and the angle at K to the angle at Z/, the two triangles NMK, HDL are" equal (E. 26. 1.) to each other, and NK is equal to HL ; therefore HL is greater than GK ; but the polygon EDF\s the half of the rectangle (E.41. 1 4 and E. 1.12.) contained by HL and its whole perimeter ; for it is equal to the aggregate of all the equal triangles into which the figure is divided by straight lines drawn from H to the angles ; in the same manner, the polygon ABC is the half of the rectangle contained by GK and an equal perimeter ; and because HL is greater than GK, the former of these rectangles is greater than the latter ; Wherefore, also, the polygon DEF is greater thark the polygon ABC. $8i COR. A regular polygon of any given 58 THE GEOMETRICAL INVESTIGATION number of sides is greater than any other polygon of a less number of sides and of equal perimeter, if the regular polygon of a given number of sides, and of a given perimeter, be a maximum, (Art. 37. and 33.) PROP. XXI. 39. Theorem. Of all equal regular polygons that which has the greatest number of sides has the least perimeter. Let A and B be two equal regular polygons of which A has the greater number of sides ; the peri meter of A is less than that of B. Let a third polygon C be supposed to have the same number of sides as A, and the same perimeter as #; then C (Art. 37.) is greater than B, and, therefore, greater than the equal polygon A\ its perimeter is, therefore, (Art. 23.) greater than that of A\ i. e. the perimeter of B is greater than that of A. r i PROP. XXII. 40. Problem. A regular polygon being given, to describe about a given circle another polygon similar to the given polygon. Let ABC be the given polygon, and DEFthe given circle ; it is required to describe about the circle D EF a poly gon similar to the given polygon ABC. OF MAXIMA AND MINIMA. 59 Find (E. 1.3.) the center H of the given circle; join H and any assumed point Z), in the circum- JL> ference; find G (Art. 35.) the center of the circle described about the polygon ABC\ join G, ^f and G, C and G, 7, &c. ; then (E. 28. and 27. 3.) the angles at the center CGA, AGI y &c. are equal to each other ; bisect each of these equal angles (E. 9. 1.) by the straight lines GK, GQ, &c.; at the point H in HD make (E. 23. 1.) the angle DHE equal to the angle KGQ, the angle EHR equal to the angle DHE, and so on, as often as the polygon has sides ; through Z), E, R, &c. draw the straight lines (E. 11. 1.) LP, LM, MN, &c. at right angles to HD, HE, HR, &c. ; the figure LMNOP is a similar polygon to the given poly gon ABC, and is circumscribed about the circle DEF. For it is circumscribed about the given circle, because (E. Cor. 16. 3.) each of its sides touches 60 f HJE GEOMETRICAL INVESTIGATION the circle. And because the angles at E and D are right angles, the two angles ELD, EHD are together (E.32. 1.) equal to two right angles; in the sarrie manner, the two angles EMtl, RHE are shewn to be together equal to two right angles ; therefore the two former angles are together equal to the two latter ; and the angle DUE is equal to the angle EHR, by the construction ; therefore DLE is equal to EMR, and so the test of the angles at M, N, 0, &c. are shewn to be equal to each other, therefore the polygon LMNO is equi angular ; and since it has the same number of angles as the polygon ABC, any one of its angles is equal (E. 32. 1.) to any one of the angles of the polygon ABC. Again, because LE is equal (E. Cor. 36. 3.) td LD, and HE to HD (E. Defc 15. 1.) and LH \s common to the two triangles LEH, LDH, the angle EHL is equal to the angle DHL; therefore EHL is the half of EHD ; in the same manner EHM may be shewn to be the half of the angle EHR; but the angle EHR is equal to the angle EHD, therefore EHM is equal to EHL ; and the angles at E are right angles, and HE is com mon to the two triangles HEM, HEL ; therefore ML is the double of EL ; and in the same manner LP may be shewn to be the double of LD ; and LE is equal to LD , wherefore ML is equal to LP ; and thus all the sides of the figure may be shewn to be equal to each other; and it has been OF MAXIMA AND MINIMA. 61 already proved to be equiangular ; therefore it is similar to the given polygon ABC. PROP. XXII L 41. Theorem. A circle is equal to the half of the rectangle contained by its circumference and its radius. Let A BCD be the given circle, and let F be the half of the rectangle contained by its circum ference and its radius. The circle ABCD is equal to the rectangle F. For if it be not equal, it is either greater than it, or less. If it be possible, let F be less than the circle ; therefore, as is shewn in the demonstration of E. 2. 12. a polygon may be inscribed in the circle, which shall be greater than F; let ABODE 62 THE GEOMETRICAL INVESTIGATION be such a polygon, so described, and from tbe center G draw (E. 12. t.) GH perpendicular to anyone of its sides, and join G, D ; and, since* tbe circumference of the circle is greater than the perimeter of the inscribed polygon, and its radius GD is greater than GH, (E. 17. and 19. 1.) the rectangle contained by the circumference and the radius of the circle is greater than that contained by GH and the perimeter of the inscribed poly gon ; but this latter rectangle, as hath been shewn in the demonstration of Art. 37-, is the double of the polygon ; therefore F is greater than the in scribed polygon ABCDE ; and it is also less; which is absurd. But, if it be possible, let F be greater than the circle. Then it may be shewn, by reasoning, as in E. 2. 12., that a polygon, KLMNX, may be described about the circle, less than F: join the center G and any one of the points O, in which a side of the circumscribed polygon touches the circle; and since the peri meter of the polygon KLMNX is greater than the circumference of the circle, the rectangle con tained by the perimeter of KLMNX and GO, which is the double of the polygon, is greater than the rectangle contained by the circumference of the circle and GO; wherefore the circumscribed * It is one of the Xapfiavoptva, or axioms, of ARCHIMEDES, that the circumference of a circle is greater than the perimeter of any rectilineal figure inscribed in it j and less than the perimeter of any rectilineal figure described about it. OF MAXIMA AND MINIMA. 63 polygon is greater than F; and it is also less ; which is absurd. Therefore the circle A BCD can neither be greater nor less than F; i. e. it is equal toF. 42. COR. The circumferences of circles are to one another as their radii (E. 2. 12. and 22. 6.). PROP. XXIV. 43. Theorem. A circle is greater than any regular polygon, the perimeter of which is equal to the circumference of the circle. Let the perimeter of the regular polygon ABC be equal to the circumference of the circle DEF; the circle DEF is greater than the polygon ABC. Find H(E. 1.3.) the center of the given circle, and (Art. 35.) G the center of a circle described 64 THE GEOMETRICAL INVESTIGATION about ABC-, about the circle DEF describe (Art. 40.) a polygon LMNOP similar to ABC-, join H, D, and frorn G draw (E. 9. 1.) GK bisect^ ing the angle AGC. Since the perimeter LMNOP is greater than the circumference of the circle DEF, it is also greater than the perimeter of the polygon ABC, by the hypothesis ; and the two polygons have the same number of sides ; therefore LP is greater than AC ; and LD, the half of LP, is greater than AK, the half of AC; also the triangle AGK, as was shewn in Art. 40., is equiangular with the tri angle LHD, and, therefore; (E. 4. 6.) LD : DH i: AK : KG ; wherefore (E. 14. 5.) DH is greater than, KG; and, because DH is greater than KG, and the circumference of the circle is equal (hypoth.) to the perimeter of the polygon ABC, the rectangle contained by DH, and the circum ference of the circle DEF is greater than the rectangle contained by GK and the perimeter of ABC; i. e. (Art. 41.) the double of the circle is greater than the double of the polygon ; therefore the circle is greater than the polygon. 44. COR. 1. A circle is greater than any recti lineal figure, the perimeter of which is equal to its circumference, if the regular polygon of a given perimeter and a given number of sides be a maxi mum (Art. 33.). 45. COR. 2. The circumference of a circle is less than the perimeter of any equal regular polygon. This is proved in the same manner as Art. 39- OF MAXIMA AND MINIMA. 65 46. COR. 3. The circumference of a circle is less than the perimeter of any equal polygon, if the perimeter of the regular polygon of a given area, and a given number of sides, be a minimum, (Art. 34.) SCHOLIUM. It was demonstrated by GALILEO that a circle is a mean proportional between any two regular and similar polygons, one of which is isoperimetri- cal with the circle, and the other circumscribed about the circle. The proposition is equally true, although the two polygons be not regular, pro vided that they are similar to each other ; and hence Art. 43. and 45. may be easily deduced. The relation, also, between the surfaces of isope- rimetrical regular polygons which have not the same number of sides, and the relation between their perimeters, when their surfaces are equal, may, by means of this proposition, be collected from the comparison of similar figures, which ad mit of being circumscribed about the same circle. But, besides that the properties of these circum scribed polygons form a distinct subject, sepa rately investigated in the next section, the method according to which Art. 37. is demonstrated, was preferred, as a specimen of the more ancient Geometry. If a given quantity of land, supposed to be a plane surface, were required to be enclosed by a E 66 THE GEOMETRICAL INVESTIGATION fence of given dimensions, according to any regular figure, with the least quantity of materials, it is manifest from Art. 45. that the fence must be made circular ; and it is easy to compute the exact saving which would accrue from the adoption of this form rather than that of other regular figures. If, for example, the quantity of surface to be enclosed be four acres, and r denote the side of a square rood, or 34.78 yards nearly, O the perimeter of an oblong having its sides as 4 to 1, and containing four acres, T the perimeter of an equal equilateral triangle, S that of an equal square, H that of an equal regular hexagon, and C the circumference of an equal circle, O = 20 x r T- 18.236 x r S=l6 x r ; .-. O-S=4r=139. S 12 nearly. yards H = 14. 889 x /-;. .- H= l.lllr = 38.6*4; yards and O- H= 177.76. yards C= 14.179 xr; .\S- C= 63.33; yards ,- and H-C= 24.69. The difference between the perimeter of the re gular hexagon and that of the equal circle is not, perhaps, so great as, without the calculation, might be expected. But, besides that the advantage is less than might have been conjectured, the cir cular form could seldom be adopted, unless the surrounding space were waste. Where subdivisions are wanted, the figures which fill space about a OF MAXIMA AND MINIMA. 67 given point must be had recourse to; these, as will be shewn in the third part of this work, are the equilateral triangle, the square, and the re gular hexagon ; and there is a considerable differ ence between the lengths of the perimeters of these figures in the above example. 47. DEF. An arch of a circle is any part of the circumference; a chord is any straight line in a circle, terminated both ways by the circum ference ; a sagltta is a straight line joining the bisection of the chord and the bisection of the arch which the chord subtends ; and is, therefore, (E. 3O. 3. and Cor. 1. 3.) in the circle, a segment of that diameter which is at right angles to the chord. PROP. XXV. .48. Theorem. Any given sector of a circle has to the trilineal figure contained by its arch, and the semi-chord and sagitta of the double of its arch, a greater ratio than a right angle has to the angle of the given sector. Let ACD be the given sector ; also let the arch 68 THE GEOMETRICAL INVESTIGATION ADK be the double of the arch AD ; and let AK be the chord, and BD the sagitta of the arch ADK ; the sector ACD has to the trilineal figure ABD a greater ratio than the right angle ABC has to the angle of the sector ACD. First, let the sector ACD be less than a quad rant; draw (E. 11. 1.) AF at right angles to AC, and BH at right angles to AF- AB is at right angles (Art. 46.) to CD ; therefore (E. 8. 6.) the triangles CAP, CAB, FAB, BAH, FAR, are similar to each other ; from the center A at the distance AB, describe the circle GBE ; the trili neal figure EBF has (E. 8. 5.) a greater ratio to the figure EBH than to the sector EAB ; and, therefore, (Art. 36. Note,) componendo, the triangle FBH has to the figure EBH a greater ratio than the triangle FAB to the sector EAB. But because the angle ACD is equal to the angle BAE(E. 33. 6. and 11.5.) the sectors ACD t EAB are to each other as their respective circles; i. e. (E. 2. 12.) as the square of AC to the square of AB ; and (E. 19. 6.) the two similar triangles ABC, ABH are to each other in the same ratio ; wherefore (E. 1 1. & 16. 5.) the sector ACD is to the triangle A CB as the sector ABE to the triangle ABH ; and (E. 17. 5.) the figure ADB is to the triangle ACB as the figure BE His to the triangle ABH; also the triangle ACB is to the triangle FAB as the triangle ABH to the triangle FHB (E. 22. 6.) ; therefore (E. 22. 5.) ex cequali, the figure ADB is to the triangle FAB as the figure EBH to the triangle FHB ; but the OF MAXIMA AND MINIMA. 69 triangle FHB was shewn to have a greater ratio to EBHthm the triangle FAB to the sector EAB-, therefore the triangle FAB has a greater ratio to the figure BAD than it has to the sector EAB\ and, therefore, (E. 10. 5.) the sector EAB is greater than the trilineal figure DAB ; EAB has, there fore, a greater ratio to BAG than DAB has; but DAB has a greater ratio to BAG than it has to ABC-, much more, then, has EAB a greater ratio to BAG than DAB has to the triangle BAC. But the sector EAB is to the sector BAG as the angle EAB to the angle BAC; wherefore the angle FAB has a greater ratio to the angle BAC than the figure DAB to the triangle BAC-, and the tri angle BAC has to the figure DAB a greater ratio than the angle BAC to the angle FAB ; therefore, (Note to Art. 36.) componendo, the sector DCA has a greater ratio to the figure DAB than the angle FAC to the angle FAB ; and FAC is a right angle, and FAB is equal to the angle of the sector ACB ; therefore the sector DCA has to the trilineal figure DBA, a greater ratio than a right angle has to the angle of the sector. But if the sector A CD * be greater than a quad rant ; draw, as before, AE perpendicular to AC, and from A as a center, at the distance AC, de scribe a circle, and let it meet AE in E, and AB produced in F; the angle A CD is greater than the angle CAE ; and the two circles ADK, CEF are See the figure in the following page. 70 THE GEOMETRICAL INVESTIGATION equal ; therefore the sector A CD is greater than the sector EAC ; wherefore the sector A CD has a greater ratio to the triangle ABC than the sector ACE has to it ; much more then has the sector A CD a greater ratio to the triangle ABC than the sector ACE has to the sector ACF 9 which is greater than ABC-, but (E. 33. 6.) the sector ACE is to the sector ACF as the angle EAC to the angle CAF; therefore the sector ACD has a greater ratio to the triangle ABC, than the angle EAC has to the angle CAF*, and (convertendo and componendo) the sector ACD has a greater ratio to the trilineal figure ABD, than the angle EAC has to the angle EAF; but the angle EAC is a right angle, and the angle EAF is, therefore, equal to a right angle together with the angle CAB; as also (E. 32. 1.) is the exterior angle ACD j therefore the angle EAF is equal to the angle ACD ; and the sector ACD has a greater ratio to ABD, than a right angle has to the angle of the sector ACD. OF MAXIMA AND MINIMA. 71 PROP. XXVI. 49. Theorem. Of circular segments, the arches of which are equal, the greatest is the semi-circle. Let the segment ABC be a semi-circle, and let its arch ABC be equal to the arch DEF of any other circular segment ; the semi-circle ABC is greater than the segment DEF. First, let the segment DEF be less than a semi-circle; find (E. 1. and 25. 3.) the centers, G and H 9 of the two circles of which the two given figures are segments; draw (E. 11. 1.) the straight lines GB, at right angles to AC, and HKE, at right angles to DF ; through H draw (E. 31. 1.) the diameter LHM parallel to DF, and join H, D; then (E. 3O. 3.) AGE and LHE are quadrants, and (Art. 42. and E. 15. 5.) the arch LDE is to the THE GEOMETRICAL INVESTIGATION arch AB as HL to GA ; but, by the hypothesis, AB is equal to the arch DE ; wherefore LE is to DE as HL to GA ; but (E. 33. 6.) LE is to DE as the quadrant L/fZ? to the sector DHE ; and (E. 2. 12. and 15. 5.) the square of HL is to the square of GA as the quadrant LHE to the quad rant AGB; wherefore the quadrant LHE has. to AGB a ratio the duplicate of that which it has to the sector DHE ; and DHE is a mean propor tional between the two quadrants LHE and AGB. But (Art. 48.) the sector DEH has a greater ratio to the trilineal figure EDK than the right angle LHE has to the angle DHE, i. e. a greater ratio than the quadrant LHE has to the sector DHE, and a greater ratio than that of the sector DHE to the quadrant AGB; therefore (E, 8. 5.) the quadrant AGB is greater than the figure DKE, and its double ABC is greater than DEF the double of DKE ; that is, the semi-circle is greater than the segment which has an equal arch, and which itself is less than the half of the circle to which it belongs. But if DEF be greater than a semi-circle, it may be shewn, as before, that the quadrant LHE is to the sector DHE as the sector DHE is to the quadrant AEG ; and, therefore, (Art. 48.) the sector DHE has a greater ratio to the figure DKE than the right angle LHE has to the angle DHE, i. e. a greater ratio than that of the quad rant LHE to the sector DHE ; and, therefore, greater than that of the sector DHE to the OF MAXIMA AND MINIMA. ?3 quadrant AGE -, wherefore the quadrant AGE is greater than the figure DEK, and its double AEC y greater than DEFthe double of DEK. 50. COR. The space which can be enclosed by a given finite line,, together with an indefinite straight line, the given line being disposed in the form of a circular arch, is greatest when the arch is a semi-circle. 51. Postulate. A plane figure of any kind having been described upon a finite straight line, let it be granted, that a similar and equal figure may be described^ or be supposed to be described, upon an equal straight line. PROP. XXVII. 52. Theorem. If space be to be enclosed by a given finite line, together with an indefinite straight line, and if the f nclosed space be not made a semi-circle, a greater space may be found under the same conditions. Let the figure ACEB be contained by the finite line ACEB of given length, and the part 74 THE GEOMETRICAL INVESTIGATION BA of the indefinite straight line BX; if the figure ACEB be not a semi-circle, a greater space may be found, contained by a line of the given length, and part of an indefinite straight line. Let ACEB be not a semi-circle ; if the straight lines drawn from A and B, to any the same point C in the perimeter ACEB, are in every such point, at right angles to each other, ACEB (E. 31. 1.) is a semi-circle; but it is not, by the hypothesis; therefore there is some point in the perimeter ACEB, to which, if straight lines be drawn from A and B, they will not be at right angles to each other; let C be that point; join A, C and B, C; then, since the angle BCA is not a right angle, draw (E. 11. 1.) CD at right angles to CB, and (E. 3. 1.) make it equal to CA ; upon CD let the figure CND be supposed (Art. 51.) to be described, similar and equal to CMA ; and join B, D; then the triangle BCD is greater (Art. 8.) than the triangle BCA ; and if there be added to BCD the two figures EEC, CND 3 and to BCA the two figures EEC, CM4, the whole figure BE CND is proved to be greater than the whole figure BECMA, because CND is equal to CMA i i. e. a greater figure has been found than BECMA, BECMA not being a semi-circle. 53. COR. l. If there be any space, en closed according to the above conditions, which is a maximum, it is a semi-circle. 54. COR. 2. If the semi-circle be the greatest space which can be so enclosed, then the figure OF MAXIMA AND MINIMA. 75 contained by two given finite lines, one of which is straight, is a maximum, when it is a segment of a circle, having the given finite straight line for its chord. Let the space ANB be contained by the finite straight line AB, and the finite line ANB dis posed in the form of an arch of a circle, the re- maining part of which, BCD A, may be found by E. 25. 3. ; and let AOB be any other figure con tained by AB and the boundary AOB equal to ANB-, draw the diameter of the circle AC; then if the semi-circle ABC be a maximum under the above specified conditions, the segment ANB is greater than the figure AOB. For then the whole figure A NBC is greater than the whole figure AOBC; and if from both be taken the common part ABC, there remains the segment ANB greater than the figure AOB. 55. COR. 3. The same supposition being made, the circle is greater than any other plane figure of equal perimeter; and the circumference 76 THE GEOMETRICAL INVESTIGATION, &C. of the circle is less than the perimeter of any other equal plane figure. First, let the circle A BCD*, and the plane figure EFGH have equal perimeters, the circle is the greater. For draw any diameter AC of the circle ; and let EFG be taken in the perimeter of the other figure, and supposed equal to the semi -circumference of the circle ; then, (Art. 53.) according to the sup position, the semi-circle ABC is greater than the figure EFG, and the semi-circle ADC is greater than the figure EHG ; wherefore the whole circle is greater than the whole figure EFGH. But let now the circle ABCD be supposed equal to the figure EFGH; if in the former case it be greater than EFGH, in this case its circumference is less than the perimeter of that figure ; for it cannot be equal to that perimeter, because then the circle would be the greater figure of the two ; neither can it be greater than the perimeter, because (Art. 42. and E. 2. 12.) the circle of which it is the circumference would be still gneater, and would not be equal to the figure EFGH. See the figure in p. 75. ON MAXIMA AND MINIMA PART I. SECTION II. 77} Jeift sdt PROP. I. -IT & ( f)6. Problem. IF two given straight lines touch a given circle, to draw the shortest tangent to the same circle, on either side of the straight line joining the given points of contact, which is termi nated by the two given tangents, b 9 Let the two straight lines AB, CD, which are parallel, or the two AB, AC which meet in the point A, touch the given circle BECF in B 78 THE GEOMETRICAL INVESTIGATION and C; it is required to draw the shortest line, on either side of the straight line BC joining B and C, a H O which shall touch the circle, and be terminated by the first two tangents, AE, DC, or AE, AC. First, let the straight lines ^jB^and CD be parallel ; find (E. 1. 3.) the center A: of the circle BECF; draw the diameter (E. 31. 1.) EKF parallel to AE, or DC, and through the points E and F draw (E. 17. 3.) GH, and /L touch ing the circle ; GH is shorter than any other tangent to the circle on the same side of BC ; and /L shorter than any other tangent on the same side with itself of EC, both being terminated by AE and DC OF MAXIMA AND MINIMA. 79 For draw PNO touching (E. 17-3.) the circle in any other point than F; and because (E. 18. 3.) the angles at E and F are right angles, GE is parallel (E. 28. 1.) to IF, and the figures GF, EL, GL are parallelograms ; and, therefore, (E. 34. 1.) the angles at / and L are right angles ; therefore (E. 17. l.) the angles XPI 9 XOL are each of them acute; and (E. 19. l.) PX is greater than IX; and XO greater than XL-, wherefore PX, together with XO 9 is greater than IX, together with XL ; i. e. PO is greater than IL. But let, now, the two given tangents not be parallel, and let them meet in A ; find (E. 3. 1.) the center K of the circle BECF; join A, K, and let AK meet the circumference in E y and again, 80 THE GEOMETRICAL INVESTIGATION when produced, in JP; through E and F draw (E. 17. 3.) GH and IL touching the circle ; GH and IL are the shortest tangents on each side of BC. For, join K, B and K, C ; and let PNO and QR be any other tangents on each side of BC, terminated by AB and AC, and meeting GH and IL in the points X ; draw (E. 31. 1.) LM parallel to AB. And because (E. 18. 3) the angles at B and C are right angles, and KB is equal to KC, and ^fi (E. 36. 3. Cor.) to AC, therefore (E. 4. 1.) the angle BAK is equal to the angle CAK\ and AD is common to the two triangles ADB, ADC-, wherefore (E. 4. 1.) the angle ADB is equal to the angle ADC, and they are (E. Def. 10. 1.) both right angles ; hence, because (E. 18. 3.) the angles GEF, EFI are right angles, GH and IL are parallel (E. 28. 1.) to BC: Again, because the angle GAE, as hath been shewn, is equal to the angle HAE, and the angle AEG to the angle AEH, and AE is common to the two triangles AEG, AEH, GE is equal to EH; and in the same manner IF may be shewn to be equal to FL. But the two triangles PXI, LXM (E. 15. 29. 1.) are equiangular ; wherefore (E. 4. 6.) IX is to IP as XL is to LM-, but IX is less than XL, be cause F is the bisection of IL ; therefore (E. 14. 5.) IP is less than LM-, but the angle APO, which is equal (E. 28. 1.) to the angle LMO, is greater (E. 16. 1.) than the angle AIL, which is equal to the angle ALI; and A LI is greater than LOM; OF MAXIMA AND MINIMA. 81 wherefore LMO is greater than LOM, and (E. 19. 1.) LO is greater than LM ; much more then is LO greater than P/; but LC is equal (E. 36. 3. Cor.) to LF, and is, therefore, equal to IF-, and if to LO and IP be added the equals LC and IF, OC will be greater than FI and IP ; therefore ON, which is equal to OC, is greater than jF7and IP ; add PNto ON, and PB to FI and IP, and OP will be greater than FI and /Z? ; and, therefore, greater than the double of IF, i. e. than IL. And RQ, in both the cases, may be proved to be greater than GH 9 in the same manner as PO is proved to be greater than IL. 57. COR. 1. The two shortest tangents, so drawn, are bisected in their respective points of contact, and are perpendicular to the straight line joining the points of their contact ; which straight line, produced, passes through the intersection of the two given tangents, when they cut one another. 58. COR. 2. The perimeter of a triangle de scribed about the given circle, and contained by two given tangents which meet, and the greater of the two shortest tangents, so drawn, is less than the perimeter of any other triangle described about the same circle, and having the same vertical angle. For IB, IL, and LCare, together, the double of IL ; and PB t PO, and OC are, together, the double of PO ; but IL has been shewn to be less than PO; wherefore IB, IL, and LCare together less than PB, PO, and OC; add to both AB and F 82 THE GEOMETRICAL INVESTIGATION AC, and the perimeter of the triangle AIL is mani festly less than that of the triangle APO. 59. COR. 3. The perimeter of the quadrilateral rectilineal figure IGHL, described about a circle, and contained by any two given tangents, and the two shortest tangents included between them, is less than that of any other such figure PQRO, contained by the two given tangents, and any two other tan gents which meet them, and are not the shortest. For, GH is less than QR, and IL than PO; but OR, together with QP, is equal to PO, toge ther with QR, because (E. 36. 3.) OC is equal to ON, BP to PN, and CR to RX also 1G, toge ther with Lffy is, in the same manner, shewn to be equal to IL together with GH; therefore the whole perimeter of IGHL is less than that of PQRO. PROP. II. 60. Theorem. The perimeter of the equi lateral triangle described about a circle, is less than that of any other triangle described about the same circle. Let ABC be a triangle described about the circle GHM : Either the side BC is bisected at. right angles in the point of contact G, or else (Art. 57. and 58.) another triangle, of less peri meter, may be described about the given circle, having its side opposite to the angle A bisected in the point of contact. But let BC be so bisected in G; and, first, let the angle BAC be greater OF MAXIMA AND MINIMA, 83 than any one of the angles of an equilateral tri angle. Describe (E. 3. 4.) about the circle GHM the equilateral triangle DEF, touching the circle in the points G, H, and M. Then, if A, G be joined by the straight line AG, the angle AGB is equal (Art. 5?.) to the angle AGC, and (E. 19. 3.) the center of the circle is in AG ; in the same manner, if D, G be joined, the center of the circle may be shewn to be in DG -, wherefore DA and AG are in the same straight line ; and since, by the hypothesis, the angle EDF is less than the angle BAC, the angle XDA, which (E. 4. 1.) is the half of EDF, is less than the angle XAG, which is the half of BAC-, therefore (E. 16. 1.) XAG is the exterior angle, and the point D is further from BC than the point A is. Produce 84 THE GEOMETRICAL INVESTIGATION BA till it meet DFiu K; then (Art. 58.) the peri meter of the triangle FBK is greater than that of FED-, take from both the common parts FE, and FK, and there will remain EB, together with BK, greater than ED, together with DK; but the angle XAG was shewn to be greater than the angle XDA ; whence DAK, which is equal (E. 15. 1.) to XAG, is greater than ADK, which is equal to XDA; and (E. 19. 1.) DK is greater than AK; if, therefore, AK be taken from the aggregate of EB and BK, and DK be taken from the aggregate of ED and DK, there will remain EB, together with BA, greater than ED ; add to both EG; and GB, together with BA, is greater than GE, together with ED ; i. e. half of the perimeter of the triangle ABC is greater than half of the perimeter of DEF ; wherefore the whole perimeter of ABC is greater than the whole perimeter of DEF. But, if the vertical angle of the given triangle be less than any one of the equal angles of an equilateral triangle, the angle at its base (E. 5. and 32. 1.) will be greater, because ABC is iso sceles ; therefore (Art. 56. and 58.) another tri angle may be described about the circle of less perimeter, and having the angle at the base of the given triangle for its vertical angle; but the perimeter of such a triangle has been shewn to exceed that of an equilateral triangle ; much more, then, does the perimeter of the given triangle ex ceed that of an equilateral triangle. OF MAXIMA AND MINIMA. 85 PROP. III. 6l. Theorem. The perimeter of the square de scribed about a given circle, is less than that of any quadrilateral rectilineal figure described about the same circle. Let PQRO be any quadrilateral rectilineal figure described about the circle BECF; if PO and QR be not bisected in their points of contact, draw (Art. 56. and 57.) the two tangents IL and GH y which are bisected in the points of contact F and EI and if GI and HL be not bisected in their points of contact, draw SW and 77^ in the same manner, so as to be bisected in the 86 THE GEOMETRICAL INVESTIGATION points Y and Z, where they touch the circle ; the perimeter of the figure PQRO is greater (Art. 59.) than that of the figure IGHL ; and the perimeter of IGHL is greater than that of WSTV\ the perimeter, therefore, of PQRO is greater than that of WSTV, and it is evident, from the de monstration of the former part of Art, 56, and Art. 59, that WSTV \* a square. PROP. IV. 62. Theorem. If the sides of a polygon de scribed about a circle be each of them bisected in their several points of contact, the polygon is equilateral and equiangular. Let AB, BC, CD be any three adjacent sides of a polygon described about the circle EFG, and kt them be bisected in their points of contact E, F, and G; the polygon ABCD is equilateral and equiangular. OF MAXIMA AND MINIMA. 87 Find (E. 1. 3.) the center of the circle K, and join K y E, and K, B, and K, F; then (E. 36. 3.) BE is equal to BF, therefore BA, which is the double of BE) is equal to BC, which is the double of BF; and in the same manner BC may be shewn to be equal to CD, and the polygon to be equilateral. It is also equiangular ; for the angles at F (E. 18.3.) are right angles; BF is .equal to FC, and KF common to the two triangles KBF, KFC-, therefore (E. 4. 1.) the angle KBF is equal to the angle KCF; and in a similar manner it may be shewn that the angles EBK and KBF are equal, as also the angles KCG and KCF; the angle EBF is, therefore, the double of the angle KBF, and FCG is the double of KCF; but the angle KBF was shewn to be equal to the angle KCF; therefore the angle ABC is equal to the angle BCD ; and in the same manner the remaining angles may be shewn to be equal ; therefore the figure is also equiangular, as well as equilateral. PROP. V. 63. Theorem. If a polygon described about a circle be not equilateral and equiangular, another polygon of the same number of sides may be de scribed about the same circle, which has a less perimeter. Let PQRO (Fig. Art. 6l.) be any polygon described about the circle BECF; if any of its 88 THE GEOMETRICAL INVESTIGATION sides, PO, be not bisected in its point of contact N, draw (Art. 56. and 57-) the tangent IL 9 which is bisected in its point of contact F\ it was shewn, in Art. 56, that IL is less than PO, and PI less than LO, and the remaining part of the perimeter is common to the two figures IQRL, PQRO ; therefore the perimeter of IQRL is less than that of PQRO ; and, thus, if every one of the sides of PQRO be not bisected in its point of contact, i. e, (Art. 62.) if the figure be not equilateral and equi angular, another polygon, of the same number of sides, may be described about the circle, which has a less perimeter. PROP. VI. 64. Theorem. Any polygon, which is described about a circle, is equal to the half of the rectangle contained by the perimeter of the polygon, and the radius of the circle. For it may be divided into as many triangles as it has sides, each of which is equal (E. 41. 1.) to half of the rectangle contained by a side and the radius of the circle ; therefore the whole polygon, which is equal to the aggregate of these triangles, is also equal to half of a rectangle, which is the aggregate of the rectangles, i. e. (E. 1 . 2.) to half of the rectangle contained by the perimeter of the polygon and the radius of the circle. 65. COR. All isoperimetrical polygons de- OF MAXIMA AND MINIMA. 89 scribed about the same circle, are equal to each other, whatever be the number of their sides. PROP. VII. 66. Theorem. Of all triangles described about the same circle, that which is equilateral is the least. For (Art. 60.) the perimeter of the equilateral triangle is less than that of any other triangle de scribed about the same circle ; and, therefore, the rectangle contained by the perimeter of the equi lateral triangle and the radius of the circle, is less than that contained by the perimeter of any other triangle and the same radius ; i. e. the double (Art. 64.) of the circumscribed equilateral triangle is less than the double of any other circumscribed triangle; therefore the equilateral triangle is less than any other triangle described about the same circle. SCHOLIUM. This proposition may be thus demonstrated, in dependently of Art. 60. First, an isosceles triangle AIL (Fig. Art. 6l.) is less than any scalene triangle APO, having the same vertical angle, and circumscribed about the same circle. For LM being drawn parallel to AI the two triangles PXI, and LXM (E. 15. and 29. 1.) are similar; wherefore (E. 19.6.) P XI : 90 THE GEOMETRICAL INVESTIGATION LXM as the square of IX is to the square of XL ; but IX is less than XL ; wherefore the triangle PXI is less than the triangle LXM ; much more then is it less than the triangle LXO ; the trape zium APXL, together with the triangle PXI, is, therefore, less than the same trapezium, together with the triangle LXO ; i. e. the isosceles triangle AIL is less than the scalene triangle APO, having the common vertical angle IAL*. Secondly, an equilateral triangle, described about a given circle, is less than any other tri angle described about the same circle. For if the circumscribed triangle, which is not equilateral, be not isosceles, a less may be found, by the first case, which is isosceles ; let it, therefore, be isosceles as ABC (Fig. Art. 60.), and let DEF be an equilateral triangle described about the same circle and touching it in the same point G as the base of the isosceles triangle ; produce BA to meet DF in K. Then, by the former case, the triangle EDF is less than the triangle BKF; and since the part of KF, above the line ^C x ,is less than the part below it, it may be shewn, (as in the former case PXI was proved to be less than LXO) that the triangle BKF is less than the triangle BAC-, much more then is the triangle DEF less than the triangle * The demonstration of this case was given by Dr. STEDMAN, (Phil. Trans, of 1775.) t This proposition being so proved, it is evident that Art. 66, might fairly be deduced from it. OF MAXIMA AND MINIMA. 91 PROP. VIII. 67. Theorem. The square is the least of all quadrilateral rectilineal figures described about a given circle. This is deduced from Art. 6l. and 64, in the. same manner as Art. 66. was made to follow from Art. 60. and 64. PROP. IX. 6s. Theorem. If a polygon, described about a given circle, be not equilateral and equiangular, a less polygon, of the same number of sides, may be described about the same circle. This is manifest from Art. 63. and 64. PROP. X. 69. Theorem. Of all triangles, standing upon the same base, and on the same side of it, and having equal vertical angles, the perimeter of that which is isosceles is the greatest. Let ABC be an isosceles triangle, and APC any other triangle standing upon the same base AC, and having its vertical angle APC equal to the vertical angle ABC. The perimeter of ABC is greater than that of APC. From the center B, at the distance BA or BC S 92 THE GEOMETRICAL INVESTIGATION describe the circle ADC; produce AE and AP to meet its circumference in D and E ; and join Z>, Cand E 9 C; the angle ADC is equal (E. 21. 3.) to the angle ;/JC ; and ABC is equal to APC by the hypothesis ; but (E. 20. 3.) ^1?C is the double of ADC-, wherefore, also, APC is the double of AEC; again, the exterior angle APC is equal (E. 32. l.j to the two interior angles PEC, PCE; and APC is the double of AEC-, if, therefore, PEC be taken from both those equals, there will remain PEC equal to PCE ; therefore (E. 6. 1.) PE is equal to PC, and AE to ^/P and PC; also 1XB is equal to BC, and ^Z> to AB and #C; but (E. 15. 3.) the diameter AD is greater than AE ; wherefore ^J5 and J3C are, together, greater than AP and PC together ; and if to both AC be added, the whole perimeter ABC is shewn to be greater than that * This Proposition is deduced by Dr. HORSLEY from the 97th of Euclid s data. In order to make use of Euclid s propo sition, it is only necessary to complete the circle ADC f bisect the arch which lies below AC, and to join the points of the bisection and the points B and P. OF MAXIMA AND MINIMA. 70. COR. If a polygon, inscribed in a circle, be not equilateral, (and, therefore, also equiangular) another polygon, of the same number of sides, may be inscribed in the same circle which has a greater perimeter. PROP. XI. 71. Theorem. The perimeter of an equilateral triangle inscribed in a circle, is greater than that of any other triangle inscribed in the same circle. Let ABC be any triangle inscribed in the circle ABC. Either ABC is isosceles, or else (Art. 69.) another triangle of greater perimeter, and which is isosceles, may be inscribed in the same circle. Let, therefore, ABC be isosceles ; and either its 94 THE GEOMETRICAL INVESTIGATION vertical angle EAC is greater than any one of the angles of an equilateral triangle, or else, as was shewn in the latter part of Art. 60, another iso sceles triangle may be inscribed in the circle, having its vertical angle greater than the angle of an equilateral triangle ; and (Art. 69.) its peri meter will be greater than that of the former isosceles triangle. Let then ABC be isosceles, and have its vertical angle EAC greater than that of an equilateral triangle. Inscribe (E. 2. 4.) the equilateral triangle ADE in the circle ABC, having its base DE parallel to BC, and draw AG (E. 12. 1.) perpendicular to DE ; then (E. 29. 1.) it will be also perpendicular to EC-, and, because DE is parallel (E. 2. 4. and 27. 1.) to the tan gent at A, AG is perpendicular to it; therefore (E. 10. 3.) the center of the circle lies in AG, and AG bisects (E. 3. 3.) BC and DE, in F and G ; therefore AB and BF are equal to the semi-peri meter of ABC, and AD, DG, to the semi-perimeter of ADE : Join D, B. If DE be perpendicular to EC, EG (E. 29. 1.) is a parallelogram, and BF is equal (E. 34. 1.) to DG, but (.7. 3.) AB is less than AD; where fore AB 9 together with EF y is less than AD, together with DG ; therefore the whole perimeter of ABC is less than the whole perimeter of ADE. But, if DE be not perpendicular to EC, upon DE, as a diameter, describe the circle BHD, cutting BC in H, and AB produced in M\ join D, M\ then since (E. 31. 3.) the angle OF MAXIMA AND MINIMA. 95 BHD is a right angle, HG is a parallelogram, and HF is equal to DG; also join Z), C; the angle MBD is equal (E. 13. 1. and 22. 3.) to the angle A ED, and (E. 21. 3.) the angle DBHto the angle DAC; but D^Cis greater than DAE, or ^Z>; therefore the angle DBH is greater than the angle DBM, and (E. 7. 3.) Mfl is greater than BH: Again, because the angle AMD is a right angle, AD is greater (E. 17. and 19. 1.) than AM, i. e. than AB and jBM; much more, then, is AD greater than AB and BH, and if to AD be added Z>G, and to AB and /J//^ be added HF, which is equal to DG, AD, DG will be greater than AB, BF; i. e. the semi-perimeter of ADE is greater than the semi-perimeter of ABC; and, therefore, the whole perimeter of ADC is greater than the whole f, t-ng-v perimeter of AJBL. PROP. XII. 72. Them^em. The perimeter of the square in scribed in a circle, is greater than the perimeter of any other quadrilateral rectilineal figure in scribed in the same circle. Let A BCD be any quadrilateral rectilineal figure inscribed in the circle HF1G ; its perimeter is less than that of a square inscribed in the same circle. For, join A, C\ and (E. 1O. 1.) bisect AC in E; and (E. 11. 1.) draw the straight line FEG per pendicular to AC, which will, therefore, (E. 1. 3.) pass through the center, and be a diameter of the 96 THE GEOMETRICAL INVESTIGATION circle ; join A 9 F and F, C, and A, G and G, C; then (E. 4. 1.) the two triangles AFC, AGC are G isosceles , and, therefore, (Art. 69.) the perimeter of the figure AFCG is greater than that of the figure ABCD ; again, bisect the diameter FG in K, and draw the diameter HKIat right angles to it; also join H, F and F, /, and /, G and G, H\ and it may be shewn, in the same manner, that the perimeter of HFIG is greater than that of AFCG ; much more, then, is it greater than that of ABCD-, and (E. 6. 4.) HFIG is a square in scribed in the circle HFIG-, therefore the peri meter of the inscribed square is greater than that of any other quadrilateral rectilineal figure in scribed in the same circle. OF MAXIMA AND MINIMA. PROP. XIII. 97 73. Of all triangles standing upon the same base, and on the same side of it, and having equal vertical angles, that which is isosceles is the greatest. Let ACB be an isosceles triangle, and APB any other triangle standing upon the same base, and having its vertical angle APB equal to the vertical angle ACB ; the triangle ACB is greater than the triangle APB. About the triangle ACB describe (E. 5. 4.) the circle ACB ; and because the angle ACB is equal to the angle APB, the circumference of the circle shall pass through P (E. 21.3.); otherwise, the exterior angle of a triangle would be equal to the interior opposite angle, which (E. 16. 1.) is absurd. From the point Cdraw (E. 17. 1.) the straight line Cr, touching the circle ACB ; then (E. 32. 3.) the angle TCB is equal to the angle CAB, and, 98 THE GEOMETRICAL INVESTIGATION therefore, (E. 5. 1.) equal to the angle CBA; wherefore (E. 27. 1.) CT is parallel to AB , and AP produced will meet CT; let it be produced to meet CT in T, and join TE\ therefore (E. 37. 1.) the triangle ACE is equal to the tri angle ATB\ but the triangle ATE is greater than the triangle APE, which is a part of it; wherefore, also, the isosceles triangle ACE is greater than the scalene triangle APE. 74. COR. If a polygon, inscribed in a circle, be not equilateral, (and, therefore, also equian gular) a greater polygon, of the same number of sides, may be inscribed in the same circle. PROP. XIV. 75. Problem. To cut off from the circum ference of a given circle an arch, such that the rectangle contained by its chord and sagitta shall be a maximum. Let ABC be the given circle; it is required to cut off an arch from the circumference of ABC, such that the rectangle contained by its chord and sagitta shall be a maximum. Find (E.3. 1.) the center K of the given circle, and draw any diameter AKE ; bisect (E. 10. 1.) KE in D, and through D draw (E. 11. 1.) the chord BDC perpendicular to AE ; the rectangle contained by the chord EC and DA, the sagitta of the arch EAC is greater than the rectangle contained by the chord and sagitta of any other arch. OF MAXIMA AND MINIMA. 99 For, take any other point P in the circumference If and draw the chord (E. 12. 1.) PMR perpendi cular to AE, and (E. 29. 1.) it is parallel to BC-, join A) B, and K, B and B, E; also draw (E. 1 7. 3.) BH touching the circle in B, and let BH meet RP, produced, in Q, and AE, produced, in H. The two right-angled triangles BDK> BDE have the side BD common, and the two other sides DK, DE, which are about the right angles in each, equal; therefore (E. 4. I.) BK is equal to BE, and BK is also (E. 15. Def. 1.) equal to KE -, wherefore the triangle BKE is equilateral and (E. 5.1. Cor.) equiangular. Again, because (E. 31. 3.) the angle ABE is a right angle, and BD is perpendicular to AE, the angle BAD is equal (E. 8. 6.) to the angle DBE ; and in the same manner KBD may be shewn to G2 100 THE GEOMETRICAL INVESTIGATION be equal to DHB ; but KBD is equal to EBD, the two triangles BDK, BDE having been proved to be equal ; wherefore the angle BAD is equal to the angle BUD ; and the angles at D are right angles, and BD is common to the two triangles BDA, BDH; wherefore (E. 26. 1.) DA is equal to DH. And because QM is parallel to BD, HD is to DB as HM to MQ (E. 4. 6.) ; wherefore (E. 22. 6.) the rectangle contained by HD and AD, i. e. the square of AD is to the rectangle contained by AD and DB, as the rectangle con tained by HM and MA is to that contained by MQ and MA; but (E. 5. 2.) the square of AD is greater than the rectangle contained by HM and MA ; therefore (E. 14. 5.) the rectangle contained by AD and DB is greater than the rectangle con tained by AM and MQ ; much more, then, is it greater than that contained by AM and PM; and, therefore, the double of the rectangle con tained by AD and DB, i. e. the rectangle con tained by AD and BC, is greater than the double of the rectangle contained by AM, and PM, i. e. than the rectangle contained by AM and PR*. * Since the area of a parabola is varied as the rectangle con tained by its axis and terminating ordinate, and since the axes of different parabolas, cut out of the same cone, are varied as the sagittas of the arches cut off; by the planes of the parabolas, from the circular base of the cone, it is manifest, from this proposi tion, that the greatest parabola, which can be cut out of a given cone, is that, the plane of which divides the diameter of the base of the cone into two parts, which are to each other as 3 to 1. OF MAXIMA AND MINIMA. 101 PROP. XV. 76. Theorem. Of all triangles inscribed in the same given circle, that which is equilateral is the greatest. Let ABC be the given circle, and APR any triangle inscribed in it ; if APR be not isosceles, another greater triangle may be inscribed in it (Art. 73.) which is isosceles; but let APR be isosceles; then, the construction of Art. 75. being made, the rectangle contained by AD and EC is greater than that contained by AM and PR, and, therefore, (E. 41. 1.) the triangle ABC is greater than the triangle APR. But the angle HBD was shewn (Art. 75.) to be equal to the 102 THE GEOMETRICAL INVESTIGATION angle BKD which is one of the angles of the equi lateral triangle BKE ; and (E. 32. 3.) HBD is equal to the angle EAC\ wherefore EAC is equal to the angle of an equilateral triangle ; and ABD was shewn (Art. 75.) to be equal to HBD, and is, therefore, also equal to BAC-, and (E. 32. 1.) the triangle ABC is equiangular, and, therefore, (E. 6. 1 . Cor.) equilateral ; and it has been shewn to be greater than any other triangle inscribed in the same circle ABC. Otherwise : The proposition being reduced, by means of Art. 73, to the comparison of the equilateral triangle ADE and the isosceles triangle AB C inscribed in the same circle, and D, C being joined, the two triangles ANBaudDNCare(E. 15. Land 22. 3.) similar to each other ; wherefore (E. 19, 6.) Z>A T Cis toANB as the square of DC to the square of AB; but the angle DAC is greater, and the angle ADC is less, OF MAXIMA AND MINIMA. 103 than an angle of the inscribed equilateral triangle ; therefore DC is greater than AC (E. 19. 1.) or than, its equal, AB ; the triangle DNC is there fore, greater than the triangle ANB- y add to both the triangle ANC, and the whole triangle ADC is greater than the whole triangle ABC-, but (Art. 73.) the triangle ADE is greater than ADC; much more, then, is ADE greater than ABC. PROP. XVI. 77 Theorem. The square inscribed in a circle is greater than any other quadrilateral rectilineal figure inscribed in the same circle. Let ABCD be any quadrilateral rectilineal figure inscribed in the circle HFIG. It may be shewn to be less than the square HFIG inscribed in that circle, by the help of Art. 73, in the same manner as Art. 72. was deduced from Art. 69. 104 THE GEOMETRICAL INVESTIGATION PROP. XVII. 78. Theorem. Of two regular polygons de scribed about a circle, that which has the less number of sides has the greater perimeter, and is the greater. Let AB and CD be the sides of two regular polygons R and S 9 described about the circle PEG, both touching it in the same point E and let R have a less number of sides than S ; the perimeter of 12 is greater than that of S 9 and the polygon R is also greater than S. Find (E. 1.3.) the center K of the circle, and join K, A and K, B, and K, C and K, D, and K, E\ it follows, from Art 62, and E. 18. 3. and 29. 1, that KE bisects the angles AKB and CKD; and because the regular polygon R has a less number of sides than S, it has also a less number of equal angles, at the center K, subtended by its OF MAXIMA AND MINIMA. 105 tides; wherefore (E. 15. 1. Cor. 2.) each of those angles is greater than each of the equal angles subtended by the sides of the other polygon S; i. e. the angle AKB is greater than CKD ; and, therefore, AKE is greater than CKE } and AE greater than CE, and the side of R greater than that of A^; and (Art. 36.) the ratio of AE to CE exceeds that of the angle AKE to CKE; there fore (E. 15. 5.) the ratio of AE to CD exceeds that of AKB to CKD, or (E.33.6.) that of the arch PEG to the arch HEL ; therefore*, also, AB has to PEG a greater ratio than CD has to HEL, and (E. 15. and 11. 5.) the perimeter of R has a greater ratio to the circumference of the circle than the perimeter of S has to the same circum ference ; because the perimeter of R is the same multiple of CAB that the circumference is of FEG\ and the perimeter of S is the same mul tiple of CD that the circumference is of HEL ; wherefore (E. 8. 5.) the perimeter of R is greater than that of Si and, it is manifest, also, from Art. 64, that R is greater than S. 79. COR. An equilateral triangle is the greatest of all regular figures described about the same circle, and has the greatest perimeter. * Let A : BC" C : D; then A : CCT B : D. For, find (E. 13. 6.) a fourth proportional to D, C, and B ; then, because E: B :: C : D, and A : B C" C : D, A has a greater ratio to B than D has; and, therefore, (E. 10.5.) A C" E and A : E : C, i.e. (E. 16. 5.) CT B : D. 106 THE GEOMETRICAL INVESTIGATION PROP. XVIII. 80. Theorem. If the hypotenuses of two dis similar right-angled triangles be equal, that tri angle which has the less acute angle adjacent to its base, shall have the greater base. Let ABC, DEFbe two dissimilar right-angled triangles, having the hypotenuses AB, DE equal >f ? a to each other, but the angle ABC less than the angle DEF-, the base BC is greater than the base EF. Upon AB, as a diameter, describe the circle AGB ; at the point B, in AB, make the angle AEG (E. 23. 1.) equal to DEF, and join A, G; then because (E. 31. 3.) the angle AGB is a right angle, and AEG equal to DEF, and AB to DE, BG is equal (E. 26. 1.) to EF, but (E. 7. 3.) BC is greater than BG ; wherefore, also, l?Cis greater than OF MAXIMA AND MINIMA. 107 PROP. XIX. 81. Theorem. Of two regular polygons in scribed in a circle, that which has the greater number of sides has the greater perimeter, and is the greater. Let AE and CD be the sides of the regular polygons V and W, inscribed in the circle ABE; and let W have a greater number of sides than V- y the perimeter of W is greater than that of V, and W is greater than V. First, let W have one more side than V. Find (E.I. 3.) the center K of the circle, and join K, A, and K, B, and K, C, and K, D ; also draw (E. 12. 1.) the radius KPQE at right angles to AB or CZ>, which are supposed to be placed 108 THE GEOMETRICAL INVESTIGATION parallel to each other. And, because W has one more side than V, there is one more angle at the center K subtended by the sides of W\ there fore (E. 15. l.Cor. 2.) each of the equal angles at the center subtended by the sides of W, is less than each of the equal angles there subtended by the sides of V ; i. e. the angle CKD is less than the angle AKB\ also, since (E. 3. 3.) KE bisects AB and CD at right angles, it also (E. 4. 1 .) bisects the angles AKB, CKD, and the arches AEB, CED; and, because J^has one more side than V, and that the polygons are equilateral, and the arches cut off by the sides of each (E. 28. 3.) equal, the arch AC is contained as many times in CE 9 as ACE is in the semi -circumference ; join A, C, and insert (E. 1. 4.) successively the chords CF, FG, GE each equal to AC. Through Fand G draw (E. 31. 1.) FH, GI parallel to AB or CD ; and from C, F, G, draw (E. 12. 1.) CL, FM, GN perpendicular to AP, CQ, FH respectively ; also produce FC, and let it meet PA produced in O. Then (E. 16. 1.) the angle LAC is greater than the angle AOC\ and (E. 29. 1.) the angle AOC is equal to the angle MCF\ therefore LAC is greater than MCF; but AC is equal to CF; and (Art. 80.) CM is greater than AL ; wherefore (E. 8. 5.) AC has a greater ratio to AL than FC to CM\ and in the same manner CF may be shewn to have a greater ratio to CM than FG to FN-, and FG to have a greater ratio to FN than OF MAXIMA AND MINIMA. 109 GE to GI-, wherefore* AC, CF, FG, and GE, together, have a greater ratio to AL, CM, FN, and GI, together, than CF, FG, GE, together, have to CM, FN, and GI taken together; and (Note to Art. 78.) AC, CF, FG, and GE, to gether, have a greater ratio to CF, FG, and GE together, than AL, CM, FN, and GI together, have to CM, FN, and GI together; but (E. 28. 3.) the arches AC, CF, FG, and GE are equal to each other, because their chords are equal ; and, therefore, the arch ACE has to the arch CFE the same ratio as the aggregate of the chords AC, CF, FG, GE to the aggregate of the chords CF, FG, GE-, and AL, CM, FN, and GI are together (E. 34. 1.) equal to AP; and CM, FN, and GI are together equal to CO, ; wherefore the arch * The proposition, here assumed as true, has been proved, ia Euclid s manner, by the more ancient Commentators upon his Elements. It may be thus, for the sake of conciseness, demon strated algebraically. Let a : b C~c : d, and c : d CT e :/, &c. j then is a + c -f- e : b + d+ffc + e: d+f. a c For, because a : b CT c : d, and c : dC" e :f, - CT ~ , and C C - CT -; .*. a . dC* c . b and a ./C" e . ft ; ,*. a . c?-f- 2 .fCT c . b-\- d f e.b; add to both c . d-\-e . d-\-c .f+e ./; and a . c?+c . d-\-e . d-\- a.f+c.f+e.fCTb.c+d.c+f.c+b.t+d.e +/.-, i.e. ^ a + c "f~ e . c "4" c i 110 THE GEOMETRICAL INVESTIGATION ACE has to the arch CFE* a greater ratio than AP has to CO, ; and the arch AEB has to the arch CED a greater ratio than AE has to CD, each being in this case doubled ; therefore, also, the arch AEB has to the chord AE a greater ratio, than the arch CED to the chord CD ; and the perimeter of V is the same multiple of AE that the circumference of the circle is of AEB; and the perimeter of W is the same multiple of CD that the circumference is of CED ; wherefore the circumference has a greater ratio to the perimeter of ^than to that of W\ and the perimeter of /^is, therefore, less (E. 8. 5.) than that of W. Again, it is manifest, by reasoning as in Art. 64, that W is equal to half of the rectangle contained by its perimeter and KQ, and that V is equal to half of the rectangle contained by its perimeter and KP, which is less than KQ; hence W is plainly greater than V. Let, now, X be a regular polygon inscribed in the same circle, and having one more side than W\ it may be shewn in the same manner to be greater than W, and to have a greater perimeter ; there fore it is greater, also, and has a greater perimeter, than V \ and thus the proposition may be proved * Whatever be the curve ACE, it has been shewn by BARROW that this property obtains; the greater arch ACE has always a greater ratio to the less arch CE, than tlie ordi- nate AP has to the ordinate C2. OF MAXIMA AND MINIMA. 11! whatever be the number of sides of the regular in scribed polygon which is compared with V. 82. COR. An equilateral triangle is the least of all regular figures inscribed in the same circle, and has the least perimeter. Ill a*/. um;/M ^ ,.; .. -ni isIasM atfe io febi* to Y*f,/ !><(* sff MAXIMA AND MINIMA PART I. SECTION III. PROP. I. 83. Problem. J_N the greater of the two sides, containing the right angle of a scalene right- angled triangle, to find a point from which if two straight lines be drawn, the one to the opposite angle, the other to the hypotenuse, their aggregate shall be a minimum. Let ACB be a scalene right-angled triangle, right-angled at C, and having the side EC greater than CA ; it is required to find a point in BC 9 OF MAXIMA AND MINIMA. 113 from which if two straight lines be drawn, one to the point A, and the other to the hypotenuse AB, their aggregate shall be a minimum. Produce ^Cto />, and make CD equal to AC-, from D draw (E. 12. 1.) DE perpendicular to AB, JP . and let it cut BC in F; join F, A; the aggregate of FE and F*A is a minimum. For, take any other point P in BC; join P, A 9 and draw PQ perpendicular to BA ; then PQ will be the shortest (E. 17. and 19. 1.) of all straight lines drawn from P to BA; also, join D, P, and JO, Q. Then (E. 4. 1.) DF is equal to FA, and DP to PA ; and, therefore, DE is equal to JF and FA together; and DP, PQ, to AP,PQ, but (E. I?, and 19. 1.) DQ is greater than DE; and (E. 20. 1 .) DP and PQ are together greater than DQ; much more, then, are DP, PQ greater than DE ; i. e. AP and PQ are greater than AF and FE ; arid P is any point whatever, but F, H THE GEOMETRICAL INVESTIGATION in 5C; therefore the aggregate of AF, FE is a minimum. 84. DEF. The center of the circle inscribed in, or that of the circle described about, any regular polygon, is called the center of that polygon. Remark. It appears from E. Prop. 13. 4, in which proposition the reasoning is general, that the center of a circle, inscribed in any regular polygon, is also the center of a circle described about the same polygon. PROP. II. 85. Theorem. If from any point, in a regular polygon, straight lines be drawn perpendicular to the several sides, their aggregate shall be equal to the aggregate of the perpendiculars drawn from the center to the sides : And if perpendiculars be drawn to the sides, from any point without the polygon, their aggregate shall be greater than that of the perpendiculars drawn from the center to the sides. Let ABCDEF be any regular polygon ; find (Art. 35.) its center K ; and let P be any point within the polygon, and Q, any point without it ; the aggregate of the perpendiculars drawn from K, to the several sides of the figure, is equal to that of the perpendiculars drawn from P to the sides, and is less than that of the perpendiculars drawn from Q to the sides. OF MAXIMA AND MINIMA. 115 First, the polygon may be divided into as many triangles as it has sides, both by drawing lines from A , and from P, to each of its angles ; and it is manifest (E.37. 1.) that each triangle, having its vertex at K, is equal to a right-angled triangle which has one of the equal sides of the figure for a base, and its altitude equal to the perpendicular drawn from K to that side ; and, if these altitudes be placed in the same straight line, it follows from E. 37. 1. that the aggregate of these triangles is equal to a right-angled triangle, having one of the equal sides of the polygon for its base, and the aggregate of the altitudes for its altitude : In the same manner it may be shewn that the aggre gate of the triangles, into which the figure is divided by straight lines drawn from P, is equal to a right-angled triangle having a side of the poly gon for its base, and the aggregate of the per pendiculars drawn from P for its altitude; but these two right angled triangles are equal ; for each of them is equal to the whole polygon ; also H3 116 THE GEOMETRICAL INVESTIGATION their bases are equal; wherefore (Art. 19.) their altitudes are equal ; i. e. the aggregate of the per* pendiculars drawn from K to the sides, is equal to that of the perpendiculars drawn from P to the sides. But, if Q be a point without the polygon, let .straight lines be drawn from Q to each of the MEN angles of the figure ; then, it is manifest that the aggregate of the triangles thus formed, each having its vertex at Q and one of the sides of the figure for its base, is greater than the polygon, and, therefore, greater than the aggregate of the tri angles having their summits at K\ and it may be shewn, by proceeding as in the former case, that, therefore, the aggregate of the perpendiculars drawn from Q to the sides, is greater than that of the perpendiculars drawn from K to the sides. OF MAXIMA AND MINIMA. 117 PROP. III. 86. Theorem. The aggregate of the straight lines, drawn from the center of a regular polygon to each of its angles, is less than that of the straight lines drawn from any other point what ever to each of the angles. Let ABC be any regular polygon of which K is the center, and P any other point ; the aggregate F " of the straight lines, drawn from A" to the angles, is less than that of the straight lines drawn from P to the angles. Join K, A, and K, B, and K, C, &c. ; and also P, A, and P, B and P, C, &c. ; through A, B, C, &c. draw (E. 11. l.) FD, DE, EF y &c. perpendi cular to KA, KB, KC, &c. respectively. Since K is the center of a circle described about ABC, the figure * DEF is a regular polygon of the same * The construction and proof used by Euclid in Prop. 12. B. 4, is applicable to the description of any regular polygon about a circle, in which a similar polygon has been inscribed. 118 THE GEOMETRICAL INVESTIGATION number of sides as ABC, and of which K is also the center; from P draw (E. 12. 1.) PG, PH, PI, &c. perpendicular to DE, EF, FD, &c. respec tively; then (E. 17. 1.) the angles PEG, PCH, PAI are each less than a right angle, and, therefore, (E. 19. 1.) PA, PB, PC, &c. are to- gether greater than PI, PG, and PH taken together ; but (Art. 85.) the aggregate of KA, KB, KC is not greater than that of PI, PG, PH whether the point P be within, or without, the polygon DEF y wherefore the aggregate of PA, PB, PC, &c. is greater than that of KA, KB, KC, &c. PROP. IV. 87- Problem. In a given quadrilateral recti lineal figure, to find a point, from which if straight lines be drawn to each of the angles of the figure, their aggregate shall be a minimum. Let ABC be the given quadrilateral figure ; it is required to find a point within it, from which if straight lines be drawn to A, B } C, and Z), their aggregate shall be a minimum. OF MAXIMA AND MINIMA. 119 Join A, Cand B, Z>; and let AC cut ED in the point E ; EA, EB, EC, and ED are together less than the aggregate of four straight lines drawn from any other point within the figure to A, B^ C, and D. For, let P be any other point, and join P, A, and P, B, and P, C, and P, Z>. Then (E. 20. 1.) .4P, and PC are together greater than AC, and jBP, and P/> are together greater than BD ; wherefore P^, PJ5, PC, and PD are together greater than AC and #D ; i. e. than EA, EB, EC, and ED. PROP. V. 88. Problem. The vertical angle of a triangle being less than the exterior angle of an equila teral triangle, to find within it a point, from which if straight lines be drawn to the three angles of the figure, they shall make equal angles with each other. Let ABC be a triangle having the vertical angle JS C JE F A less than the exterior angle EDZ of the equila- 120 THE GEOMETRICAL INVESTIGATION teral triangle DEF; it is required to find a point in ABC, from which if straight lines be drawn to A y By and C, they shall make equal angles with each other. Produce EF both ways to X and Y\ upon AB and AC describe (E. 33. 3.) the circular segments AGB and AGC, capable of containing each an angle equal to DEX or DFY, and let them cut each other in the point G ; join G, A and G, B, and Gy C ; the angles AGB } BGC, and CGA are equal to each other. For, since (E. Book 5. 1. Cor.) the angle DEF is equal to the angle DFE, and that DEX, DEF are (E. 13. 1.) equal to two right angles, to which also the angles DFE, DFY are equal, if from these equals be taken the equals DEF, and DFEy there will remain DEX equal to DFY-, wherefore the angle AGB is equal to the angle AGC. Again, (E. 32. 1. Cor. 2.) the three angles DEXy DFYy and EDZ are equal to four right angles, to which also (E. 15. 1. Cor. 2.) the angles at G are equal ; and AGB, and AGC were made equal to DEX, and DFY, therefore BGC is equal to EDZ; and EDZ may be shewn to be equal to DEX, in the same manner that DEX was shewn to be equal to DFY 9 wherefore the angles AGBy BGC y and CGA are equal to each other. OF MAXIMA AND MINIMA. 121 PROP. VI. 89. Problem. * In a given triangle having each of its angles less than the exterior angle of an equilateral triangle, to find a point from which if straight lines be drawn to the three angles, their aggregate shall be a minimum. Let ABC be the given triangle, having each of its angles less than the exterior angle of an equilateral triangle ; it is required to find a point within ABC, from which if straight lines be drawn to A, B, and C, their aggregate shall be a minimum. Find (Art. 88.) the point D, at which the straight >r This Problem was, in substance, proposed by FERMAT to TORRICELLI; the method of solution here given is that of VIVIANI. 122 THE GEOMETRICAL INVESTIGATION lines DA, DB, and DC make equal angles with each other; let P be any other point in ABC, and join P, A and P, B and P, C ; then are DA, DB, and DC together less than PA, PB, and PC together. From the center D, at the distance DB, the greatest of the straight lines, DA, DB and DC, describe the circle DEF, and let it meet DA and DC, produced, in E and F; join B, jE and E, F, and JP, By and P, -E, and P, P; then (E. 4. 1.) EB, BF, and .FZ? are equal to each other, and D (Art. 84.) is the center of the equilateral triangle EBF; wherefore (Art. 86.) DB, DE, and DF are together less than PB, PE, and PF together; but (E. 20. 1.) PA and AE are, together, greater than PE ; and PC together with CFis greater than PF; therefore DB, DE and DF are, together, less than PB, PA, AE, PC and CF together; take away the common parts AE and CF, and the remainder DA, DB, DC is less than the re mainder PA, PB, PC. PROP. VII. 90. Problem. Through a given point within a circle, which is not the center, to draw the least chord. Let A be a given point within the circle PJBQC; it is required to draw through A the least chord. Find (E, 3. 1.) the center K of the circle, and OF MAXIMA AND MINIMA. 123 join A, K; through A draw (E. 11. 1.) the chord EAC at right angles to AK\ EC is the least chord which passes through A. For, let PQ be any other chord passing through A ; from K draw (E. 12. l.) KM at right angles to PQ : then, because KMA is a right angle, KAM (E. 17. 1.) is l ess tnan a ^ght angle ; where fore (E. 19. 1.) KA is greater than MK, and (E. 15. 3.) PQ is greater than BC. PROP. VIII. 9f Problem. To divide a given circular arch into two such parts that the aggregate of their chords shall be a maximum. It is manifest from the demonstration of Art. 69, that if the given arch be (E. 30. 3.) bisected, the aggregate of the chords of its two parts will be a maximum. 124 THE GEOMETRICAL INVESTIGATION PROP. IX. 92. Problem. Through either of the points of intersection of two given circles, which cut each other, to draw the greatest of all straight lines passing through that point, and terminated both ways by the two circumferences. Let APE, ADO. be two given circles, which cut each other, and let A be one of the points of intersection ; it is required to draw through A the greatest straight line, which is terminated both ways by the circumferences ABP, ADQ. Find (E. 1. 3.) the centers Cand K of the two circles, and join C, K\ through A draw (E. 31. 1.) the straight line BAD parallel to CK ; BAD is the greatest of all straight lines passing through A^ and terminated both ways by the circumferences of ABPandADQ. For, let PAQ, be any other straight line so ter- OF MAXIMA AND MINIMA. 125 urinated, and passing through A ; from the points C, K 9 draw (E. 12. 1.) the straight lines CE and KF perpendicular to ED, and CM and KN perpendicular to PQ, and through K draw KL parallel to PQ; therefore (E. 28. 1.) LKNM, and CKFE are parallelograms, and (E. 34. 1.) KL is equal to NM 9 and CK to EF, but because the angle CLK is a right angle, (E. 17. and 19. 1.) CK is greater than KL ; wherefore, also, EF is greater than MN; but (E. 3. 3.) BA is the double of EA, and ^4Z) the double of AF\ wherefore BD is the double of EF; in the same manner it may be shewn that PQ is the double of MN; and EF has been proved to be greater than MN-, there fore, also, BD is greater than PQ ; and is a maximum, PROP. X. 93. Problem. If two semi-circles lie on con trary sides of the same straight line, and the radius of the greater be the diameter of the less, to draw the greatest straight line perpendicular to the diameter, and terminated both ways by the two curves. Let ABD, AECbe two semi-circles lying on contrary sides of the straight line AD, and let the radius of ABD, the greater, be the diameter of AEC, the less ; it is required to draw the greatest straight line perpendicular to AC, and terminated by the curves AB, AEC. 126 THE GEOMETRICAL INVESTIGATION Find (E. 3. 1.) K the center of the circle AEC, and draw (E. 11. 1.) KE at right angles to AC-, produce CA to F, and make AF equal to AC-, join F^ E\ draw EG perpendicular to FE, meet ing AC in G, and through G draw HGI perpen dicular to AC ; HGI is the greatest of all straight lines which can be drawn perpendicular to AC, and which are terminated by AB and AEC. For, let PQ be any other straight line so drawn ; draw (E. 17. 3.) the two straight lines XY and ZW touching the circles ABD, AEC, in the points H and / respectively ; produce PQ, to meet XY in the point R, and ZW in the point S\ and join C, H, and K, /; then, (E. 8. 6.) FK : KE n KJ^ i KG and, dividendo, AF : KE :: GC : KG (E. 17. 5.) i. e. HC : KI :: GC : KG (by the construct.) and HC : GC KI : KG (E. 16. 5.) OF MAXIMA AND MINJMA. 127 Therefore the two triangles HOC, KGI, which have the right angles at G equal, have also the sides about two other angles proportionals ; and, therefore, (E. 7. 6.) the angle CHI is equal to the angle HIK ; to CHI add the right angle CHY, and to HIK add the right angle CIZ, and the whole angle YHI is equal to the whole angle HIZ y and they are alternate angles; wherefore (E. 27. 1.) XFis parallel to ZW\ and (E. 28. 1.) HI is parallel to RS\ therefore (E. 34. 1.) HI is equal to RS-, but RS is greater than PQ; where fore, also, /// is greater than PQ. Otherwise : After having found the center K, of the circle AEC, trisect (E. 1O. 6.) the straight line KC, and through 6r, the point of trisection nearest to K, draw HGI perpendicular to AC\ HGI is the maximum required. For, join C, H and K, /; and since, by the construction, and by the hypothesis, GC is the double of KG, and CH is the double of KI 9 there fore, HC : KI :: GC : KG. And the remainder of the demonstration is then the same as in the former method of proof. 94. COR. 1. Of whatever kind the two curves are, the maximum described in the above propo sition may always be found, if two tangents can be drawn to the curves, which are parallel to each other. * - THE GEOMETRICAL INVESTIGATION 95. COR. 2. Hence, if the two curves be para bolas, and the axis be divided in the ratio of the parameters, the aggregate of the two semi-ordi- nates, so drawn, will be greatest when it meets the axis in that point of division. PROP. XL 96. Theorem. If there be three magnitudes, of which the first is greater than the second, the first shall have to the second, a greater ratio than the first together with the third has to the second together with the third. Let A, B, C be three magnitudes, of which A is greater than B ; A has a greater ratio to E than A together with C has to B together with C. For, C has a greater ratio (E. 8. 5.) to B, than it has to A : therefore (Note to Art. 36.) C together with B has a greater ratio to B, than C together with A has to A; and (Note to Art. 78.) C to gether with B has a greater ratio to C and A, together, than B has to A ; or A has to E a greater ratio than A together with C has to B together with C. 97. COR. If D be a magnitude greater than C, A has a greater ratio to B than A and C, together, have to E and D together. For, the aggregate of A and Chas (E. 8. 5.) a greater ratio to that of B and C, than to that of B and D ; much more, then, is the ratio of A to B OF MAXIMA AND MINIMA. IftT 129 greater than that of the aggregate of A and C to the aggregate of B and D. PROP. XII. Hi 98. Theorem. If a straight line be drawn from the vertex of a triangle cutting the base, it shall have a greater ratio to either of the segments which it cuts off, than the side adjacent to the other seg ment has to the base. Let ABC be a triangle, and AD a straight line drawn from A, cutting the base AC in D. AD has a greater ratio to DC than AB has to BC. From AB cut off AE equal to AD ; then (E. 20. 1.) AD together with BD is greater than AB ; take AD from AD, BD, and AE, which is equal to AD, from AB, and the remainder BD is greater than BE ; wherefore (Art. 96.) AD has a greater ratio to DC than AB has to BC. 130 THE GEOMETRICAL INVESTIGATION PROP. XIII. 99. Problem. Of all equal triangles standing upon the same base, to find that in which the ratio of the greater side to the less is a maximum. Let AP E he any triangle standing upon the given base AE ; of all the triangles which can be described upon AE as a base, equal to APB, it is required to determine that in wbich the ratio of the greater side to the less is a maximum. Through P draw (E. 31. 1.) the, straight line XY parallel to AE, and (E. 39. 1.) it will be the locus of the vertices of all the triangles standing upon the base AE^ and on the same side of it as APE, which are equal to APE ; bisect AE in D (E. 10. 1.) and through D draw (E. 11. 1.) DK perpendicular to AE, and let it meet XY in K ; join K, A ; and from the center K, at the distance KA, describe the semi-circle LAF, and let it cut XY in L and F, join A, L and E, L, and A, F OF MAXIMA AND MINIMA. 131 and B, F; in either of the two equal triangles ALB, AFB, the ratio of the greater side to the less exceeds the ratio of AP to PB. For draw (E. 12. 1.) FG perpendicular to XY\ then FG (E. 16. 3. Cor.) touches the circle LAP in F, and (E. 29. 1.) cuts AB produced at right angles ; join P, G; PG is greater (E. 17. and 19. 1.) than GF; also the angle PAB is greater than the angle BPG-, for the two angles cannot be equal ; otherwise the two triangles PGA, PGB would be similar, and (E. 4. and l6. 6.) the rectangle AG, GB would be equal to the square of GP ; but it also (E. 36. 3.) is equal to the square of the less line GF; which is absurd. Neither can the angle PAB be kss than the angle BPG ; for then if a straight line be drawn (E. 23. 1.) from P, making with BP an angle equal to the angle PAB ; it may in the same manner be shewn, that a less rectangle than AG, GB, is equal to a greater square than that of GF, which is absurd ; wherefore the angle PAB is greater than BPG, and if PH be drawn, making the angle BPH equal to PAB, the point H is beyond the point G. The two triangles AGF, BGF are similar, as are also the two triangles A HP, GHP; wherefore (E. 4. 6.) AF : FB :: AG : FG. Therefore (E. 22. 6.) the square of AF : the square of FB :: the square of AG : the square of FG :: AG ; J5G, because AG, FG, and BG are proportionals; therefore (E. 11. 5.) the square of 12 132 THE GEOMETRICAL INVESTIGATION AF : the square of FB :: AG : GB>, and, in the same manner, it may be shewn that the square of AP : the square of PB:. AH: BH-, but (Art. 96.) the ratio of AG to GB is greater than the ratio of Alt to BII-, wherefore (E. 13. 5.) the square of AF has a greater ratio to the square of FB than the square of AP has to the square of PB ; but if Q be the side of a square, which is a fourth proportional to the squares of AF, FB, and AP, Q is less than PB ; and (E. 22. 6.) AF : FB :: AP : Q ; wherefore the ratio of AF to FB is greater (E. 8. 5.) than that of AP to PB. PROP. XIV. 100. Problem. Through any of the angular points of a given rhombus, to draw the shortest line terminated by the two sides, produced, which contain the opposite angle. Let A BCD be the given rhombus, and C one A P of its angular points; it is required to draw OF MAXIMA AND MINIMA. 133 through C the least line, which is terminated by AB and AD produced. Join A, C, and through C draw (E. 11. 1.) the straight line ECF perpendicular to AC ; ECF is the shortest line which can be drawn through C > and which is terminated by AB, and AD produced. For, let PCQ be any other straight line drawn through C, and terminated, in P and Q, by AB and AD produced. Draw EG perpendicular to AE, and let it meet AC produced in G ; join G, F; and draw GM perpendicular to PQ. Then (E. Def. 32. and Prop. 8. 1.) the angle BAG is equal to the angle DAC, and the angles ACE, ACFj are right angles, and AC is common to the triangles ACE, ACF; therefore (E. 26. 1.) the angle AEC is equal to the angle AFC, and EC is equal to CF- 9 and CG is common to the two right-angled triangles ECG, FCG ; therefore (E. 4. 1 .) GF is equal to GE, and the angle GFE to the angle GEF; but the angle AFE was before shewn to be equal to the angle AEF; wherefore AFG is equal to AEG, and is a right angle. From the center G, at the distance GE, describe the circle NER, which will touch AE in E, and AF in F; let it cut PQ in ^Varid S ; then, because the angle at M is a right angle, GC (E. 17. and 19. 1.) is greater than GJ/; wherefore (E. 15. 3.) EF is less than NS ; much more, then, is EF less than PQ. 101. COR. Hence it is manifest that the base of an isosceles triangle is less than any other 134 THE GEOMETRICAL INVESTIGATION straight line, drawn from any point, in either of the two equal sides, through the bisection of the base, so as to meet the other side produced. PROP. XV. 102. Theorem. If the base of an isosceles triangle be divided into two unequal parts, the base is the least of all straight lines which can be drawn through the point of section, from the side that is nearest to it, so as to meet the other side produced. Let ABC be the given isosceles triangle ; let the base EC be divided into two unequal parts in OF MAXIMA AND MINIMA. 135 the point Z), and let PQ be a straight line drawn through D, from any point P, in the side AC, which is nearest to D, meeting AB produced in Q : PQ is greater than BC. For, since the angle APQ is greater (E. l6. 1.) than the angle C, that is (E. 5. 1. and hypothesis) than the angle ABC, therefore(E. 13. l.)the angle QBD is greater than the angle DPC: Make, there fore, (E. 23. 1.) at the point JB, in CB, the angle CBG equal to the angle DPC. And, first, if the angle A be not less than a right angle, the angle DPC, and therefore, also, its equal DBG, is (E. l6. 1.) an obtuse angle : so that (E. 32. 1. and 19. 1.) GD is greater than BD, and DC than DP : and, by the hypothesis, BD is greater than DC. But (E. 15. 1. and con struction) the triangles GBD, DPC, are equi angular: wherefore (E. 4. 6.) GD : DB :: DC : DP. And GD has been shewn to be greater than DC. Wherefore (E. 25. 5.) GD together with DP is greater than BD together with DC-, that is, GP is greater than BC: much more, then, is PQ greater than BC. But, secondly, let the angle A be less than a right angle. From D, as a center, at the distance DC describe a circle, and let it cut AC in E; join E, D ; and produce ED to meet AB produced in F. Then (.17. 1. and hypothesis) since the 136 THE GEOMETRICAL INVESTIGATION angle ABC is an acute angle, the angle CBF is (E. 13. 1.) obtuse : whence (E. 32. 1. and 19. 1.) DFis greater than DB ; but, by the construction, JED is equal to DC: wherefore, the whole EF is greater than the whole BC : and. it may, in the same manner be shewn, that a straight line drawn through D, from any point between E and A, to meet AB produced, is greater than EF, and consequently greater than BC, But let, now, PQ, be drawn through D, from any point P, between E and C, and let it meet AB produced in Q. Then, the same construction being made as in the former case, the angle GBD is equal to DPC y and the angle BGD to DCP; but (E. 16. 1.) the angle DPC is greater than DEC, that is (E. 5. 1.) than DCP; therefore, the angle GBD is greater than DGB, and (E. 19. 1.) the side GD is greater than BD. It is manifest, then, that PQ may be shewn to be greater than BC, exactly in the same manner, as PQ was shewn to be greater than BC in the former case. SCHOLIUM. If a straight line be drawn through a given point, which divides the base of an isosceles triangle unequally, from the side which is thejiirther from that point, to meet the side which is the nearer to it, OF MAXIMA AND MINIMA. 137 produced, the line so drawn is not necessarily greater than the base. For, let ABC be the given isosceles triangle, and I) a given point in the base EC, nearer to AC than to AE. And, first, if the angle A be not less than a right angle, it is manifest (E. l6. 1. 32. 1. and 19. 1.) that EC is greater than any straight line that can be drawn from C to AE: If, therefore, from any point R, in AE, as a center, at a distance equal to EC, a circle be described, it will cut ^Cproduced, in some point S: let R, D and S be supposed to be in the same straight line, and let R, S be joined. Then, it is mani fest, that of all straight lines drawn from R to 138 THE GEOMETRICAL INVESTIGATION meet AC produced, those which cut EC between D and C will be less than RS, that is, than J5C; and that those which cut EC between B and D, will be greater than EC. Again, let the angle A be less than a right angle. From C draw CN at right angles to AE. Then it may be shewn, as before, that EC is greater than any straight line, drawn from C to meet AE be tween N and E ; and, if any point R be assumed between JVand B, a straight line RS may be drawn, equal to EC and cutting EC in some point Z), nearer to AC than to AB, as before : so that any straight line, drawn through Z), to meet AC pro duced, from any point between A and A 7 , is greater than EC-, and of any straight lines, drawn from R, to meet AC produced, cutting BC> those which cut EC between D and C are less than BC\ whilst those which cut it between E and D are greater than EC. It may be further shewn, by the same mode of reasoning, that, in all cases, and wherever the point D is assumed, the straight line drawn through D perpendicular to AB, and terminated, on the other side, by AC produced, is less than any straight line drawn through Z), from any point in AB, between the intersection of that perpendicular and the vertex A, to meet AC produced. Again, whatever be the species of the isosceles triangle ABC, and wherever the point Z), nearer to AC, be assumed, if through Z), the straight line OF MAXIMA AND MINIMA. 139 HI be drawn (Art. 2.) so as to be bisected in Z>, then is HI greater than BC. For, draw HM and AK perpendicular to EC ; then (E. 5. 1. 26. 1. and hypothesis) BK is equal to the half of BC, and the angle BAK to CAK. Also, if DL be drawn paraHel to CA, cutting AK in X 9 then (Art. 2.) AL is equal to LH. But (construction, and E. 29. 1.) the angle LAX is equal to LXA ; wherefore (E. 6. 1.) LX is equal to Z^4; and it is, therefore, equal to LH. Also, (construction, and E. 29. 1. and E. 6. 1.) LB is equal to LD : from these equals take away the equal parts LH and LX\ and the remainder HB is equal to the remainder XD. Whence (E. 26. 1.) the two right-angled triangles HMB 9 XKD are equal ; so that BM is equal to KD ; add to each of these equals MK, and B K is equal to MD ; but (E. 17. 1. and 19. 1.) HD is greater than MD; therefore HD is greater than BK, which is equal 140 THE GEOMETRICAL INVESTIGATION to MD\ and HI the double of HD, is greater than BC> the double of BK. PROP. XVI. 103. Problem. To draw the shortest tangent to a given circular arch, which shall be terminated by the semi-diameters, produced, that pass through the extremities of the arch. Let LCM be a given circular arch, and KX 9 OF MAXIMA AND MINIMA. 141 KY, the two produced semi -diameters, which pass through its extremities L and M: it is required to draw the shortest tangent to LCM, which shall be terminated by KX and KY. Bisect the arch LCM (E. 30. 3.) in the point C; and through C draw (E. 17- 3.) the tangent FCG: FCG is the least straight line which can touch the arch LCM, and be terminated by KX and KY. For, let PQ be any other tangent to the arch LCM, which is terminated by KX and KY\ and join K, C: then (E. 18. 3,} the angles KCF, KCG are right angles ; the angle CKF is equal (con struction, and E.27.3.) to CKG-, and KC is common to the two triangles KCF, KCG; where fore (E. 26. l.) the side KF is equal to KG-, so that the triangle KFG is isosceles. It is manifest, therefore, from Art. 102, that PQ is greater than FG : and, in the same manner, it may be shewn, that any other tangent is greater than FG. 104. COR. 1. Hence, if it be required to divide a given circular arch into two parts, so that the aggregate of the tangents of the parts may be a minimum, it is evident that the given arch must be bisected. 105. COR. 2. Of all triangles which have the same vertical angle, or equal vertical angles, and the perpendiculars, let fall from the vertex of the 142 THE GEOMETRICAL INVESTIGATION base, also equal, that which is isosceles has the least base. PROP. XVII. 106. Problem. To draw the shortest tangent to a given circle, which shall be terminated by two given parallel straight lines, that are situated on contrary sides of the center. Let ACED be the given circle ; K, its center, and WX, YZ two given parallel straight lines, on OF MAXIMA AND MINIMA. 143 contrary sides of the center K : it is required to draw the shortest tangent to ACBD, that shall be terminated by /^.Yand YZ. Through the center K draw (E. 31. 1.) the diameter AKE parallel to either of the two parallels WX, JTZ; and through B draw (E. 17. 3.) the tangent EBF: EBF is the shortest of all tangents to the given circle, that are terminated by WX and YZ, and that are situated on the same side of the center K. For, let PQ be any other tangent on that side of the center, terminated by WX and YZ\ and through P draw PG parallel to EF, then is EG a parallelogram, and (E. 34. 1.) PG is equal to EF: but (E. 18. 3. and 29. 1.) the angle PGQ is a right angle; wherefore (E. 17. 1. and 19. 1.) PQ is greater than PG, or than EF. And in the same manner may a tangent to the circle, at the other extremity A, of the diameter AB, which (E. 34. 1.) is equal to EF, be shewn to be less than any other tangent on that side of the center. PROP. XVIII. 107. Problem. To draw the shortest tangent to a given circle, which shall be terminated by two given straight lines that meet one another, and that are equally distant from the center, so as that the tangent and the point of intersection of the given lines, shall be on contrary sides of the center 144 THE GEOMETRICAL INVESTIGATION Let the two given straight lines AZ and which meet in A, be equally distant from the center 7i, of the given circle ELM. It is required to draw the shortest tangent to the circle, which shall be terminated by AZ and AW, and which shall be on the contrary side of K, that A is. Join A and the center K*\ produce AK to meet the circumference in C; and through C draw (E. 17. 3.) the tangent HCI HCI is the least tangent, which was to be drawn. For, let RS be any other tangent to the circle, on the same side of K that /// is, and terminated, also, by AZ and AW. From K draw (E. 12. 1.) KB and KD perpendicular to AZ and AlV, re spectively: then, because (E. Def. 4. B. 3.) KB and KD are equal, and that the angles at B and D are right angles, and that AK is the common hypotenuse of the two right-angled triangles ABK, ADK> therefore (E. 47. 1.) AB is equal to AD, and (E. 8.1.) the angle BAK is equal to DAK\ wherefore, since the two right-angled triangles ACH, ACI, are (E. 18. 3.) right-angled at C, the angle ARC is equal (E.32. 1.) to AIC\ so that the triangle A HI is isosceles. It is manifest, there fore, from Art. 1O2, that RS is greater than HI: and, in the same manner, may any other tangent on the same side of K, be shewn to be greater than ///. * See the figure in p. 140. ot MAXIMA AND MINIMA. 143 PROP. XIX. 108. Problem. To describe a circle which shall touch a given straight line, and pass through two given points, both on the same side of it, and in the same plane with it. Let CD be a given straight line, and A, B two given points without it, both on the same side of CD; it is required to draw a circle through A and B, which shall touch CD. ^ ^ Join A, B\ and first/ let AB be parallel to CD. Bisect (E. to. 1.) AB in L, through L C JP If H draw (E. 11. 1.) LH perpendicular to AB or CD; join A, H ; at the point A, in HA, make (E. 23. 1.) the angle HAK equal to the angle AHK, and join KB-, then (E. 6. 1.) KH is equal to KA, and (E. 4. 1.) KA is equal to KB ; from the center K, at the distance KA, or KB, or /TH, describe the circle AHB ; it shall pass through the three points A, H, and B, and (E. 16. 3. Cor.) shall touch CD in H. But if AB be not parallel to CD, let ^#, pro* 146 THE GEOMETRICAL INVESTIGATION duced, meet CD in the point D. Upon AB as a diameter describe the circle ABE, and from D draw (E. 17. 3.) the straight line DE touching it in E; from DC cut off DH (E. 3. 1.) equal to DE, and describe (E. 5. 4.) the circle AHB pass ing through the three points A, H, and B. The circle AHB, which passes through A and B, touches CD in H. For (E. 36. 3.) the rectangle contained by AD and DB is equal to the square of DE, and, therefore, is equal also to the square of DH, because DH was made equal to DE -, wherefore (E. 37. 3.) the circle AHB touches CD in H. 109. COR. AB subtends a greater angle at the point H, in the straight line CD, than at any other point whatever in CD. For, let P be any other point in CD ; P is with out the circle AHB ; join A, P, and B, P ; let BP cut the circle in Q; also join A, Q. The angle AHB is equal (E. 21. 3.) to the angle AQB ; but the exterior angle AQB is greater (E. 16. 1.) than the interior opposite angle APB ; wherefore, also, AHB is greater than APB. PROP. XX. 110. Problem. To find a point in the circum ference of a given circle, at which any given straight line drawn from the center, but less than a radius of the circle, shall subtend the greatest angle. OF MAXIMA AND MINIMA. 147 Let CPD be the given circle, and AB a given JP, D or* straight line, drawn from the center A, less than a radius of the circle ; it is required to find the point, in the circumference, at which AB subtends the greatest angle. From the point B draw BC perpendicular (E. 11. 1.) to AB, and let it meet the circum ference in C; join A, C ; the angle ACB is greater than any other angle subtended by AB, at any other point in the circumference. For, take any other point P in the circum ference, and join P, A and P, B ; and upon AC as a diameter, describe the circle ACB, * which * If two circles meet each other, and the point which is common to both circumferences lie in the same straight line with the centers of both, the circles shall be in contact with each other. This is evident, by drawing from the common point a straight line perpendicular to the line joining the centers, which (E. 16. 3. Cor.) will touch them both; and the converse of E. 31. 3. is readily proved by a reductio ad absurdum. K2 148 THE GEOMETRICAL INVESTIGATION will pass through B, and touch the circle CDP in C; let BP cut the circle CAB in Q; and join A, Q ; then the angle ACE is equal (E. 21. 3.) to the angle AQ,B, and ^/Q# is greater (E. 16. 1.) than APB ; wherefore ACE is greater than APB. PROP. XXI. 111. Problem. Two points being given, the one within, the other without, a given circle, to find a point in the circumference, to which if two straight lines be drawn from the given points, the one falling upon the concave, the other upon the convex circumference, the angle contained by them shatfl be a minimum. Let DPE be the given circle, and A, B, the two given points; it is required to find a point in the circumference DPC to which, if two straight lines be drawn from A and J5, the angle contained by them shall be a minimum. From the point B draw (E. 17. 3.) the two straight lines BD, BE, touching the circle in. D OF MAXIMA AND MINIMA. 149 and E ; join A, D and A, E ; and let the angle DBA be * greater than the angle ABE^ then is the angle AEB less than any other angle, contained by two straight lines drawn from A and B, to any other point of the circumference, the one falling upon the concave, the other upon the convex circumference. For, join D, E; and since BD and BE touch the circle DPE, they are (E. 36. 3.) equal to each other, and the angle BDE is equal (E. 5. 1.) to the angle BED ; and because DB is equal to BE, and AB is common to the two triangles DBA, EBA, and that the angle DBA is greater than the angle EBA, DA is greater (E. 24. 1.) than AE ; wherefore the angle AED is greater (E. 18. 1.) than the angle ADE ; from BDE take ADE, and from BED, which is equal to BDE, take the greater angle AED, and the remainder AEB is less than the remainder ADB. Again, because BD and BE touch the circle DPE, all straight lines drawn from B to the convex circumference must fall within BD and BE, in the arch DPE, take any point P on the same side of AB with Z>, and any point Q on the same side of AB with E, and join A, P and B, P, and A, Q and B, Q ; then (E. 21. 1.) ADB is less than APB-, and AEB has been shewn to be less than ADB ; much more, * The two points A and B are here supposed not to be in the same straight ]ine with the center of the circle ; if they be, the two angles ADB, AEB will be equal, and either of them will be less than any other angle, according to the speci fied conditions. 150 THE GEOMETRICAL INVESTIGATION then, is AEB less than APB-, and (E. 21. 1.) AEB is also less than AQ.B ; wherefore AEB is a minimum. PROP. XXII. (112.) Problem. To describe a circle which shall touch a given circle, and pass through two given points both without the given circle; the straight line joining the two given points being sup posed not to pass through the center of the circle. Let DFG be the given circle, and A, B, the two given points without it; it is required to describe a circle which shall pass through A and By and touch the circle DFG. Join A, B-, and find (E. 1. 3.) the center Cof ; describe (E. 5. 4.) a circle AFGB which OF MAXIMA AND MINIMA. 151 shall pass through the three points A 9 C, and B ; join F, G ; either FG is parallel to AB, or it is not; if it be parallel, through C draw DCH per pendicular (E. 12. 1.) to AB, and produce it to K- y join A, D and A, E ; at the point A, in EA, make (E. 23. 1.) the angle EAK equal to the angle KEA ; and at the point A, in DA, make the angle DAL equal to the angle LDA ; then if a circle be described from the center K, at the distance KA 9 it shall pass through the two points A and B, and touch the given circle externally in E, and if another circle be described from the center L, at the distance LA, it shall pass through A and B, and touch the given circle internally in D. For, join K, B and L, B ; ^B is bisected (E. 3. 3.) in //, by DH 9 and the angles at H are right angles; wherefore (E. 4. 1.) KB is equal to KA 9 and L to JL4, and (E. 6. 1.) KE is equal to KA 9 and LD to Z>^ ; therefore the circle described from the center K, at the distance KA, will pass through the three points A, E, and B ; and that described from the center L, at the distance LA, will pass through A 9 D, and B ; and the circles will be in contact, because the straight line join ing their centers passes through the points in which they meet. (Note to Art. 102.) But if FG be not parallel to AB, let them be produced so as to meet in the point M ; from M draw (E. 17. 1.) the two straight lines MD and ME to touch the given circle DFG in D and 152 THE GEOMETRICAL INVESTIGATION JE-, through A, E, and B describe (E. 5. 4.) a circle, and it shall touch the given circle in E ; also describe another circle through the three points A, D, B, and it shall touch the given circle in D. For, (E. 36. Book 3. Cor.) the rectangle BM, MA is equal to the rectangle GM, MF; and (E. 36. 3.) the rectangle GM, MF is equal to the square of MD, or ME ; wherefore,, also, the rect angle BM, MA is equal to the square of MD, or ME-, and (E. 3J\ 3.) the circle passing through A, B and E is touched by ME, which also touches DEG in the same point E\ therefore the circle passing through A, E and B touches the given circle in E ; and, in the same manner, it may be shewn, that the circle described through A, D and B touches the given circle in D. OF MAXIMA AND MINIMA. 153 PROP. XXIII. 113. Problem. *If a given straight line lie wholly without a given circle, and do not when produced, cut the circle, to find the two points in the circumference at which the given line will subtend the greatest and least angles. Let DPE be the given circle, and AB the given straight line without it ; it is required to find the two points in the circumference DPE, at which AB shall subtend the greatest and least angles. Describe (Art. 105.) the circle AEB passing through A and B, and touching DPE externally in E, and also the circle ADB, touching DPE in ternally in D y AB subtends a greater angle at E, and a less angle at D> than at any other points in the circumference of * This is the 34-th Proposition of the 6th Book of PAPPUS j but he has given no construction for describing a circle to pass through two given points, and also to touch another circle. 154 THE GEOMETRICAL INVESTIGATION For, join A, D and B, D, and A, E and B, E ; take any other point P such, that A y P and B, P, being joined, AP falls upon the concave circum ference ; and any other point Q such, that A, Q and Q, 5, being joined, AQ falls upon the convex cir cumference ; produce AP to meet the arch DB in jR, and let AQ cut the arch AEB in S ; and join R 9 B, and Q, B; the angle AEB is equal (E. 21. 3.) to the angle A SB ; but (E. 16. 1.) ASB is greater than AQB ; wherefore AEB is greater than AQB ; and AEB is also (E. 21. 1.) greater than APR. Again, ARE is less (E. l6. 1.) than APR; but (E. 21.3.) ADR is equal to ARB; wherefore ADR is less than APR ; and AEB is the greatest, and ADR the least, of all angles subtended by A B at the circumference of the given circle DPE. PROP. XXIV. 114. Problem. To divide a given finite straight line into two parts, such that the rectangle, con tained by the whole line and one of the parts, shall most exceed the square of that part. Let AR be the given finite straight line ; it is required to divide it into two such parts that the rectangle, contained by AR and one of the parts, shall most exceed the square of that part. OF MAXIMA AND MINIMA. 155 Bisect AB (E. 10. 1.) in C, the excess of the rectangle AB, BC above the square of BC is a maximum. For. take any other point P in AB. The rect angle AB, BC is equal (E. 3. 2.) to the square of CB, together with that of AC; and the rectangle AB, BP is equal to the square of PB, together with the rectangle AP, PB; but (Art. 6.) the square of AC is greater than the rectangle AP 9 PB\ wherefore the excess of AB, BC above the square of BC is greater than the excess of AB, BP above the square of PB. PROP. XXV. 115. Problem. To divide a given finite straight line into two parts, such that the aggregate of the squares of the two parts may be a minimum. Let AB (Fig. to Art. 114.) be the given finite straight line ; it is required to divide AB into two parts, such that the aggregate of their squares may be a minimum. Bisect (E. 10. 1.) AB in C; the squares of AC, and CB are, together, less than the squares of any other two parts, taken together, into which AB can be divided. For, take any other point P in AB ; then (E. 9, 2.) the squares of AP, PB are, together, equal to the squares of AC, CB and PC, together; 156 THE GEOMETRICAL INVESTIGATION wherefore the squares of AP, PB are greater than the squares of AC, CB. PROP. XXVI. 116. Problem. To divide a given circular arch into two such parts, that the rectangle con tained by their chords shall be a maximum. Let ACE be the given circular arch ; it is re quired to divide it into two such parts, that the rectangle contained by their chords shall be a maximum. Bisect (E. 30. 3.) the arch ACE in C, and join A 9 C, and B, C; the rectangle AC, CB is a maximum. t , For, take any other point P in the arch ACB 9 and join A, P, and B, P, and A, B. Then (E. 29. 3.) AC is equal to CB, and, therefore, AP and PB are unequal ; but (Art. 69.) AC and CB, i. e. the double of AC, are, together, greater than AP and PB ; therefore AC is gVeater than the half of AP, PB, and the square of AC greater than the square of the half of AP, PB ; and (Art. 6.) the square of the half of AP, PB is greater than OF MAXIMA AND MINIMA. 157 the rectangle AP, PB ; much more, then, is the square of AC, i. e. the rectangle AC, CB, greater than the rectangle AP, PB. t <\\ fXflMlf.I ,ff .3)^*^ PROP. XXVII. 117. Problem. Of all triangles having the same vertical angle, and their bases all passing through the same given point, to find that in which the rect angle, contained by the segments into which the given point divides the base, is a minimum. Let XAY be the vertical angle, and B the given point in the base, common to all the tri angles ; it is required to find a triangle having A for its vertical angle, and its base passing through the point B, and divided by it into two segments, such that the rectangle contained by them is a minimum. Bisect (E. 9. 1.) the angle XAFby the straight line AC, and from B, draw (E. 12. 1.) BC at right 158 THE GEOMETRICAL INVESTIGATION angles to AC, and produce it both ways to meet .^ff and XY, in D and E, the rectangle DB, BE is a minimum. For, draw (E. 11. 1 .) DK perpendicular to AD, and join KE : Because, by the construction, the angles at C are right angles, and the angles DAC, EAC are equal, and AC common to the two tri angles A CD, ACE, AD (E. 26. 1.) is equal to AE : Again, because AD is equal to AE, and AK common to the two triangles ADK, AEK, and the angle DAK is equal to the angle EAK^DK is equal (E.4. 1.) to KE, and the angle AEK to the right angle ADK: From the center K, at the distance KD, describe the circle DER; and it will, therefore, pass through E, and (E. 16. 3. Cor.) touch ^JT and XY, in D and E. Let now PQ, passing through B, be the base of any other triangle, having A for its vertical angle ; PQ must cut the circle ; let it cut the cir cumference in R and S; then the rectangle PB, BQ is manifestly greater than the rectangle RB 9 BS; and (E 35. 3.) the rectangle RB, BS is equal to the rectangle DB, BE-, wherefore the rect angle DB, BE is less than the rectangle, PB, BQ ; and is a minimum. PROP. VIII. 118. Problem. Of all triangles having the same vertical angle, and their bases all passing through OF MAXIMA AND MINIMA. 159 the same given point, to find that in which the rectangle contained by the sides is a minimum. Let FAC be the given vertical angle, and D the given point in the base, common to all the triangles ; of all triangles having the same vertical angle A y and having their bases drawn through D, it is required to find that in which the rect angle contained by the sides is a minimum. Through D draw (Art. 1.) the straight line FDG, which is bisected in D, and terminated by AF and AC ; the rectangle AG, AF is a minimum. For, draw through D any other straight line BDC, terminated by AF and AC in B and C; also draw (E. 31. 1.) GI parallel to AB ; then it may be shewn, as in Art. 3. that Gl is equal to BF; and, (E. 4. 6.) because the two triangles CAB, CGI are equiangular, CA : AB :: GC : GI or BF; therefore (E. 16. 6.) the rectangle AB, CG 160 THE GEOMETRICAL INVESTIGATION is equalitf the rectangle AC, BF, but th rect angle AC, BF is greater than the rectangle AG, BF; wherefore the rectangle AB, CG is greater, also, than the rectangle AG, BF; add to both the rectangle AB, AG, and the two rectangles BA, AG and AB, GCare greater than the two AG, BA and AG, BF; \. e. (E. 1. 2.) the rectangle AB, AC is greater than the rectangle AG, AF; wherefore the rectangle AG, AFis a minimum. 119. COR. Since * triangles, which have the same vertical angle, are to one another as the rect angles contained by their respective sides, about that angle, it is evident that of all triangles having a common vertical angle and their bases all passing through the same given point, that is the least which has its base bisected in the given point. PROP. XXIX. 120. Problem. To find the greatest of all tri angles having the same given vertical angle, and * It is easily proved, by the help of E. 1. 6. that if there be four lines which are proportionals, and four other lines which are also proportionals, the rectangle contained by the first and fifth has to the rectangle contained by the second and sixth, the same ratio, which the rectangle contained by the third and seventh, has to the rectangle contained by the fourth and the eighth : And, thence, it may be deduced as a corollary from E, 23. 6. and 41. 1. that triangles, which have equal vertical angles, are to one another as the rectangles contained by the sides about those equal angles. OF MAXIMA AND MINIMA. 161 the direct distances, between the vertex and the bisection of the base in each, all equal. Let XAY be the given vertical angle, and AD equal to the direct distance between the vertex and the bisection of the base in each of the tri angles; it is required to determine the greatest triangle which has A for its vertical angle, and the direct distance between A and the bisection of its base equal to AD. Draw (E. 9. 1.) the straight line AE bisecting the angle XAY, and make AE equal to AD ; through the point E draw (E. 11. 1.) the straight line BEC perpendicular to AE, meeting AX and AY in B and C ; the triangle ABC is a maximum. For, let APQ be any other triangle having A for its vertical angle, and the bisection, jR, of its base, at the same distance from A as E is ; through R draw (E. 31. 1.) ST parallel to jBC, and through Q draw VQ parallel to AP ; then, because QV is L 162 THE GEOMETRICAL INVESTIGATION parallel to AP, the angle VQP is equal (E. 29. 1.) to the angle QPS; and (E. 15. 1.) the two vertical angles QRV, SRP are equal, and the side QR is equal to the side RP ; wherefore (E. 26. and 4. 9.) the triangle Q,RV is equal to the triangle PRS; add to both the trapezium AQRS, and the tra pezium AQVS is equal to the triangle APQ\ but the triangle ABC is greater than the tra pezium AQVS\ for the point R must fall above BC, because it is in the circumference of a circle described from the center A, at the distance AD, which touches BC in E ; wherefore, also, the tri angle ABC is greater than the triangle APQ, and is a maximum. PROP. XXX. 121. Problem. To fi nd the least of all triangles having the same given vertical angle, and the per pendiculars, drawn from the common vertex to the base in each, all equal. Let XAY be the given vertical angle, and AB equal to the perpendicular drawn from the vertex UH OF MAXIMA AND MINIMA. 163 to the base in any one of the triangles ; it is re quired to find the least triangle which can have A for its vertical angle, and the perpendicular drawn from A to its base equal to AB. Bisect (E. 9. 1.) the angle XAY by the straight line AC, and make CA equal to AB ; through C draw (E. 11. 1.) the straight line DCE perpendi cular to AC, and let it meet AX and AY in D and E; the triangle ADE is a minimum. For, let APQ be any other triangle having A for its vertical angle, and the perpendicular AT drawn from A to its base PQ, equal to AC or AB. It is manifest from E. 17. 1. that PQ cannot pass through C; let it cut DE in R ; it is further evi dent from the construction, and E. 27. 1. that DC is equal to CE ; also (E. 16. 1.) the angle PDE is greater than the angle DAE ; if, therefore, through Z), DS be drawn parallel to AE, it will meet PQ between Pand R ; and (E. 15. and 29. 1.) the two triangles DRS, QRE are similar ; there fore (E. 19. 6.) the triangle DRS is to the triangle QRE as the square of DR to the square of RE; but DR is greater than RE, and the square of DR greater than the square of RE ; wherefore the triangle DRS is greater than the triangle QRE; but the triangle PRD is greater than DRS ; much more, then, is PRD greater than QRE ; add to both the trapezium AQRD, and the triangle APQ is shewn to be greater than the triangle ADE ; therefore the triangle ADE is less than any other such triangle; i. e. it is a minimum. 164 THE GEOMETRICAL INVESTIGATION Otherwise : Let XAY be the given vertical angle, and AB equal to the perpendicular drawn from A to the base of any one of the triangles : Then, it is mani fest, from the hypothesis, that if from A a center, at the distance AB, a circle be described, the bases of the triangles will be tangents to it : And, if DE be drawn touching the arch contained between AX and AY in it s bisection C, DE (Art. 105.) is the least base: Therefore, since the triangles have equal altitudes, the triangle ADE is (E. 1. 6. Cor.) the least triangle. -M lodftift er ir ;ft ni Rt\ Ji^ jj J I ; > rigi*!* PROP. XXXI. 122. Problem. Of all equal triangles, having their vertical angles also equal, that which is isosceles has the least perimeter. Let ABC be any one of a set of equal triangles, each having its vertical angle equal to BAC, and let the triangle ABC be scalene ; its perimeter is OF MAXIMA AND MINIMA. 165 greater than that of an equal isosceles triangle having its vertical angle equal to BAC. Find (E. 13. 6.) AD a mean proportional be tween AB and AC, and produce AC to E, so that AE is equal to AD ; join D, E, D, C and B, E ; and about the triangle DEC describe the circle BDCF; its circumference will pass beyond the point E ; for it cannot pass through E 9 otherwise the angle DEC would (E. 21. 3.) be equal to the angle DEC, but DEC is equal (E. 5. 1.) to the angle ADE, and, therefore, (E. 16. 1.) greater than DEC; neither can the point E be beyond the circumference ; for then the exterior angle of a triangle would be less than the interior opposite angle, which (E. 16. 1.) is impossible; produce, therefore, BE to meet the circumference in F 9 and join D, F and C, JP; and because, by the construction, AB : AD :: AE : AC, therefore (E. 17.6.) AB : BD :: AE : CE, and (E.2.6.) DC is parallel to BE ; therefore (E. 29. 1.) the angle DCE is equal to the angle CBF, and (E. 26. and 29. 3.) DB is equal to CF; also the angle DEC is equal (E. 21. 3.) to the angle DFC, and the angle DCB (E. 27. 3.) equal to the angle CDF, and DC is common to the two triangles DEC, DFC, wherefore (E. 26. 1.) BCis equal to DF; and DB was shewn to be equal to CF; therefore DB, JSCare, together, equal to DF, FC, together; but (Art. 13.) DF, FC are, together, greater than DE, EC; wherefore DB, jBCare also greater than DE, EC-, add to both AD, and AC; 166 THE GEOMETRICAL INVESTIGATION then AB, EC y AC are, together, greater than AD, DE, EA-, i.e. the perimeter of the scalene tri angle ABC is greater than that of the equal isosceles triangle ADE y having the same vertical angle. 123. COR. An isosceles triangle is greater than any other triangle of equal perimeter, and having also an equal vertical angle. PROP. XXXII. 124. Problem. Through a given point in a given circle, to draw a chord which shall cut off the least segment. Let PBQ, be the given circle, and A a given point within it ; it is required to draw through A a chord which shall cut off the least segment. Find (E. 1. 3.) the center K of the circle; join A, K-, and through A draw (E. 11. 1.) the chord CAB perpendicular to AK ; the segment CPB is a minimum. OF MAXIMA AND MINIMA. 167 For, through A draw any other chord PAQ, ; and from the center K draw (E. 12. 1.) KM per pendicular to PQ, ; and because AMK is a right angle, AK is greater (E. 17. and 19. 1.) than KM-, from KA cut off (E. 3. I.) KN equal to KM; and through N draw (E. 1.1.) the chord RNS at right angles to NK; therefore (E. 14. 3.) RS is equal to PQ; and (E. 28. 26. and 24. 3.) the segment RPS is equal to the segment PBQ; but the segment CPB is less than the segment RPS; wherefore, also, the segment CPB is less than the segment PBQ ; i. e. the segment cut off by CB is less than the segment cut off by any other chord, PQ, passing through the point A. SCHOLIUM. Many more propositions, of the same kind, might have been added, if those already given in this Section had not been thought sufficient, both in number and variety, to recommend the subject to the attention of the Student in Geometry. In the several branches of Natural Philosophy a great number of problems, relating to Maxima and Minima, will offer themselves to his invention ; and he will do well to exercise his ingenuity in discovering geometrical solutions to them. They will seldom be either more difficult or more tedi ous, when treated in this manner, than when they are solved by fluxional calculation * 168 THE GEOMETRICAL INVESTIGATION The following questions may serve to exemplify what is here recommended. r (I.) Of all truncated right cones, of the same base and of the same altitude, and moving in a fluid in the direction of a common axis, to find that which meets with the least resistance. Let OC be the semi-diameter of the given base, and OD the common altitude, of all the truncated right cones. * Bisect OD in Q, and join CQ; pro duce QD to S, and make QS equal to QC; join C, S; from D draw DF perpendicular to the common axis OS; the resistance on the truncated right cone generated by the revolution of the trapezium FDOC is a minimum. For, draw DM perpendicular to CS; and from the center Q, at the distance QO, or Q Z>, describe the circle OKD cutting CQ in K; join O y K and K, D 3 and produce OK to meet CS in L ; * This is the construction given, without an investigation, by NEWTON, in the Scholium to the 34th Proposition of the Second Book of the Principia. OF MAXIMA AND MINIMA. 169 then (E. 5. 1.) the angle QKD is equal to the angle QDK, and the angle QCS to the angle QSC; wherefore (E. 32. 1.) the exterior angle CQO is the double, both of QKD and QCS; therefore the angle QKD is equal to the angle QCS, and (E. 28. 1.) KD is parallel to CS ; but (E. 31. 3.) OKD is a right angle ; wherefore OLS (E. 29. 1.) is also a right angle. By the common principles of the motion of a body in a resisting medium, the resistance on the surface of the truncated cone is to the resistance on its base as the aggregate of the squares of DM and CL is to the square of CO ; i. e. as the aggregate of the squares of KL and LC, (E. 34. 1.) or of the square of CK (E. 47. 1.), to the square of CO; and if CPDO be the trapezium by the revolution of which any other truncated right cone of the same given base and altitude is generated, and OR be drawn at right angles to CP, cutting the circle OKD in N, and C, N be joined, the resistance on this truncated cone is to the resistance on its base, as the square of CN is to the square of CO ; but (E. 8. 3.) CK is less than CN, and the square of CK is, therefore, less than the square of CN; wherefore (E. 10. 5.) the resistance on the surface of the truncated cone generated by the revolution of CFDO, is less than that on the surface of any other such cone. (II.) A given weight being appended to a point in the axis of a cylindrical lever, at a given dis- 170 THE GEOMETRICAL INVESTIGATION tance from one of its extremities which is the center of motion, and the diameter and specific gravity of the lever being also given, to determine the length of the lever, so that the force applied at its other extremity, to raise the given weight, may be a minimum. Let the given distance from the center of motion at which the weight is hung be called 2 a, and let 2 x be assumed the length of the lever \ call the force applied at the extremity of the lever, which is to be a minimum, P; let the weight of one inch of the lever be unity, and let the given weight be (A) such unities ; therefore the weight of the whole lever will be 2#; and from the Principles of Mechanics, Px 2x=2x. x + A. 2a .; /. P .x= #*+ A . a. Let y be a fourth proportional to x, a and A, so that (E. 16. 6.) A.a^xy, .*. P.x=x*+xy, /. P = x + y. But the rectangle contained by x, and y is a given quantity ; wherefore (Art. 26.) the sum of x and y is a minimum when x=y ; that is, P is a minimum when a? is a mean proportional between A and a. ij ; (III.) In the same manner, if it be inquired what curve, by its revolution, generates a surface best adapted for a Speaking, or a Hearing Trumpet, the question will be found to depend upon the same geometrical proposition as the preceding; OF MAXIMA AND MINIMA. 171 A a a quantity of the form x + must be a mini mum ; that is, x must be a mean proportional between A and a ; A, x, and a being three conti guous portions of the air included in the tube, which is an elastic fluid ; whence it is manifest that the curve sought is the Logarithmic curve. (IV.) Lastly, An horizontal space being given, to find the perpendicular altitude through which a heavy body must fall from rest, so that after wards describing the horizontal space, with its acquired velocity, the whole time of its motion shall be a minimum. feet Let (a) denote the horizontal space, (ni)l6~, and (x) the required perpendicular altitude ; then, from the Principles of Mechanics, the time of the body s motion is denoted by 7= + \ ..... ; - ; and this V m quantity is to be a minimum ; therefore, also, its square ~ 4- + - must be a minimum, m m 4mx (. which it will be when x + is least ; that is, as 4 X before, when x = - . This proposition is best demonstrated syntheti cally, by the help of the parabola. It is, however, 172 THE GEOMETRICAL INVESTIGATION, &C. not unworthy of remark, that the solution of this, and of a great variety of problems of the same kind, may be reduced to the simple geometrical theorem, which asserts the perimeter of a square to be less than the perimeter of any other equal rectangle. ON MAXIMA AND MINIMA PART II. ON THE ALGEBRAICAL INVESTIGATION Of MAXIMA AND MINIMA. SECTION I. ON THE BINOMIAL THEOREM. JL HE Binomial Theorem forms the basis of the algebraical investigation of Maxima and Minima. Unless, indeed, it be first legitimately established, neither the Principles of Fluxions, nor those of any other equivalent system, which furnishes the same rules of computation, can be freed from very weighty objections. It is, therefore, most inti mately connected with this part of our subject. If, however, what has already been written and 174 ON THE BINOMIAL THEOREM. published concerning this theorem, might be con sidered as satisfactory, it need not have occupied a place in this Treatise. But the truth is, that the commonly received demonstrations of it have no just pretensions to logical exactness. It is, indeed, surprising that men of great talents and attain ments have either overlooked the defects of the proofs which they have given of this theorem, or else have preferred the publication of an imperfect chain of reasoning, to the confession of their in ability to be rigorously exact. BARON MASERES alone seems to have been scrupulously anxious to attain absolute precision on this point ; and, where his laborious endeavours fail of success, he is the first to acknowledge the failure. These assertions can be justified only by a brief review of the methods of investigating the Binomial Theorem, which are commonly thought to be the best. The proof, of which LANDEN was the author, seems to have been for a long time in great esteem with the algebraists of this country. It was exhi bited, under its most advantageous form, in the Philosophical Transactions of the year 1796 ; from which it has been copied, and re-copied, in several publications that have appeared since. The principal objections under which it labours are these. 1. There is no previous definition laid down m of what is meant by the expansions of (a x) n , (ax)~ m , and (aa?)"~*> nor is any enquiry made ON THE BINOMIAL THEOREM. 175 into the general nature of those expansions, to justify the assumption of the series with which the proof sets out. 2. The proof proceeds upon the supposition of m a numerical equality between (a x)" 9 (a x)~ m , (# #)"*, an d their respective expansions con tinued without limit, which equality, does not exist. There is not, necessarily, even an approximation in value between those quantities and their several expansions, when particular numbers are put in the places of a and x. 3. The sophism of shifting the hypothesis is next introduced; that is, results are obtained by making a supposition in one part of the demon stration, which, if it had been made in a preceding part, would have wholly stopt the process. The value of x is made equal to that of y in one equation, after building upon another equation, in which if that supposition had been made, the latter equation could not have been obtained. 4. The equality of the coefficients of the cor responding powers of x in two series of the form A + Ex + CV + &c. = a + bx + ex 2 + &c. is asserted ; and as this is asserted without proo it is to be supposed that the common method of inferring that equality, by making x O, which proceeds upon a petitio principii, is deemed suf ficient. One, or more of these principal objections may 176 ON THE BINOMIAL THEOREM. be made to most of the other proofs of the Binomial Theorem which are best known. That published by LAGRANGE in his Theorie des Fonctions Analytiques, has been rendered much less objectionable by the learned author of the Principles of Analytical Calculation, in whose work it appears. Still, it hardly seems judicious to employ the symbol of equality, where no equality exists, although the reader is forewarned of it. This is, in reality, not merely an extension, but a change, of the meaning of the sign. There is, perhaps, some impropriety in denoting by it the relation subsisting between a variable quantity and its limit; still less, then, ought to be used where no such approximation necessarily takes place. There are degrees of inequality, but none of equality. An objection of such a nature, however, is comparatively light. But the proof is made to depend on the assumption, that if f- !2 (a -f- x -f- z) ~ n be expanded as a binomial, first by considering a 4- x as one quantity, then by taking x 4- z as one quantity _, the resulting series shall be identical, whatever the index of the expansion be, whether integral or fractional, positive or negative. That this is true when the index is a positive integer, will be readily granted ; for it may be intuitively perceived. But can it fairly be assumed to obtain in any other case ? How is it that the mind assents to any general proposition ? It in stantaneously verifies the included assertion, or ON THE BINOMIAL THEOREM. 177 negation, by having recourse to definitions, or to some obvious instances to which the proposition is applicable, and in which it is manifestly true, in dependently of any particularity belonging to any one of the instances considered. If there be not this perception of the agreement, or disagreement, of ideas, no genuine conviction can follow. The plausibility of the enunciation of a proposition may, indeed, be such as to win a hasty consent from the indolent ; but, without that necessary verification, there can be no real knowledge. That " the same result must be had, when the same algebraical operation has been performed on the same quantity " is a proposition, at first sight, sufficiently plausible ; but in the application made by LAG RANGE of that assertion, the quantity ope rated upon cannot, strictly speaking, be said to be the same quantity in all the cases ; and when the proposition is more precisely enunciated, it be comes necessary to resort to the usual method, in^ order to judge whether it be true or false. Now + is the index of several operations, and there is none of them which manifestly produces two identical series, when (a + x) + z and a -f (x -f z) have been subjected to it. In so simple a case as that in which the index is - 1, the resulting series are far from being manifestly the same ; and they are in no case necessarily so, excepting that in which the index is a whole positive number; for then only is there a numerical equality between M 178 ON THE BINOMIAL THEOREM. (a + x + z) m and its whole expansion . The assertion is, undoubtedly, true ; but it requires, and admits of a proof, as much as the Binomial Theorem itself; and it does not appear that any advantage would be gained by previously establishing the truth of this assertion. The faults, here imputed to LANDEN and LAGRANGE, seem to have arisen from the de sire of being concise, where conciseness is not attainable, and where precision should have been chiefly aimed at ; and from the affectation of gene ralizing too hastily, where all the included parti culars do not readily occur to the mind, and where they have scarcely enough in common, to furnish the basis of a demonstration equally applicable to each of them. A proof very lately published, in the second part of the Philosophical Transactions, for the year 18 16, which sets out from the assumption, that (a + x) m . (a + 3/) w = [(a -f x) . (a+y)J n , whether m be positive or negative, integral or fractional, is evidently liable to some of the objections which have been urged against the methods of LANDEN and LAGRANGE. The equation thus assumed for the foundation of a general proof, is intended to comprehend some cases, in which, strictly speaking, no equality obtains. Or, if it be otherwise explained, it supposes, in the form of the results of the opera tions, designated by the index m, an identity which is far from being self-evident. ON THE BINOMIAL THEOREM. 179 Considered with respect to exactness, Professor ROBERTSON S proof of the Binomial Theorem, has great merit ; and this is the place to acknowledge that his manner of inferring the form of the coeffi cients of the expanded binomial, when the index is a positive fraction, has been adopted in the following attempt. The author had, indeed, pre viously applied the same principle, in the case of a negative index; but, before he had perceived its application to the former case, it fell in his way to see the Professor s demonstration. It is re markable that Mr. Robertson himself is not exact, when he investigates the expansion of (a x)~ m . He does not obtain it without the same shifting of the hypothesis, the same reduction of the value of x to 0, a supposition implicitly excluded in the previous reasoning, which has been noted in Landen s proof. He also places signs of equality between quantities which are not equal. After what has been said, the reader will not look for conciseness in the following demonstra tion ; but the Author has studiously endeavoured to avoid the imperfections which have here been pointed out. His original intention was merely to supply the defects, and to rectify the errors of other writers better known than himself. It appeared to him to be necessary, both to the correction of Landen s proof, and to the demon stration of the assumption made by Lagrange, t,hat the ratio of any two successive coefficients, in M2 180 ON THE BINOMIAL THEOREM. 4- *!L every case of the expansion of (a x) ~ n should be shewn not to exceed any finite ratio ; but the in vestigation of this property, he found, led so directly to the law of the coefficients in each case, that there was no advantage in deducing it sepa rately, in order, afterwards, to form one general demonstration. He could not, however, proceed satisfactorily, in his inquiry into the nature of the series obtained by algebraic evolution, without the help of the Polynomial Theorem. This theorem, therefore, he was obliged previously to demon strate ; the demonstration of it followed, with the greatest ease, from the method which he had em ployed for the simple involution of the binomial ; and it gave him, more than he at first expected to follow immediately from it, namely, the law of the i coefficients of the expansion of (a x) . He has since found that he had been anticipated in this step by Dr. HUTTON. Complete precision has, undoubtedly, been aimed at in the following proof ; the reader will judge whether it has been attained. One objec tion, however, may probably be foreseen, namely, that some of the conclusions here drawn, may seem not to be arrived at without the help of induction. This term properly denotes the in ferring some general proposition from observing that it is true in a multitude of separate instances; rio necessary connexion being perceived between the instances themselves and fhe common pro- ON THE BINOMIAL THEOREM. 181 perty. It is thus that the laws of motion are col lected, and other axioms in Physics. Induction, in this, the proper sense of the word, is wholly in admissible in abstract mathematical science. But there is nothing of this kind in the inference, drawn from a partial division of unity by 1 - ,r, that the m th term of the quotient must be x? n ~ l ; the con nexion between the subject and the predicate, the. form, and the law of continuation, of the series, is intuitively known ; and the mind is as fully satisfied of the truth of the assertion, as it can be of that of any other proposition in Euclid s Elements. The same may be said of the equations which determine the f terms of the involution of a binomial, or a polynomial, raised to the m th power. When a series admits of being continued without limit, as the expansion of (ax)~ m and 4- (a x) ~ n , only a finite nnmber of terms can then be exhibited; but if, in all cases, the law of the formation of the terms be found for any two suc cessive terms whatever, the p* and the (p -f- l) th , it may be fairly concluded to obtain in them all - 9 and this law beiqg once determined, the series may be continued to any number of terms what ever, without Induction, ART. i. DEF. The word Function in the fol lowing articles, is used to designate any algebraic 182 ON THE BINOMIAL THEOREM, expression, containing one or more variable quan tities, mixt, or not, with constant quantities ; such an expression is called the Function of the variable quantity, or quantities, which it contains. 2. DEF. The Limit of a Function is a constant quantity, from which the function may be made to differ less than any other given quantity, but to which it can never be equal. M.- c : MU ; 4dh$flpi ^ .Itowli? 1 - tttbeil8!viljrl*ai bnuh 3rffi4HMh( iitf<f$d \ PROP. I. 3. Theorem. If, in a series of quantities, con tinued indefinitely, each term be the half of that immediately preceding it, any term whatever is greater than the sum of all the terms which follow it. Let K be any term whatever of such a series ; It K iC therefore , , -> & c * are tne succeeding terms ; but, by the common rule, investigated in the Elements of Algebra, 2 K is the limit of the K K progression K + 4. -}- &c^fad wifinitum ; K K that is, 2 K is greater than K -f : - -f - -f- &c. to whatever number of terms the series be continued ; take K from both, and there remains K greater than -f - -f - + &c. ad infinitum. 248 ON THE BINOMIAL THEOREM. . 183 4. COR. If each term of a series be less than the half of that immediately preceding it, any term whatever is greater than the sum of all the terms which follow it. PROP. II. 5. Theorem. In a series of quantities, either finite or continued indefinitely, of the form A Bx -f Cx* Dx* + &c., in which the value of x is arbitrary, and the coefficients A, B, C, &c. are constant quantities, none of which has to that preceding it a ratio greater than any ratio assign able, x may be taken such that any term shall be indefinitely greater than the sum of all the terms of the series which contain higher powers of x. Let the constant coefficients A, B, C, &c. be supposed to be all positive, and to go on increas ing from A, so that the second is greater than the first, the third than the second, and so on ; and let Rx p , Sx p+l be the two successive terms in which the ratio of the latter coefficient to the former is greatest ; then, since this ratio, by the 8 hypothesis, is finite, is a finite quantity, and, therefore, also -~- is a finite quantity ; but the yalue of x is arbitrary ; let, then, x become less 184 ON THE BINOMIAL THEOREM. n n than ;r r ; therefore S . x is less than --: and 2i *3 2 n S.xP +l less than ~.<r^; that is, the term SV+ 1 is less than the half of the preceding term ; but, by the supposition, no other coefficient has to that of the term immediately preceding it, so great a ratio as S has to R ; wherefore, the second term is less than the half of the first, and every term iu the series, after the first, is less than the half of the pre ceding term ; therefore, (Art. 4.) any term of the series is greater than the sum of all the remaining terms; and, if this be the case, when the coefficients are supposed to be all positive, and to go on increasing from A, it will, a fortiori, be the case, when any Other supposition is made, relative to the magnitudes, and the signs of the coefficients ; wherefore x may always be so taken, as that any term of the series shall be greater than the sum of all the terms that follow it : and, it is manifest, that by continually diminishing the arbitrary value of #, any term may be made to become indefinitely greater than the sum of all the following terms*. * All the coefficients, A, J3 9 C, &c., of the series being sup posed to have finite and constant values, the proposition may be proved, independently of Art. 4. in the following manner. First, the value of x may be taken such, that the first terra A of the series A + Bx -f- C#* + &c. shall have to any other term, as 2 x p , a ratio greater than any assigned ratio, as that of r to unity. For, ON THE BINOMIAL THEOREM. 185 6. COR. The first term (A) of a series, such as that described in Art. 5, is its limit : and as A may be of any finite value whatever, it is plain that the sum of all the terms, but the first, of such a series For, let s denote any number greater than r. And, if x, the value of which is arbitrary, be taken equal A : QX? :: ~ : x 9 ^A A : : >wJ i-twtod That is, the first term A has to the term Six* a greater ratio than that of r to unity, which is any given ratio. Next, let m denote the number of all the terms of the series but the first, and t any number whatever : let the coefficients be supposed to be all positive and to go on increasing from A t and let x be taken so that A : B x > mt : 1 ; the possibility of which has been demonstrated. Wherefore, Now, it is plain from what has been proved above, and from the supposition made relative to the magnitudes of the coefficients, that x must be taken of a still smaller and smaller value, in order that A may have to C x*, Dx 3 , &c. a ratio greater than that of m t to unity : when, therefore, x is taken so that A may have to the last of the terms a ratio greater than that of mt to unity. will still be greater than t . B x, t . C x z , t . D # 3 , and each of the following terms ; .-. ~ H- -+&c. (m terms) > (t . Bx+t .Ctf + t. DjrH&c,) 7ft 1* i. e. 186 ON THE BINOMIAL THEOREM. may be made to become less than any given finite quantity : Further, A is also the limit of a series of the form A Bx + Cx* &c. Rz y R being a function, either of the arbitrary quantity z, or of x and , containing only positive powers, and not containing any coefficient which has, to that of the term immediately preceding it, a ratio greater than any assignable. For, if z 9 the value of which is arbitrary, become equal to x, this series may be made to agree with the series of A Bx -fc- Cx~ -f &c., from which it differs only in form ; there being no real differ ence between two perfectly arbitrary quantities > however they may have originated, or by whatever notation they may have been expressed. PROP. III. 7. Theorem. If two series, of the form de scribed in the last proposition, be equal to each other, the first term of the one is equal to the first i. e. Wherefore, Bx -j- Cx* -f D x 3 -f &c. may be made less than -- , however great t is. Whence it is evident, that the sum of all the terms but the first may be made indefinitely less than any assignable quantity. In the same manner it may be shewn, that x may be so taken, as that any term of the series, shall be indefinitely greater than the sum of all the other terms, which contain higher powers of * And if this be true when the coefficients are all positive, and go on increasing from A, it will, a fortiori, be true in all other cases. ON THE BINOMIAL THEOREM. 1ST term of the other, and the coefficients of the same powers of the arbitrary quantities, in each, are severally equal. First, let a bx + ex 1 &c, qxP = ARx -f Co?* &c. Q# /; let bx + car* &c. + qx? = + ?r, and Bx+Cx* &c. Q^ = , II; therefore a TT - A U. If it be possible, let a and A be unequal, and let their difference be 5, which is, therefore, a finite and constant quantity. Then a^A=^=Il^7r, whatever be the value of a?; but (Art. 6.) x may be taken such that TT and II shall each of them be less than any finite quantity ; in which case their difference II ^ TT will manifestly be less than any finite quantity sup pose, therefore, x to be so taken, as that II and TT shall each of them be less than S; wherefore TL ^TT is less than $ ; but II ^ TT is equal to 5 ; which is absurd; therefore a = A; and bx + cx + kc. qxP = Bx + Cx z &c. QxP; divide, now, this equation by x, and + b + cx &c. + qx?~ l = B + Ca? &c. + Q^- 1 ; ^ may, therefore, be shewn to be equal to J2, in the same manner as a was proved to be equal to A ; and the same may be shewn, in the same manner, of the rest of the coefficients of the corresponding terms of the two series. 188 ON THE BINOMIAL THEOREM. 8. COR. i. If ax p + bx + CX* -f &C. + * J* + &C. + i 7T 1 + |tf# -h 6cc. then 1 = 1, B=:b, and the rest of the coefficients of the same powers of x, in the two series, are equal to each other; the same limitation being laid down with respect to the ratio of any two successive coefficients in the three series, as in Prop. II. For, let the given equation be multiplied by 1 + &* 4 &c. H +&C. ( r 3 cc. i E and ft = -B ; 7; but J5 has been proved to be equal to b ; /. j3 6 + 7 = j3 B + 7, and c = C In the same manner, d may be shewn to be equal to D, and the rest of the coefficients, of the one ON THE BINOMIAL THEOREM. 189 series, to the rest of the coefficients, of the other, each to each. For, if p and P be any two corresponding co efficients, in the two originally given series, the coefficients of the same power of x, in the two series resulting from the multiplication, will mani festly be and P -f j3 . O y . N -f &c. + H ; but /3 . o, 7 . n, &c. will have been proved to be equal to /3 . O, 7 . N 9 &c. respectively; wherefore (Art. 7.) P = p. 9. COR. 2. The same limitation being made, with respect to the coefficients as before, if a bx + co? a &c. lx k vz = ABx+Cx* &c. L^* F* ; if, also, the values of x and s be both of them arbitrary, and independent of each other; and if v and /^be functions of z, or of a? and z, containing only positive powers, then shall a = A, b = ^, &c. = &c. /= L. For, first, the series a bx + cz* &c. /a*, is equal to A+Bx + Cx* &c. L^, whatever be the value of x : If not, there is some certain value which put for x in the two series will 190 ON THE BINOMIAL THEOREM. render them unequal. Let therefore, that value of cT be substituted in the given equation, and let the difference between the two series, which involve only powers of x and constant quantities, be denoted by S : Therefore, S = (V <*. v) .2 = 0: Now this equality is to obtain whatever be the values of x and z. But, if the value of x continue to be as it was last assumed, the value of z (Art 6.) may be so taken as that (V -^- v) z shall be less than the constant quantity o ; wherefore, the equality does not obtain, in the equation whatever be the value of z ; which is contrary to the original supposition. The two series a bx -\- ex* &c. + lx k y A Bx + Car &c. Lof, cannot, therefore, have any finite difference : that is; they are equal to one another : whence (Art. 7.) a = A, b = /?, c = C 9 &c. = &c. I = L. SCHOLIUM. The value of x in Art. 7- 8. 9. is supposed to be absolutely arbitrary, so that it may be made either greater or less than any finite quantity ; but the proposition is true of two such series, which are equal to eacb other, when x has any particular value, and are also equal, when x has any less value ON THE BINOMIAL THEOREM. 191 whatever ; even although they would not be equal if x had a value greater than that particular value : the demonstration, indeed, proceeds only upon this latter supposition. It is, however, implied, in all the cases, that whatever value x has on the one side of the equa tion a b x + ex* + &c. = A Bx -f Cx* -f &c. it shall have the same value on the other. In order, therefore, to render the demonstration of Art. 8. unexceptionable, it may be necessary to shew that there is some one value of x, which being sub stituted for it on both sides of the equation, both TT and n become less than any proposed finite quantity. Now, it is evident from Art. 5. that there is some value of x, which renders TT less than the finite quantity $ : let this value be denoted by /. It is evident, likewise, that there is some value of .r which renders n less than 3: let that value be denoted by L. If, then, either of these two values, as L, be greater than the other, it is manifest, that if / be substituted, instead of L, for x, II will still, a fortiori, be less than $ : so that the same value of x renders both II and TT less than the given finite quantity S. 1O. Definitions. I. The expansion of (a x) m is the series arising from the multiplication of (a x).(ax) to m factors. It is manifest,, from the actual process of this 192 ON H BINOMIAL THEOREM. multiplication, that the resulting series is of the form a m bx+ ex* Sic. x m ; and that the coefficients b, c, &c. are independent of the value of x. It is further evident, from the first Principles of Universal Arithmetic, that if two numbers be divided, each into any parts whatever, the product of the one, multiplied by the other, shall be equal to the sum of the products arising from multiplying the parts of the ,one by the parts of the other ; and hence it follows, that, if specific numbers be substituted for a and x in (a + x) m , and in its expansion, already obtained according to any given value of m, the number, which is the sum of a and #, shall, when raised to the m b power, be equal to the sum of the terms of the expansion. The sign of equality is, therefore, rightly placed between (a x) m and its expansion, when the index (m) is positive and integral. The same is true of the series .-/ 4- Bx 4- Cx* + Dx } 4- &c. ,:f Lx* which arises from the actual multiplication of (a 4- bx 4- or -f &c. 4- /#*) . (a 4 bx -f ex* 4 &c. to m factors, and which is called the expansion oi (a + lx + ex* + &c. 4- lx*) m . i II. By the expansion of (a x} is meant a series, of the form fix 70?* $X* &C. ON THE BINOMIAL THEOREM. 193 such, that if any number (p -f- l) of its terms be raised to the n ih power, that product is of the form ax + * + * * whatever be the value of x. And the expansion of (a x) is the series arising from raising the ex- i pansion of (a -f- x) 7t to the w th power ; it is, there fore, of the form A Ex Cx z Dx^ + &c. in which expression, the coefficients A, B, C, D, &c. are independent of the value of x. III. The expansion of (a x}~ m is the series arising from the actual division of unity by the expansion of (a x) m ; which series is of the form a + b x -f c o,- 2 + d x 3 + &c. involving only positive powers of x, and having its coefficients a , b , c , &c. independent of the value of x. There is, also, an arithmetical equality between 1 ( *) and a f b x + c * a + d x s + &c. + - (a x) m This is manifest, from actually dividing 1 by a + b x + cx z + &c. + x m , which is the expansion of (a Hh x) m , and is also equal to (a x) m . N 194 ON THE BINOMIAL THEOREM. IV. The expansion of (a x)~^ is the series arising from the division of unity by the expansion of (a x)* ; and it is, therefore, of the form A B x C x* D x* &c. ; and the coefficients A , B f , C , D , &c. are also independent of the value of x. PROP. IV. 11. Theorem. The two first terms of the ex pansion of (l + x) m are 1 + mx. For, Art. 10, (1 -f x) m - I -\- bx + ex* + dx* + &c. + x m ; .e. when, therefore, the index of the power is in creased by unity, the coefficient of the second term of the expansion is increased by unity; i then, the coefficient of the second term of the ex pansion of (1 -f- x)* be 2, that of the second term of the expansion of ( 1 + x) m is m ; but the coeffi cient of the second term of the expansion of (1 -f- x)* is known, from actual multiplication, to be 2 ; wherefore the coefficient of the second term of the expansion of (1 + x ) m ls m* ON THE BINOMIAL THEOREM. 195 12. COR. 1. The expansion of (1 -#) m is of the form 1 - mx + c x* - d x 3 + &e. (Art. 10.) 13. COR. 2. The expansion of (a+x) is of the form a m + ma m ~ l x + &c. For, Therefore, f a _|_ x] m x ex* a?* a m m a "aT a m% multiply both sides of the equation by a m ; then (a + x) m = a m + ma m ~ l x + &c. In the same manner, it may be shewn that the expansion of (a + x)~~ m is of the form or ma~ m ~ l x + &c. ; or, that it is the series arising from dividing the ex- (x\ "* 1 + ^ ) by a m . PROP. V. 14. Problem. To find the expansion of (a +x) m , the index (m) being positive, and a whole number. N2 196 ON THE BINOMIAL THEOREM. It follows, from Art. 10. that, . (A).... and (B) . . . . Let z/, the value of which is arbitrary, be taken equal to x + z, the quantity z having also an arbi trary value ; and let x -f- z be substituted for y in the equation marked (B). Then ( 1 + x + z) m = I + b . (x -\- z) + c . (x -f- #)* + d. (x +z) s + &c. + (x + z) m = 1 ^. bx + ex* +da? +.&c.> 4 x /-Art. 10. and 11. ; ^ + rf But Let now the equation (A) be taken from the equation (C), and m.(l+a:) m - l * + c.(l+x) m -*z*+ &c. +z m = r 62 + 2 cr% -f 3 dx*% + &c. = ) + c*- ( +dz* ON THE BINOMIAL THEOREM. 197 Every term of the last equation is divisible by z ; let it, therefore, be divided by % ; and m.(l +a?) w - 1 + c(l 4- a?) w - a * + &c. + z m ~ l = & + 2 e;c + 3 dx* + &c. } = < -f- c z + &c. + </* ; 5 therefore, (Art. 9.) m. 1 -f x m ~ l =64- 2c^ + 3Zr* + 4 ex? + &c. = (l + #) . (b-\- 2cx + 3dx z +4ex 3 -f &c.) i. e. m . (l-t-x) m , or w-f mbx-{-mcx*-{-mdx s -t- &c.-f-/w,r w a ( -f b x-i- 2cx*-\-3dx*-\- &c. 5 And, since this equation is true, whatever be the value of x, therefore, (Art. 7.) m = 6, 2c-}-b = m.b 9 3d+2c=m.c, rq-}- (r l) p ~ mp; therefore -b m "" l - m ~ l 2 2 ,_ m 2 _ wi 1 m - 2 _ ,/ m ~ 3 _ m 1 m- 2 m- 3 4 234 g m - ( r ~ 1 )^^. m ~ T t . ^LzJziz!) 198 ON THE BINOMIAL THEOREM. Therefore the expansion of (l-r-,r) ?n is l+mx -f m.~- ^ and (Art. 13.), that of (a + x) m \s a m -{-ma^ l x + m. 15. COR. 1. The expansion of (a~x) m is a m - ma m ~ l x + m.^~ a m -"~x* ~ &c. For, it is evident., from the actual involution of (a-x) m , that the coefficients will be the same as those of the expansion of (a-\-x) m , and that the signs of its terms will be alternately positive and negative. ON THE BINOMIAL THEQR&jt. 199 16. COR, 2. Since, (Art. 14.) if b, c, d, &c, be the coefficients of the first, second, third, &e. powers of x, in the expansion of (1 + x) m , + ti i i ^ 2f 1 i". s f^ ^ r <N CO *xs ^s _ ;i CO \ A ^S^ 1 5 ^ i c* 7 " c i ^* mt _ 1 CO S s S: & ro + + H 4. 4. * j. & M r V co -f ^ H * * :* | ^ i, S J 1 L ^ CO | ! i f^ a I 1 co fe ?t S h <D S - h -3 + i _. i *. . + t* k s "H s$ 5^ ^ ifcCs v rH -H 1 " ir ^ rH 1 C* | c^ J I 1 ^ i h bD g i- 1 - M h hs<i ?^ ^^ ^ ^^ s a + *+ ^ + 4 1 ^ F- g - II II >s " 200 ON THE BINOMIAL THEOREM But also,, Art. 14, (l -f x) m+n = + * ~ therefore, (Art. 7.) b + b f ~ m }- n, & , w - , , m+n-l m+n-2 = (w + W ).-__ . ___ ; ^ ; and the rest of the coefficients of the different powers of x in the product of the two series mul tiplied together, are severally equal to the coeffi cients of the same powers of x in the expansion of (1 + x) m+n ; therefore the former set of coefficients are reducible to the form of the latter : and that they admit of being reduced to this form does not at all depend upon m and n being integers. w / If - were put for m, and - for n. in the two q ,9 series multiplied together, the resulting coefficients would be of the same form as before, and might, therefore, be resolved in the same manner ; so that if ON THE BINOMIAL THEOREM. 201 ?. 2 be multiplied by f r -n f--o f r -2) -; y t* j v if j s i v f ^+5-l V] ^ 2 + J I | I hrK & the several coefficients of the different powers of x in the product, are reducible to the form And (Art. 13.) if i + ^.ia q ~* x* + &c. be multiplied by L T i * J ^ L a j,, a> + -s- us x + -A^-\ a> +&c - the resulting series will be of the form 202 ON THE BINOMIAL THEOREM. SCHOLIUM. If the problem solved in Art. 14. had been the limit of our enquiries, in this part of the subject, a plainer course would have been chosen, than that which has there been pursued. The mode of demonstration actually employed in that article, has, indeed, been adopted, for the sake of giving uniformity to the whole investigation of the form of the expansion of (a + x) - , in all its several cases. The following proof of the Binomial Theorem, in the case of a positive and integral index, is perhaps less circuitously derived from first prin- ciples_, than any other. It is founded principally on the Doctrine of Permutations ; than which no part of Algebra is more free from difficulty and obscurity. The problem is, to determine the form of the product of (a -\- x) . (a -} x) . (a + x), &c. when there are m factors. Let the multiplication be supposed to proceed, without any algebraic addition of the several partial products thence successively arising; putting, always, the letter which is the multiplier before that which is the multiplicand. Thus, =a + x .(a+x) ax -f- xa -f xx. ON THE BINOMIAL THEOREM. 203 = (aa -f- ax 4- xa + xx) . (a+x) = aaa -f aax + axa + axx + #a -f \-xax + ##0 4- ###j &c. = &c. Thus, the process of multiplication consists, en tirely, in placing, at each step, first a and then x be fore each of the permutations, which, together, form the next preceding product ; and thus, it is very evident, that the square consists of all the different permutations that can be made out of aaxx, taken two by two ; the cube, of all that can be made out of aaaxxX) taken three by three ; and so on ; the m ih power j of all that can be made out of a a a . . . (m) bbb ... (m) taken m together. It is, also, equally manifest, that the partial products, or several collec tions, each containing m of those 2m quantities will be of the forms, rfm fj m ~* f f > <v>w 2 /2 R?(* &"C fl^ T^ 1 ~~^ fl T m ~~^ f^ and that there will be as many products, of any one of those forms, as there are permutations of the letters which compose it. The form a m ~~ 3 a?, for example, will occur as often as there are different permutations of the letters aaa . . . (m- 3) xxx, taken m together. Hence, from the ^ell-known theory of permutations, (a+x) m = a m + ma m ~- l x + m.?^-- a m ~ z x* + &c. &c. -f 2 * JL m , ax m ~~ l -4- x m 204: ON THE BINOMIAL THEOREM. COR. The number of all the terms, in the product of (a + x) .(a+x).(a + x) (m\ when no addition has been made of the several partial products successively arising, is equal to 2 m : for it is 4, or 2 2 , in the product of (a + x) . ( + #), and it is doubled at every succeeding step. It is evident, therefore, that the sum of the coefficients, in this case, is equal to 2 W ; for each coefficient is unity ; so that their sum is equal to the number of terms., that is, to 2 m . And, as the sum of the coef ficients, in this dilated form of the product, is neces sarily equal to the sum of the coefficients in that more compact form of it, which is expressed by the Binomial Theorem, therefore, m 1 , m I m-2 2 W = m + m . + m - ^- 3~~ + In the same manner may be investigated the expansion of the polynomial (a + b+c+d + &c.) m , wben the index m is integral and positive. Thus the demonstration of the Binomial and Polynomial Theorems, in the simplest case of them, is made to depend on the solution of this question : c If there be p quantities all of one kind, and q quantities all of another kind, and if p + q = m, what is the number of permutations of which these quantities admit, taken m together?" To this question the following theorem, deducible from ON THE BINOMIAL THEOREM. 205 the common rules relating to combinations and permutations, affords a ready answer. If M be put for the number of permutations of which m quantities admit, taken all together, P for the number of permutations of which p quantities admit, and Q for the number of which q quan tities admit ; if m = p + q ; and if C be put for the number of combinations of which m quantities admit, when p or q of them are taken together, then, M P x Q x C. For, (tFbocTs Algebra, Art. 229, 230.) M = m. (m 1) .(m - 2) ...... l. 1.2 ............ q .-. M = P x Q x C. Hence, if all the JP quantities be of one kind, and the q quantities be all of another kind, P= 1; Q= 1; and M = P x Q x C = m.(m - l).(?n - 2) ---- fro - (y - i)] 1.2.3 2 which, therefore, is the general form of the coef ficient of the q ih term of the expansion of (a+x) m . In the same manner it may be shewn, in general. that if p -f q + r -f- &c. = m t 206 ON THE BINOMIAL THEOREM. and if there be p quantities of one kind, q of another, r of a third kind, and so on, the number of permuta tions of these quantities, taken m together, is equal to m . (m - 1 ) (m - 2) 1 (1 .2 p) .(1.2 #).(! . 2 r) &c. " PROP. VI. 17. Theorem. If A, B, C, D, &c. be a series of quantities, such that .,. 3 r then A B + C &c. P= - T> For A = , and, therefore, m Hence, 4. R-, r- r-4. 2C ^5+6=0+ and thus it is evident, that -" A+B+ C &c. P ifp terms be added together. ON THE BINOMIAL THEOREM. 207 18. COR. Hence, ) = 4C;3D + m(C+B+A)=: PROP. VII. ig. Problem. To find the expansion of (a + fix + 73?* + $x* + &c. + cr O, the index m being positive, and a whole number. Let, Art. 10., (a + ftx + yx* + $x* + &c. + ex ) = A + Ex + Cx* + Dx 3 + &c. + a- m x mr , and (a -1- /3y 4- 7 y* + ^ 3 4. &c. + cr/) m = ^f + By + Cz/ ? + Dy\ &c. + <r "" r ; and, the value of y being arbitrary, let y=x + a ; then, a + /3# + 7^4- Sx 3 H-&C. 4- 4- &c. 4- &c. + &c. /- ^ -f Bx 4- Co:* 4- Dx* + &c. 4- <r ft ^ wr -v ^ j 4- &c. 4- &c. + &c. f 4- a m z mr ; 3 that is, if S be put for and V% for the rest of the terms of the first SOS ON THE BINOMIAL THEOREM. member of the equation, all of which contain z as a factor, (S + V%) m = A -f Ex + Cx* -{- Dx* + &e. + &c. + &c. therefore, (Art. 13.) S m -f mS m ~ l Vz + c^"" 2 /^*2 a + &c. = A + Bx + Cx* + JQ + Cx a + &c. But S m = A+ Bx + Cx* + Dx* -f &c. therefore, m S m ~ ] Fz + c S m ~" V~ %? + &c. = C = 5 Bz &c. + &c. Every term of this last equation is divisible by z; let it, therefore, be divided by 2, and, B 4- 2 C# + Bjp& t +Ea? 4- &c. + C^ + &c. 4- &c. i + Dz * "Sii - ^^ Now the only part of T 7 ", which does not contain any power of z, is /3 + 27 < r-f3S ( r a + &c. H therefore, (Art. 9.) ON THE BINOMIAL THEOREM. 209 mS m (($ 4- 27*? 4- S.B + 2 Cx + . 4- x* + &c.) &c. 4- T(rx r ~ l ) 4 Ex* + &c.) 4- 4- c* H CO 4- H 4- * S 4- O 4- H CO 4- O 4- f c? 0.05 + 2 VK* .CO TF CO X + bJD O cq + + II ll :te&i ^ "J 8 gj Q^ ~ H K c^ r + Q o B a. CO 4- . ON THE BINOMIAL THEOREM, \Vherefore, (Art. 7.) ~ 2). 7 .C y -r- + 4me.A 4- (2m- 3). 7 .> S.C &c. = &c. + ((p-i)wi- I)TT. B + pmp.A, Q being the coefficient of the (p + l)* h term Now it is manifest, that A = a OT ; , - A_< ,> , -T &c. = &c, and, all the assumed coefficients having been thus investigated, the expansion of the polynomial is found. 20. COR. 1. If j3, S, &c. the coefficients of the alternate terms of a 4. 0,r -h 7 r 2 4- $tf 4- &c. be negative, it is manifest, from the equations last ON THE BINOMIAL THEOREM. 211 obtained, that the values of B, D, F, &c., the co efficients of the alternate terms in the expansion, will also be wholly negative. 21. COR. 2. The first (p + l) terms of the ex pansions of (o + fix + 7#* -j- &c. + p&) m , and of (a + fix + 7-*r + &c. + paf +<ra" +1 -I- &c.)> will be the same. For the same equations deter mine the coefficients Q, P, &c. in both cases. 22. COR. 3. If a, /3, 7> &c. be each unity, the equations which determine the coefficients of the expansion of (1 + x + ff + &c.) tt become J5 = m . A ~ m 2C = (m-l).B + 25 = (m+ l).S r*=O ~ 2) . C H- 4 C = (m +2) . C (Art. 17.) = (m- 3) . D + 6 D = (m + 3) . Z>, &c. = &c. 212 ON THE BINOMIAL THEOREM -. / fit! o^ffi \\iti 23. COR. 4. The expansion of (a +x)i is of the form a-+a*+ .- tt n v 2 in which the law of formation of the coefficients is the same as that in the expansion of (a + x) n . For, let a 4- /3o? + 7** + &c. + p& be the (p + 1) first terms of the expansion of j>+ o?) ; then, (Art. 10. Def. II.), I (a + fix +.*yX* + &C. 4* TTc^" 1 > pa*)* =: = 4- ^ 4- * 4- *+ &c. 4- ft.^ +1 + &c. 4- /^. And, comparing this with the equation which de termines the coefficients of the (p 4-1}- term of the expansion of a polynomial, d=a, jB=?l, C, Z>^ &c. P, Q, are each equal to nothing ; whence pax 0=0+0 + &c. +((/>- DW- I).TTX 1 +pnp.a . .a= (I -(p I).W but /5 and ?r are any consecutive coefficients of the i expansion of .(a 4-*)*; wherefore the law of the ON THE BINOMIAL THEOREM. formation of the coefficients is the same as that of the coefficients of (a + $) m , substituting -form i i (Art. 14.) ; and the expansion of (a + x)" is a~ l + 24. COR. 5. The expansion of (a + x)~* is the same as the series obtained by treating a + x ac cording to the rule for algebraically extracting the n th root of a compound quantity; that is, by taking a for the first term of the series, as being (1 x n a~ l ) , or a, from a + x ; dividing the first term of the remainder by na* , and making the quotient the second term of the series; which will, therefore, be x then subtracting from a + x the w th power of the terms already found, and repeating the same process. For the three first terms thus obtained being; o i a n + fix + 7 of, it follows from Article 21. that \a* + fix + 7 o:V =a 4- x -f 0+ Dtf 4- &c. ; 214 ON THE BINOMIAL THEOREM, and (Art. 19.) i 3 a n . D = (n 2) . /3 x -f (2 n ~ l) . 7 ; the next term of the series is as before. In the same manner, the remaining terms of the series obtained by this rule, may be shewn to be identical with those of the expansion r of (a + x)* . And thus, after/) + 1 terms have been found (1 a" + j3a? + yx* + &c. whence, JL I? a" . Q = ((p 1) . TZ - l) therefore, In the same manner it may be shewn that the ex- r pansion of ON THE BINOMIAL THEOREM. 215 (a-x) n is cf -~a* *#+- . Y- (a* * ^ - &c. w ra ( 2 ) 25. COR. 6. The expansion of ~-i m Yn~ t 2 -2 -a n *+-.) (a* a? 2 H-&c. 5 w w f 2 ) in which series the coefficients are formed accord ing to the same law as those of the expansion of (a -f #) w . For the expansion of (a -f #) is (Art. 10. and 23.) the product of !>-> - w fffs-J - C aW ^+&c.r x&c. n n 2 to m factors : but (Art. 1 6.) the product of . n n 2 ON THE BINOMIAL THEOREM. 7? ft 2 wherefore, also, by the same Article, * i a n + - a" j: -f &c. w, IS and C I i i_i 7 ia w + -a n x + &c. i /i s And it is manifest (from Art. 20. and 23.) that m the expansion of (a - #)" is the same, with the ex ception of its even terms having the negative, instead of the positive, sign. tn 26. COR. 7 If w= w. the expansion of (a 4- x)~ n , becomes a + x, or the w th root of (a + x) m ; for every term, after the two first terms of the series, then vanishes, and the two first terms become a+3E. Also, if S be put for the sum of any ON THE BINOMIAL THEOREM. number (p+l) of the first terms of the series, 2 x* + &c. and a, /3, 7, S, &c. be put for the first term, and the successive coefficients of the powers of x in m the expansion of (a-\- x)" , then, S . S . Sac. (ni) = a -}- PX -}- yx z -{- &c. + a m x mp ; . &c. to n factors, = i. e. (S. S. . . (n)) .(S.S... (n)) . &c. to m factors = (a+/3oH-7**+&c.)x (a+/3 < r + 7 < r : *-{-&c.) &c. (w), because (Euclid 16. 7.) m times w is equal to n times m. Wherefore (Art. 10. Def. II.) (a + x + * + * + &c. + Rtf + * +&c. + f f l x*P) m =s = (a + j3 a? + 7 a?* + &c. + q m x m P} n : m whence, the expansion of (a-f #)~* is a series, such that if any number (p+ 1) of its terms be raised to the n th power, that product is of the form (a -f x) m + O + + &c. + ma m ~ RxP + 1 + &c. For, (Art. 21.) the first;? + 1 terms of the ex pansion of (a + fix + yx* -f &C.) % will be the same, whether p -f 1 terms, or more, be m taken. The expansion of (a + #) is, therefore, the 218 ON THE BINOMIAL same as that obtained by the rule for extracting the n th root of a compound algebraical quantity. 27. COR. 8. The expansion of - m a m-\-\ b 2 w + 2 c g - a 2 a 3 * a" J in which series the coefficients are formed according to the same law as those of the expansion of ( + o?) w ; a being the first term, b the whole coefficient of x in the second, c that of x* in the third, &c. For, (Art. 10. Def. III.) 1 7/ , <> _ Q ^ (i+*)- "(i +*y Also, l 1 + 07 ~ But . & c . (to m factors) = 7-; rr. ; 1+071+07 (1+ #) w ^ * If b, c, d, e, &c. be the coefficients of the powers of ,r, in the expansion of (1 + x) m , and unity be actually divided by that expansion, it will be found that b =b d = c b- b c + d &c. = &c. Which equations contain the law of the formation of the co efficients in the expansion of (1 + -i)-" 1 , although it capnot readily be deduced from them. ON THE BINOMIAL THEOREM. 219 . i _ b x + C a - d a* + &c + Q ^ But (Art. 22.) being put for the coefficients of* 2 , a?, &c. ; /. (Art. 8.) V = B; c = C, d = Z>; &c. = &c. Wherefore^ the expansion of is 1 -mx + ,b x^-~ c a?+ &c. 2 3 ij? And, by substituting - for x in the equation last investigated, the expansion of (a+x)~ m may be shewn to be ma m + I I m + 2 c a~ m -- # H -- ~*. -^-^-^ . - a 2 a 3 a Also, since =1 1 + a?- the expansion of (a - x) m may in the same manner be shewn to be 220 ON THE BINOMIAL THEOREM. ?/i+l V ra + 2 c , a~" ^1 y>7 ff> I - _ yi* I _____ I*-* J_ /V"f* -p //i* a, -f t*. T . i/ ~r -v^ a 2 a 3 a 28. COR. 9. The expansion of (a x) n is a * m m hi 4- 2 qpS^V+.-S . .^T~ C ar 3 + &c. n 2 a 3 For (Art. 26.) when unity is divided by m-l , . m m "" X ^ + &c 2 the coefficients of the different powers of x in the quotient can be reduced to the form m -f 1 m 4- 1 m + 2 ,, ^ ? 2 2 3 "* And when unity is divided by m m I n , 4- a? + - . j - f a? &c., w n {, 2 J the coefficients of the several powers of x in the quotient will be the same as before, except that - must be written for m ; wherefore, these coefficients may be reduced to the form n n 29. COR. 10. Thus, whether the index m of (a+x) m be positive or negative, integral or frac tional, it may be shewn, that the coefficients of the ON THE BINOMIAL THEOREM. 221 expansion of (a + x) m are formed according to the same law ; and, that in no case of the expansion of a binomial, is the ratio of the latter of two conse cutive coefficients, to the former, greater than any that can be assigned : And, therefore, the value of x may be taken such, that the difference between any one term, and the sum of all that follow it, shall be less than any given finite quantity. For, let p and q be any two consecutive coeffi cients ; it has been demonstrated, in every case of the expansion of (a?\-x) m , that + n = m- (r - l) .p ^ .1 c <1 + m + (r l) therefore = -= - - - -; p r YYl __ f which quantity is always less than -^ - - ; and, if r be greater than m, it is less thari 2. Therefore the ratio of q to p can never be so great as that no ratio can be assigned which is equally great; and, therefore, (Art. 6\) the value of x may be taken such, as that any one term of the expansion shall become the limit of all the terms which follow it. . ON MAXIMA AND MINIMA. **^ PART n. ^*- * rf + rf^rff^-r* SECTION II. ON THE EQUATION WHICH SERVES TO DETERMINE THE VALUE OF ANY FUNCTION OF ONE OR MORE VARIABLE CtUANTITIESj WHEN IT IS A MAXIMUM OR A MINIMUM. > Ki ifbiiiv/ bain^e^- $d J5o oilt fim PREVIOUSLY to the estimation of continued quantity, it is necessary to make some hypothesis respecting the generation of variable magnitudes. BARROW enumerates eight different modes in which quantity may be supposed to be generated. Its increase and decrease by motion, which is the foundation of the doctrine of Fluxions, is readily conceived in a vague and general manner. But there is no inconsiderable difficulty in deducing, THE ALGEBRAIC INVESTIGATION, &C. 223 logically, from that primary notion, the rules of algebraic computation, without which mere theory is of little value. Motion implies velocity ; velocity requires the consideration of time ; and to any enquiry concerning the nature of time we are not yet enabled to return a much more satisfactory answer than that of AUGUSTIN, so often cited, " Si nemo qucerat, scio ; si quis interroget, nescio" All that seems necessary, in the first instance, in the place of the fluxional hypothesis, is to express, in as general a manner as is possible, the condition of a quantity being variable according to a certain law; i. e. of its admitting any variation whatever with respect to magnitude, so as to continue to be the same species of magnitude, as it was before that variation took place. If % denote a line vari able at pleasure, and z f any addition or diminution of which it is capable, then % % will denote the condition of that line : and if az m denote a sur face, or a solid, variable at pleasure, but always retaining its species, expressed in terms of a variable line z, a (z z ) m will be a general representative of its value. Of this kind is the notation adopted in the following section. There is little real difference, in the methods used by the ancient and modern authors, who have treated the investigation of Maxima and Minima algebraically. The reasoning of LAGRANGE has been principally followed, in the most important propositions belonging to this part of the subject. 224 THE ALGEBRAIC INVESTIGATION His rules of computation are the same with those of LEIBNITZ, and all the writers on Fluxions. It was the demonstration only of these rules, and not the rules themselves, which needed to be improved. Js^ Jon 9115 3 w ) n i i J 1o o ; 1 . ..fJiU fi I. 30. DEF. A quantity is said to be constant, when it is always of the same value throughout any calculation : A variable quantity is that which may have any value within certain limits : An arbitrary quantity is that of which the value may be supposed, in any calculation, either greater or less than any given finite : quantity : In the lan guage of Algebra, a function containing only one variable quantity, is called a Maximum, when the variable quantity can neither be increased nor diminished, without the value of the function being thereby made less than it was before: And it is called a Minimum, when the variable quantity can neither be increased nor diminished, without the value of the function being thereby made greater than it was before. 31. DEF. Variable quantities being denoted by the last letters of our alphabet, the same letters, with an accent placed above them, are used to express arbitrary quantities, by which those varia ble magnitudes are supposed to be increased or diminished. OF MAXIMA AND MINIMA. 225 Thus z * , , z + * express different suc cessive values of the variable quantity represented by #. 32. DEF. If, in any algebraic function, con taining one or more variable quantities, for each of the variable quantities be substituted the sum of that quantity and its arbitrary increment, and the function be then expanded, that part of the expan sion which contains the first powers only of the several increments, is called the Derivative, or First Derivative of the function ; that containing the second powers of the increments, multiplied by 1 . 2, is called the second Derivative ; and that con taining the third powers, multiplied by 1 . 2.3, i& called the third Derivative , and so on. Thus % is derivative of z ; 2 x x is the derivative ofV. PROP. I. 33. Problem. To find the derivative of any power of a variable quantity. Let z represent any variable quantity ; it is re- 4- m quired to find the derivative of z~. + ^ The expansion of (* + ) ~ * is (Art. 28.) = m + 2-1 x -z~ n z + &c. n And, because the second term is the only one P 226 THE ALGEBRAIC INVESTIGATION which contains the first power only of % , the deri vative of z~~ l is (Art. 32.) ~ n Hence the derivative of any power of a variable quantity is found by the following rule : Multiply the proposed power by a number which is equal to its index, and then diminish its index by unity, and multiply the result by the derivative of the variable quantity. It is manifest that the second derivative is ob tained from the first, the third from the second, and so on, by the same rule. 34. COR. 1. The derivative of the sum of any number of powers of variable quantities is the sum of the derivatives of the several powers. Let x n 4- w 4: &C. = F; then the derivative of V is \^ 35. COR. 2. The derivative of (a + bz cz 1 4- &c.) w is Let a 4- bz + c** 4 d& 4 &c. = x\ K (Art. 33.) OF MAXIMA AND MINIMA. 227 bz + 2czz f + f 3dz s z + &c. = x ; and the derivative of (a 4- bz + cz 3 + dz* + &c.) wl is equal to the derivative of x m ; i. e. (Art. 32.) to mx m ~ l x , or to PROP. II. 36. Problem. To find the derivative of the product of two or more powers of variable quan tities. 1. Let it be required to find the derivative of x m . z n . The product of (x -t- x ) m . (z + .z f ) n , or of 5T -f n . IS n . mx ~ l x +mx m ~ l . nz n ~ l . x . z /. (Art. 32.) the derivative of x m z n is " """ P 2 228 THE ALGEBRAIC INVESTIGATION 2. In the same manner the derivative of x m z n y f ma be shewn to be x m z n .py p ~ l .y + x m yP .nz n ~ l z + z n yP .mx m ~* x . And the derivative of two, or more, powers of variable quantities is found by this rule ; Multiply the derivative of each of the powers by the product of the rest, and take the sum of the resulting products. Or, take the derivative of the given product, upon the several suppositions that each of its factors is variable, whilst the rest are constant ; and add the results. 37. COR. The derivative of (a + bx + cx* + &c.)>* x (A + Bz + Cz~ + &c.) n is (a-f bx + cx z + &c.) m .(A+Bz + Cz*-\-&c.) n - 1 . (B + 2Cz + &c.) z + (A + Bz + Cz* + &c.) M . m(a + bx + ex* -f &c.) m ~ l . (b + 2cx + &c.) x (Art. 34.) and is, therefore, found by the same rule as that of x m z n . PROP. III. 38. Problem. To find the derivative of a fraction, the numerator and denominator of which are two variable quantities. gf* Let it be required to find the derivative of -j- . z" (x + v Y 1 The quotient of -t > or OF MAXIMA AND MINIMA. HT 229 m. I _*>-=* + &c. .^-!-^--^ + &c. is the same as the product of .*" * 2 + &c.^ 4- wo?*- 1 x + m . + n **-** + n . . z n ~*z" + &c. or of (7W 1 ^m + mx *n-l x > + m 1__; ^-2^ * + & C . \ (^- n - z-* n .nz n ~ l z + &c.),- (Art. 19.) ; which product is x m z~ n - jf. z-* n .nz n ~ l z + &c. -f a^.wia^" 1 ^; .-. (Art. 32.) the derivative of **#* x m is z~ n .mx m ~ l x - a?* 1 *"" 1 * *, w^;"" 1 z or ~* x x m nz n ~ l . z Hence the derivative of a fraction, the nume rator and denominator of which are powers of two different variable quantities, is found by this rule ; Multiply the derivative of the numerator by the denominator, and also the derivative of the deno minator by the numerator; subtract the latter product from the former, and divide the result by the square of the denominator. 30 THE ALGEBRAIC INVESTIGATION 39. COR. The derivative of a fraction of the f orm (a + ftg + cg +fec.)* is follnd by the same 1 (A + bx +CV+&c.) B rule. PROP. IV. 40. Theorem. If the function of a variable quantity be a maximum, or if it be a minimum, its.first derivative is equal to nothing. Let z m be any * function of a single variable quantity. The expansion oi m - I m , 2 m _% , . , and the expansion of.. 0/ it 67 hab fjflJ. (. - * 2 5 therefore (Art. 28.) t * The word function is used in the sense of the definition in Artir.; so that % may represent binomial or polynomial t quantities; such, for example, as*-a?% ax + x+~> &c. ; and is the derivative of those quantities; z m may likewise i fv \ a? represent (aa:-^ 4 ) 3 ; ^ * &c - OF MAXIMA AND MINIMA. 231 (2.) *", and (3.) *+(m*- z ) + ( m". - 2^- a /a ) + &c. represent any three consecutive values of the func tion : let V be the value of z m when it is a max imum, or when it is a minimum ; then, (Art. 30.), if V be a maximum, it must always be greater both than the first, and than the third of the three con secutive values; and if it be a minimum, it must always be less, both than the first and than the third of those values, however little z be, and whether it be positive or negative. Let z be taken (Art. 29.) such that m ,z m ~ l z f shall be greater than the sum of the succeeding terms, in both the expansions ; therefore the sum of all the terms but the first will then be positive or negative, accordingly as the second term mz m ~ l z f is positive or negative ; that is, accordingly as z f is positive or negative, for the sign of mz m ~ l is necessarily the same in both the expansions; whether, therefore, z f be positive or negative, + (mz m ~ i z f ) and - (mz m " l z f ) have, necessarily, contrary signs : the three conse cutive values, therefore, upon the supposition here made, will be of the form, (1.) V mz m - l z - S. (2.) V. (3.) V+ mz m ~*z S. $ being a quantity less than mz m ~ l z ; so that the first or third value of the function is greater or less 239 THE ALGEBRAIC INVESTIGATION than the second, accordingly as mz m ~~ l z has a positive or negative sign. It is manifest, therefore, that as long as mz m ~ l % has any magnitude, V can neither be greater both than the first and third value, nor less both than the first and third value ; that is, it can neither be a maximum, nor a mini mum, unless mz m ~ l z be equal to nothing: if, therefore, the function be either a maximum, or a minimum, its first derivative, mz m ~* z , is equal to nothing. 41. COR. 1. When the value of the function is a maximum, its second derivative is negative; and when it is a minimum, its second derivative is* positive. For, whether the function V be a maximum or a minimum,, its first derivative (Art. 40.) is equal to nothing ; therefore V is greater than when it is a maximum, and less than the sum of that same series, when it is a minimum, however little z be ; let z be taken (Art. 29.) such, that m . shall be greater than the sum of the succeeding terms of the expansion ; then, since z * is neces sarily positive, V is greater or less than OF MAXIMA AND MINIMA. 233 accordingly as is negative or positive ; if it be necessarily greater, that is, if it be a maximum, must be negative ; and if it is a minimum, m .^JL .-. " if If Ml* i 2 51 1? must be positive; therefore (Art. 32.) the second derivative must, in the former case, be negative, and it must be positive in the latter case. 42. COR. 2. But, if the second and third terms of the expansion be each equal to nothing, the value V 9 of the function, when it is a maximum, must be greater than and it must be less than the sum of that series, when it is a minimum : As before, let z be taken such, that m 1 m 2 .3/0 m. - . - s m - 3 s 3 2 3 shall be greater than the sum of all the succeeding terms of the expansion ; then, since the sign of 234 THE ALGEBRAIC INVESTIGATION z B depends upon that of 2 , and is the same with it, V cannot always be greater than the value of that series, nor always less than it, unless m _ i m 2 , m. . ^~z m -- 3 z 3 be equal to nothing; that is, (Art. 32.) unless its third derivative be equal to nothing. 43. COR. 3. In the same manner it may be shewn, if the n first derivatives of the function be each equal to nothing, n being an even number, that the \n + l) th derivative of the function must also be equal to nothing, if the function either be a maximum or a minimum ; and that if it be a maximum, the (n + 2) th derivative must be nega tive ; but that if it be a minimum, the (w-f2) th derivative must be positive. firj; 3 9.dt io 44. Problem. To determine the conditions under which the aggregate of several functions, each being the function of one variable quantity only, is a maximum or a minimum. Let XYZ&,c. be the aggregate of the functions, which is to be a maximum or a minimum, X being a function of x, Y of y, Z of ^ &c. 1. It is manifest that X+Y+Z 9 &c. all the terms being positive, will be greatest, when X, Y 9 Z, &c. are each a maximum, taken separately; OF MAXIMA AND MINIMA. 235 and least, when each of these quantities, taken separately, is a minimum. But, in determining the maximum value of the aggregate, care must be taken lest the minimum value of any one of the functions, taken sepa rately, be joined with the rest ; and, likewise, when the minimum value of the aggregate is sought, none of the maximum values of the several functions, taken separately, must be joined with the rest. 2. The binomial X JTis greatest, when X is a maximum and Y a minimum ; and it is least, when X is & minimum and Y a. maximum. 3. The trinomial X+YZ is greatest, when (X+Y) is a maximum arid Z a minimum; and it is least, when (X + Y) is a minimum and Z a maximum. Also, X Y- Z is greatest, when X is greatest and Y+Z least; and it is least, when X is least and Y+Z greatest. The solution of this case is, therefore, reduced to that of the two former cases ; and the same method is applicable, to discover when the aggre gate of any number of such functions is a max- mum or a minimum. The following example is given in order to illustrate the above proposition. The remaining examples will be taken so as that the determina tion of the maximum or minimum shall depend 236 THE ALGEBRAIC INVESTIGATION upon the solution either of a simple or of a quad ratic equation. EXAMPLE. Let it be required to determine when the aggre gate of the two functions x 3 - 3x* - 3 is a maximum. Gall the former function X, and the latter Z; the first derivative of X=3x*x - 6xx -3x , which (Art. 40.) must be equal to nothing ; there fore X* - 2X 1 = 0. / The two roots of this equation are 1 + V 2, and 1 - A/1 ; if 1 + \/~2 be substituted in (6 x - 6) . x \ the second derivative of X, the result -is positive ; therefore, (Art. 41.) when a: has this value, the function X is a minimum ; if the other root, I _ ^^ be substituted in the second derivative oiX, the result is negative ; this, therefore, must (Art. 41.) be the value of x, in order that the function X shall be a maximum. riii - Again, the first derivative of Z is jjj - 24* * + 362S - 8z ; OF MAXIMA AND MINIMA. 237 therefore, whether Z be a maximum, or a minimum, _ 6 X* + 9 z - 2 = O. The three roots of this equation are 2, 2+>/ 3, and 2 *J~3 ; and when 2 is substituted for z in the second derivative of Z, i. e. in (12s - 48Z + 36).* % the result is negative ; and it is positive if either of the other roots be substituted for z in the second derivative of Z; therefore (Art. 41.) 2 is the value of z 9 which renders the function Z a maximum. Therefore the two values of x and z, which must be taken, in order that X + Z may be a maximum, are 1 - *fli and 2 ; not 1 + A/~2, and 2 *J 3 which would make that aggregate a minimum ; nor yet 1 x/~2 and 2 >/ 3 ; nor 1 >/~2 and 2. If the proper values of x and s be substituted, the value of X + Z, when it is a maximum, will be found to be 3 + 4 v/T. SCHOLIUM. It appears, from Art. 40, that, whether the value of a function be greatest, or least, its first derivative is equal to nothing ; if, therefore, there be several values of the variable quantity of the function, which answer this condition, of rendering its derivative equal to nothing, some of them may be such as make the function a maximum, others such as make it a minimum : and it may S38 THE ALGEBRAIC INVESTIGATION be determined whether any particular value (a) of the variable quantity, thus found, belong to the greatest, or the least, value of the function, either by means of Art. 41, as in the preceding example, or by successively substituting in the function a - S, a and a + $, for the variable quantity : for then, if the first and last result be both less than the second, it is manifest that the value (a) renders the func tion a maximum ; if the first and last of these results be both greater than the second, (a) renders the function a minimum ; and if, of the first and. last results, the one be real and the other imaginary, the function may be at once a maximum and a minimum ; a maximum in one respect, and a mini mum in another. But if there be no value, of the variable quan tity, which will make the first derivative of the function equal to nothing ; or, in other words, the first derivative being equated with nothing, if the roots of that equation be all impossible, it is plain that the function does not admit either of a maxi mum or a minimum. r< It may also be determined whether any value (a) of the variable quantity, found according to Art. 40, render its function a maximum, or a minimum, by substituting for it a $ in the first derivative : for, this being done, if the first deri vative thus become positive, the whole variation of the function may be considered as positive, previously to its having the value, which is either greatest or least ; that value must, therefore, in this OF MAXIMA AND MINIMA. 239 case, be a maximum, since the function was gra dually increasing, before it arrived at that mag- nitude: on the contrary, a o being substituted for the variable quantity in the first derivative, if the result be negative, it is evident that (a) renders the function a minimum ; but if, when a $ and a -f-o have been severally put for the variable quan tity in the first derivative, the results have the same sign ; that is, the derivative being equated with nothing, if that equation have an even number of equal roots, each equal to (a), then (a) will not, properly speaking, give the function a value which is a maximum, or a minimum ; for, in that case, if the function be increasing before it had that value, it will manifestly go on increasing after having ac quired it ; and, if it be decreasing, it will go on decreasing. The same discovery might be made by the application of Art. 43. Thus it appears, that though the proposition stated in Art. 40, be true, its converse is not universally true. Further, the language of Algebra is not so perfect as always to express, clearly and com pletely, all the limitations of any proposed ques tion ; and the equation, derived from the proposi tions implied in a problem, may have a root which shall indicate a construction, or an operation, in consistent with the conditions given, and the grounds on which the reasoning was built : the solution, thus obtained, will not then be true ; although the root of the equation be real and 240 THE ALGEBRAIC INVESTIGATION positive. It is necessary, therefore, to ascertain whether the answer to a question respecting maxima or minima, and, generally, whether the result of any other application of Algebra to Geometry, be so circumstanced. This remark was made and exemplified by THOMAS SIMPSON, and it may be seen more fully explained by CARNOT, in his Geometry of Position. PROP. VI. 45. Theorem. When a function of two or more independent quantities is a maximum, or when it is a minimum, if its first derivatives be obtained, upon the several suppositions that each of the variable quantities, in its turn, is the only variable quantity in the function, each of those derivatives is equal to nothing. First, let V be a function of the two indepen dent variable quantities x and y ; and let P . x be the derivative of V 9 when y is constant, and Q . y the derivative of V^ when x is constant : if Q . ?/ be put equal to nothing, such a value of y will be obtained, in terms of x and known quantities, as will render /^ a maximum or a minimum (Art. 40.) in respect of y, whatever be the value of x ; since the two variable quantities x and y are indepen dent of each other; this value of y is, therefore, a function of x ; let its derivative be q . x ; substitute the value of y, thus obtained, in the given function, which will, therefore, become a function of x\ call OF MAXIMA AND MINIMA. NX that function X; wherefore, the given function is already a maximum or a minimum, as far as re gards y ; and will be absolutely a maximum, if X be a maximum ; and absolutely a minimum, if ^Tbe a minimum ; that is, if the first derivative of X be equal to nothing, and its second derivative be negative, V will be a maximum ; and if the first derivative of X be equal to nothing, and its second derivative be positive, V will be a minimum, (Art. 40, 41.) Now it is manifest, y being considered as a function of #, that the first derivative of X will be P .x + Q.qx ; which is, therefore, in the case either of a maximum or a minimum, equal to nothing ; and the part P x has been shewn to be equal to nothing; wherefore Q.qx , or Q.z/ is also equal to nothing, (2.) Let fiPbe a function of three independent variable quantities x, y, and z ; suppose z to be the only variable quantity, and thus find the value of s, in terms of x and ?/, and of known quantities; which will render the function a maximum or a minimum, as far as regards z ; when this value is substituted in the given function, it will become a function of x and y only; and, thus, this case will be reduced to the former. In the same manner, the proposition may be shewn to be true, when it respects a function of four, or more, independent variable quantities. The proposition may also be readily proved ex absurdo : for if any of the derivatives, taken upon THE ALGEBRAIC INVESTIGATION the supposition that only one quantity is variable, be not equal to nothing, it is evident that the function may be greater, if it is to be a maximum, or less, if it is to be a minimum, whatever may be the values of the other quantities. Or, by reasoning as in Art. 40, it may be proved, that whenever any given function of x, #, z 9 &c. is a maximum, or a minimum, the whole of its first derivative P.x + Q.tf + R.sf + &c must be equal to nothing ; but P.x + Q.y + R.z + &c. cannot necessarily be equal to nothing, whatever x 1 , if> and % be, and whatever signs belong to them, unless P . x , Q. y , R . z , &c. be separately equal to nothing : and (Art. 36.) the derivative p.x + Q.y + R.ar, &c. is the aggregate of the derivatives of the given function, taken upon the several suppositions de scribed in the proposition. ^l 46. COR. It may be shewn, as in Art. 41 , that when (V} any given function of the indepen dent variable quantities x, y, z, &c. is a maximum, the aggregate of the terms in which x r y , z 9 &c. have two dimensions, in the expansion resulting from the substitution of x + x , y + y, z 4- * , &c. in V, is negative ; and that it is positive, when V is a minimum. OF MAXIMA AND MINIMA. 243 SCHOLIUM. In the preceding proposition, the variable quan tities contained in the function, of which the greatest, or the least, value is sought, are supposed to be independent of each other : but if there exist any relations between them, expressed by one or more equations, as many of the variable quantities, as is possible, must be eliminated by means of those equations ; and then the greatest, or the least, value must be investigated, of the function containing the rest of the variable quan tities. Again, if the number of variable quantities, assumed in the calculation, exceed the number of the conditions of the question, it is evident that any functions, of each of these quantities, may be considered as constant ; provided that these arbi trary conditions, so introduced, together with the real conditions of the question, do not exceed in number the variable quantities. It may here, also, be remarked, that when the maximum, or minimum, value of a function is sought, it is not always necessary to take the derivative of the function in order to find that value. If an equation be obtained upon the sup^ position that the function is of a given magnitude, it will sometimes appear, from the form of the 244 THE ALGEBRAIC INVESTIGATION algebraical result, what is the greatest, or least, magnitude, which the function can have. The following Examples of the application of the method of investigating Maxima and Minima algebraically, are taken from the propositions of the First Part of this Treatise. EXAMPLE I. To find the least triangle which can be con tained by two straight lines, inclined to each other at a given afagle, and a third straight line, which passes through a given point between them. Let ^Fand ^Fbe two straight lines, inclined to each other at a given angle, and D a given point between them, through which the straight line FDG passes : the triangle AFG is required to be the minimum. OF MAXIMA AND MINIMA. 245 Let DE be drawn parallel to AY and let the. given line AE be denoted by a, the given line ED by b, and AF by x : Then, (E. 4. 6.) FE : ED :: FA : AG , i.e. x a : b :: x : AG. Wherefore AG = - . But it follows, from x a E. 23. 6, that triangles which have equal vertical angles have to one another the same ratio as the rectangles contained by their sides. Therefore, the triangle AFG is the least, when the rectangle AFxAG is the least ; so that the quantity - . x x ci is to be a minimum ; therefore, (Art. 40.), 2 x . x . (x a) x z . x ~i (x~af ~ =0; .. 2 or 2 ax *- a?* = O; i. e. x 2 2 ax = O ; as in Art. 3. Part I ; and the triangle = 4ab. In order to prove that this value of# renders the triangle the required minimum, let 2 $, 2 a, and 2 a + $ be successively substituted for x in *Vl^ ; and the results are x a o wherefore, the triangle is less when a? has the 246 THE ALGEBRAIC INVESTIGATION value 20, than when it has any other value either greater or less ; that is, it is then the least that it can be. The solution may also be obtained upon the principle mentioned at the conclusion of the last Scholium : for let it be required to make the tri angle equal to a given rectangle, represented by bx* 2 be ; the area of the triangle, as before, is ; bx* /. -- = 2 be; X a 2cx -f c 2 = c* 2<7C=:c 2 a . *c Whence, it is manifest that c cannot be less than 2 a ; otherwise the value of x becomes impossible ; and the triangle has the least N value, when c has the least value ; therefore, when the triangle is the required minimum, x = 2 a> and the triangle itself is equal to 4 a b y as before. ,\>^ : ~$ iol sautiriu -.y.- nujp9ir o;i- .f^nsrtf or!) EXAMPLE II. To find the greatest parallelogram which can be inscribed in a given triangle, so as to have the vertical angle of the triangle for one of its angles. Let AFG be the given triangle ; the parallelo- OF MAXIMA AND MINIMA. 247 gram AEDH, which is inscribed in it, and has the vertical angle A for one of its angles, is re quired to be the maximum. It follows from E. 23. 6, that equiangular parallelograms have to one another the same ratio as the rectangles contained by the sides about equal angles in each ; therefore the parallelogram AEDH will be the greatest when the rectangle EA x AH, or AE x ED, is the greatest. Let AF be denoted by a, FG by b, GA by c, and FD by x : Then, (E. 4. 6.) FG : GA :: FD : DE; i. e. b : c :: x : DE ; c .. DE r x ; and in the same manner it may be shewn, that 248 THE ALGEBRAIC INVESTIGATION Wherefore ex a j . F x I (b ~ x} is to be the maximum ; which will be greatest when x . (b - x) is greatest ; /. b . x 2x . x = 0; .-. x = - 3 as in Art. 4. Part I. 2i The problem may, also, be readily solved by supposing bx x 2 to be equal to a given rectangle, as was done in the preceding example, r & r EXAMPLE III. GiJ. /I . ulii-cJ SfU iv ijA,;. i.> JilO O* yV6Q jnfii ,;^ul9:I.; J>(| To find the greatest of all equiangular and isoperimetrical parallelograms. Let 2 b denote the common perimeter, and x one of the sides of any of the parallelograms ; there fore b x will denote the other side, and, as in the preceding example, the parallelogram will be greatest when x (b - x) is greatest ; it will, there fore, as in Example 2, be greatest, when # = - ; i. e. when the figure is equilateral, as in Art 6. Part I. COR. If the angle A^ of the parallelogram, be varied as well as the proportion of the sides, the parallelogram will be greatest (Art. 45.) when its sides are all equal, and when A is at the same time OF MAXIMA AND MINIMA. 249 greatest ; i. e. when the figure is a square : For, if S and S be the two sides about the angle A, the surface of the figure is expressed by S .S siri^f; and the sine of A is greatest when A is a right angle. But if the sides remain constant, and the angle, at which they are inclined to each other, be varied, it is manifest, from the expressed value of the parallelogram, that it is greatest, when it is a rectangle ; which agrees with Art. 6. Part L EXAMPLE 4. To find a point, in an indefinite straight line, from which,, if two straight lines be drawn to two given points, without the indefinite line and on the same side of it, their aggregate shall be a minimum. Let XFbe the indefinite straight line, and A, and B, two given points, without it \ a point P is to be found, in XT 9 such that AP + PB shall be a minimum. Let the perpendiculars drawn from A and B to 250 THE ALGEBRAIC INVESTIGATION XY be denoted by a and b, the portion of XY, included between them, by c, and CP by x ; then AP = rfsa* + # 2 , and J5P = <&* + (c - tf)* ; wherefore */a*-}-x 2 + +/b* + (c x)* is to be the minimum ; rp rrf (ft _^ /yi\ rg! * * \^ 0- I . ,6 " C X x a a + a?* "" 6* + (c - x .,.*{* * * r. f 9 . . (a + b) . a: = ac; . . a + b : a :: c : x; . . b : a :: c x : x, as in Art. 12. Part I. Br^ iiUocl fi | jiJfJoihiw ^jqloq ft<>v^ ; o^? EXAMPLE 5. Cl J \^ fl OJI fif/B-llJ <c.M f lu\Di\ To determine the greatest of all isoperi metrical triangles, which stand upon the same given base. Let a denote the given base, 2p the common perimeter, and x either of the sides of any of the triangles; then will 2p-a x denote the other side of that triangle. The triangle (Wbodhouse s Trig. p. 23.) is equal to OF MAXIMA AND MINIMA. 251 Jp. (p-a).(p-x) . (a+x-p), which isy therefore, to be the maximum ; and it will plainly be greatest when (p x) . (a -\rx-p) is greatest ; .-. (p - x) . x (a + x - p) . ^ = ; .-. 2p a - 2# = 0; 2p a 2 - wherefore, the triangle is greatest when it is iso sceles, as in Art. l6. Part I.; where this conclusion is deduced from another proposition, with which it is connected. It may, however^ be seen demon strated directly and geometrically, in two different manners, by Pappus and Legendre, EXAMPLE 6. -^ j * io*ii ji o nwBili To find the greatest of all isoperimetrical tri angles. It is evident, from the preceding example, that of whatever magnitude the base of a triangle of given perimeter may be, the triangle cannot be a maximum, unless it be isosceles ; let it, therefore, be required to find the greatest isosceles triangle of a given perimeter. Let 2p denote the given perimeter, and 2 x the base of an isosceles triangle of that perimeter ; O ry\ fiiL ^ O y * then will -^ , or p - x denote the side of the THE ALGEBRAIC INVESTIGATION triangle, and */(/? #)* &* tne perpendicular drawn from its vertex to the base ; therefore the triangle is equal to \/ (p xY x*.x, or to. 2p x . x } which is to be the maximum. f- . _ *" _ _ A* _ lf\ . /. f 1 - 2px 3= * Q, Tl 11 ! f^ . O // iX ^/ > . ,_/ rc l o r - 2 ^. , * il, - ~ , flljivi A \JL> - y rl ,8l .^iA ni c r ; therefore, when the triangle is greatest, it is equi lateral, as in Art. 25. Part I. :?. jl Or, the given perimeter being denoted, as before, by 2p, let x and y denote the two sides of any triangle of that perimeter, and z and s the two segments of the base made by the perpendicular drawn to it from the vertex ; z denoting that which is adjacent to the side denoted by x. then x+y + z + v=z2p, and \ (z + v) . *Jx z z z is to be the maximum ; and since therp are four unknown quantities, arid only two conditions be- longing to the question, two more conditions may be introduced. Let, therefore, ^x^-z 1 and its equal, the quantity ^/y* - v 2 , be supposed to be each of them invariable, whatever be the values of x and * , and of y and v, so that the derivatives of these quantities shall be equal to nothing. And, because J (z + v}.Jtf - a is to be a OF MAXIMA AND MINIMA. 253 maximum, and the derivative of *J x l -z l is equal to nothing, ( + V ) . J& - 55* = 0, (Art. 36. and 40.) .-. z + v f = O. Also, because cT + ?/ -f s +1; is equal to the in variable quantity 2p 9 .-. x + y + z + v - O ; but s + i/ has been shewn to be equal to nothing ; /. x + y = 0. x v Whence ,= -,. .Again, from the conditions Z 25 introduced, y . y - v . v x . x - z . z ^y* v* *J x* z 2 y . y 1 v . w = ; and a? . of . s = ; ..". z : v :: x : y. Wherefore (E. 3. 6.) the perpendicular bisects the vertical angle of the triangle, when the triangle is greatest; also the two triangles, into which it is divided by the perpendicular, have (E. 26. I.) their other sides and angles respectively equal. Hence the original equation becomes 2x -f 2z = 2p ; and z . x/or* z* is now to be the maximum ; or, since z =p - x, (p-x) . \/ 2pxp* is to be the maximum ; 254 THE ALGEBRAIC INVESTIGATION P and a? = -~ as before. EXAMPLE 7- Of all quadrilateral rectilineal figures, which have equal perimeters, to find the greatest. Let ABCD be a quadrilateral rectilineal figure of the given perimeter, the area of which is to be a maximum ; join B, D, and A, C\ then it is manifest, from Example 5, that the two sides AB, AD, and also the two sides CB, CD, must be equal to each other ; and, therefore, (E. 8. and 4. 1 .) AC must bisect BD, at right angles, in the point E. Let the perimeter be denoted by 2 a, AE by x, EB by y, and EC by z ; therefore (E. 47. 1.) AB -f BC= (a* OF MAXIMA AND MINIMA. 55 and the figure = (x + z).y t which is to be a maximum. Thus, there are three unknown quantities, and only two conditions implied in the problem ; there fore, either (0*+y )*> or(*+y 8 )*, may be supposed a constant quantity, and its derivative will, consequently, be equal to nothing. Whence (I.) *. therefore, V* .V y^ . v x .y - " 2- -f *.r/- VZL = Qi a? /. a?s -y tt a + a?-y 2 a? =0; .: (z + x) . zx = (3 + a?) . i/ 2 ; .-. z x = y\ and x : y :: y : S; wherefore (E. 6. 6.) the two triangles AEB, CEB are similar to each other, and the angle ABC is a right angle, as is also the angle ADC. Again, the two triangles ABC, ADC being (E. 8. 1.) equal to each other, the figure ABCD is the double of ABC, and is, therefore, equal to 3 256 THE ALGEBRAIC INVESTIGATION the rectangle BA x BC ; the two sides of which AB, EC are, together, equal to the given semi- perimeter a. The question is, therefore, reduced to the finding of the greatest rectangle contained by a given perimeter; but (Examples.) such a rectangle is greatest when its sides are equal. Wherefore the greatest quadrilateral rectilineal figure, contained by a given perimeter, is a square. SCHOLIUM. The same conclusion may be arrived at, if no two sides of the quadrilateral figure ABCD be supposed equal to each other, by drawing DG and CF perpendicular to AB, and CK parallel to it, and making AG, GD, GB } BF, CJFthe unknown quantities; the perimeter and area of the figure may then be expressed in terms of these quan tities ; the expression for the perimeter will be found to contain three radical quantities, which, as the number of unknown variable < quantities exceeds the number of conditions, implied in the problem, by three, may each be supposed a con stant quantity. Eliminating, upon this supposition, the several derivatives of the unknown quantities, it will first appear that AG = 0, and that, therefore, the angle DAB is a right angle; it will next be found that BF=0, and that, therefore, the angle ABC is also a right angle ; and lastly, as in the example, the area of the figure will be expressed OF MAXIMA AND MINIMA. 257 by AE x BC, AB + BC being tbe half of the given perimeter; whence, it is manifest, that when the area is greatest the figure is a square. And by the same method it may be shewn that, when the number of sides and the perimeter of any polygon are both given, the area is greatest when the figure is equilateral and equiangular. EXAMPLE 8. Of all triangles standing upon the same given base, and on the same side of it, and having their vertical angles equal and given, to find that which has the greatest perimeter. The converse of E. 21. 3. having been proved ex absurdo, it will be manifest that the locus of the summits of all the triangles, standing upon the given base, and having their vertical angles each equal to the same given angle, will be the arch of the segment of a given circle, described about any one of the triangles, of which the given base is the chord. Let the remainder of the circumference be bisected; let the straight line joining the point of bisection and the vertex of any one of the triangles, be denoted by z, and the chord of half the bisected arch by a ; also, let b denote the given base of the triangle, and x and y its two sides : Then x +y is to be a maximum. But (Simsons Euclid, Prop. D. Book VI.) R 258 THE ALGEBRAIC INVESTIGATION a(x + y) = bz; Whence, it is manifest, that (x+y) is greatest when z is greatest : but z is greatest when it passes through the center; that is, when the triangle is isosceles its perimeter is a maximum, as in Art. 69. Part I. The trigonometrical solution of the problem is as follows : Let ABC be anyone of the triangles, on the given base BC, and having its vertical angle A of the given magnitude : Let D = 180 - A y and let x = sin B ; a = sin D; b = cos D. Then, AC = 2x AB=2. sin C = 2 . sin (D - = 2 . And AB + AC, i. e. 2o: + 2 (a>/l -a?* &#) is to be -a maximum ; r;i 1 - b ~ i. e. tan | D = tan Whence (E. 32. 1.) the triangle is isosceles, when its perimeter is the greatest. OF MAXIMA AND MINIMA. 259 EXAMPLE. 9. Of all triangles inscribed in the same given circle, to find that which has the greatest peri meter. By the preceding example, the inscribed tri angle of the greatest perimeter is necessarily isosceles. Let, therefore, ARC be any isosceles triangle inscribed in the circle AD E and let AFbe drawn perpendicular to its base BC-, therefore (E. 26. 1.) AF bisects BC, and is, therefore, (E. 1. 3. Cor.) when produced, a diameter of the circle. Let 2r denote the diameter of the given circle, y the straight line AF. y the straight line BF or CF; then (E. 8.6.) R2 260 THE ALGEBRAIC INVESTIGATION y % = 2 rx a?* \ /. # = and ^fJ5* = 2r,r; .-. ^5 = (2r#) And 2^(1? -I- 2 . #, or ^5 H- /, i. e. (2 ra?) 2 + (2 r x is to be a maximum ; - r (2r,r- a-*i .-. 3 r = 2 07. Whence, and 2 5 F, or .BC = 2 (3r - wherefore, the triangle which has the greatest perimeter is equilateral, as in Art. 71. Part I. EXAMPLE 1O. Of all quadrilateral rectilineal figures inscribed in the same given circle, to find that which has the greatest perimeter. It is manifest, from Example 8, that at least OF MAXIMA AND MINIMA. 261 one of the diagonals of the greatest inscribed qua drilateral figure must divide it into two isosceles triangles; and, therefore, the other diagonal is (E. 8. and 4. 1. and E 1.3. Cor.) necessarily a diameter of the circle. Let a denote the diameter of the given circle, and x and y the two sides of the inscribed figure which are not supposed equal to each other; then (E.31.3. and 47. 1 .) a* + y* = a a , whence y = (a* #*); and 2x + 2y, or x +3/5 that is, x + (a 1 a; a )i is to be a maximum. , x .-of . . # - - | = ; (a 2 - a*)4 .-. (a 2 - a? 8 )i = J?; .-. a*-- a? = # a /. j?* = y~, and = y ; wherefore, the inscribed figure which has the greatest perimeter is a square, as in Art. 72. Part I. EXAMPLE 11. Of all triangles standing upon the same given base, and having their vertical angles each equal to the same given angle, to find the greatest. Let x and y denote the two sides of any one of the triangles, and a the tabular sine of the given vertical angle ; then the area of the triangle is equal to -^, which will manifestly he greatest 62 THE ALGEBRAIC INVESTIGATION when xy is greatest; but (Example 9.) the aggre gate of x and y is greatest, when the triangle is isosceles, i.e. when x=y ; and (Examples.) a square is greater than any other rectangle of equal perimeter; wherefore the rectangle xy cannot be greater, than it is when x=y ; and the greatest triangle standing on the given base, and having its vertical angle also given, is, therefore, that which is isosceles, as in Art. 73. Part I. EXAMPLE 12. To find the greatest of all triangles which can be inscribed in the same given circle. The greatest inscribed triangle is necessarily isosceles (Example 11.); Jet, therefore, APR be any isosceles triangle, inscribed in the given circle AEC\ and draw AM perpendicular to PR. Then, if the diameter be denoted by 2r, AM by 47, and PM by y, as in Example 9, y = OF MAXIMA AND MINIMA. 265 (2r.r ,r 2 )3; and the area of the triangle is the half of the rectangle contained by AM and PR, and is, therefore, equal to y x 9 or to (2rx- a? a )a. x, which is to be the maximum ; 3r = whence, as in Example 9. the maximum sought is an equilateral triangle, according to Art. 76. Part I. EXAMPLE 13. To find the greatest quadrilateral rectilineal figure which can be inscribed in a given circle. It is evident, from Example 11, that at least one of the diagonals, of the greatest inscribed quadrilateral figure, must divide it into two iso sceles triangles ; and, therefore, the other diagonal must be a diameter of the given circle, and must cut the former diagonal at right angles ; wherefore an inscribed quadrilateral figure, of this kind, is equal to the rectangle contained by the diameter of the circle and the other diagonal ; and it will be greatest when that diagonal is greatest ; that is, when the diagonal is equal to the diameter of the circle ; in which case (E. 6. 4.) the figure is a square, as in Art. 77. Part I. 264 THE ALGEBRAIC INVESTIGATION EXAMPLE 14. To find the greatest of all pentagons which can be inscribed in a given circle. Let ABCDE be a pentagon inscribed in the given circle AEDCB, of which the center is K; it may be shewn, by means of Example 11, that at least four sides of the figure must be equal to each other, when it is a maximum. For, first, if A, D and B, D be joined, the two sides AE and ED must (Example 11.) be equal to each other, and also the two sides JBC, CD ; join K, A and K, B, and K, C and K, D, and K, E ; then (E. 8. 1.) the two triangles EKA and EKD are equal, and also the two triangles BKC, DKC; again, if the arch EDC be bisected in jP, and E, F and F, C, and C, E and K, F be joined, the triangle EFC is greater (Example 11.) than the triangle EDC; and if the triangle EKC be OF MAXIMA AND MINIMA. 265 added to both, it is evident that the two equal triangles EKF, FKC are, together, greater than the two triangles EKD, DKC-, wherefore four times the triangle EKF, or FKC, is greater than the aggregate of the four triangles AKE, EKD, DKC, CKB-, but the arch EF, or FC, is a fourth part of the whole arch AEFDCB ; wherefore, the side AB remaining the same, the inscribed pen tagon is greatest when its four remaining sides are equal to each other. Let the pentagon, therefore, be supposed to be made up of four equal triangles and the triangle AKB ; and let a denote the vertical angle of each of the equal triangles, x its sine; b the vertical angle AKB, y its sine, and r the radius KA - 9 then 4 a and b will have equal sines, but if the one sine be positive, the other will be negative; also the area of the pentagon is equal to 4 r^x + r*?/, which is to be a maximum; or 4x^sin4a is to be a maximum. Sin 4 a = (4x 8# 3 ) . (1 - (Woodhouse s Trig. p. 4.1 .) .-. x (x - 2 a? 3 ) . (1 is to be a maximum; therefore (1 - 6* ) . (1 - x") -** + 20;* 1 - 8 . x z . (1 - x 1 ) ; 66 THE ALGEBRAIC INVESTIGATION /. 8.ff*.(l -s*)=rl - (1 - X i.e. 8 . sin 2 a . cos* a = 1 cos a. But sin a . cos a = ~ sin 2 a, (Woodhouse s Trig. p. 41.) and 1 cos a = 2 . sin* ( ) > (Woodhouse s Trig. p. 13.) .*. 2 sin* 2 a = 2 sin* - ; . a .\ sm 2 a = sm ~; 2 therefore the two arches 2 a and - must be sup- 2 a plements to each other, and 2 a + - = 180; r; / no i.e. = 180; 36o 5 but 4 a + b = 360 : .-.= .-. b = , and the pentagon is equilateral. In the same manner it might be shewn, that the perimeter of the equilateral pentagon inscribed in a circle, is greater than that of any other pentagon inscribed in the same circle. OF MAXIMA AND MINIMA. 267 SCHOLIUM. It is manifest that the process, by which the preceding problems have been solved, is the same as that which would have been employed, if they had been treated fluxionally. In reality, from the principles laid down in Sect. I. and II. Part II, all the rules of the Method of the Fluxions have been deduced by Lagrange : and by the help of one of the elementary propositions of that method, the solutions of some of the problems, which have here been proposed as examples, may be shortened. In all treatises on the Method of Fluxions, or the Differential Method, it is shewn, that if A be a circular arch, of which r is the radius, s the sine, and c the cosine, A S A = r . . c Hence, in Example 8. if a circle be supposed to be described about the triangle, which, having a given base and a given vertical angle, is to have its peri meter a maximum, and if A be the given angle, and B and C the two angles at the base, the sides will be the doubles of the sines of the opposite angles ; .*. sin J5+sin C, i.e. sin U-fsin (A + J5), is to be a maximum ; 268 THE ALGEBRAIC INVESTIGATION /, cos B cos (A + B) .-. B -- 1? x - * - f = o*; r r .*. cos J5 cos C = o ; .-. B = C; that is, the triangle is isosceles. In the same manner, in Example 9, where the perimeter of the triangle inscribed in a given circle is to be a maximum, if A, B and C be the three angles of the triangle, it may be shewn, since the triangle must necessarily be isosceles, that 2 . sin B + sin A, i. e. 2 sin jB+sin (180- 2JB) is to be a maximum ; /. 2B COSjB - 2J5 COS (180 - 2J3) = 0; . . cos B = cos (180 - 2B) ; .-. J5= 180 - 2B; 60*. o From which it is plain, that the triangle must be equilateral. Lastly, in Example 14, since it is there shewn that 4 sin a ^ sin 4a is to be a maximum ; therefore, 4 a cos. a **> 4 a cos 4 a = ; .*. cos a = cos 4a; * The cosines of B and A + jB have necessarily contrary signs. OF MAXIMA AND MINIMA. 269 .-. a + 4 a = 36o ; 360 * * 5 Whence it is concluded, as before, that the greatest of all pentagons, which can be inscribed in a given circle is equilateral. EXAMPLE 15. To draw the shortest tangent to a given circular arch, which shall be terminated by the semi- diameters, produced, that pass through the ex tremities of the arch. Let A be the given arch : then, it is manifest that any tangent, terminated by the two semi-diameters produced, is divided by the point of its contact into two segments, whicb are the trigonometrical tangents of the parts P, and Q, into which that same point divides the given arch. If, therefore, a be the trigonometrical tangent of A, x of P, and if the radius of the circle be denoted by unity, then (Woodhvuse, p. 3O.) is the trigonometrical tangent of Q ; and is to be a minimum ; .-. is to be a mini- ax + 1 mum; a#V - ax = o ; 70 THE ALGEBRAIC INVESTIGATION /. ax* -f- 2X a = 0; CL T 1 ~ 1 4- cr that is, the trigonometrical tangent of P is equal to the trigonometrical tangent of Q, or (as in Art. 103.) the given arch A is bisected, when the tangent required to be drawn is to be a minimum. EXAMPLE 16. If two straight lines touch a given circle, to draw the shortest straight line which can be termi nated by them and touch the given circle. If the two given tangents be parallel to each other, the angles which they make, with the tangent which is to be a minimum, are, together, (.29. 1.) equal to two right angles; and they are equal (E. 32. 1.) to the supplement of a given angle, if the two given tangents meet each other ; the aggregate of these angles is, therefore, in both cases, equal to a given angle; and consequently, (E. 32. 1.) the angle, subtended by the tangent, which is to be a minimum, at the center of the circle is also a given angle. So that the problem is reduced to that which was solved in Example 15 : and the tangent, required to be drawn is a mini mum, when as in Art. 56, it is bisected in the point of its contact. If, therefore, *AI and AL be the two given * See the figure in p. 79. OF MAXIMA AND MINIMA. tangents, touching the given circle BFC, the shortest tangent IL makes with them an isosceles triangle : And, if PO be any other tangent termi nated by the given tangents, since the perimeter of the triangle APO is the double of PO and of the given line AB, taken together, the perimeter is manifestly least when PO is least ; that is, when the triangle is isosceles. In the same manner it may be shewn, that the quadrilateral figure contained by two given tan gents to a circle and two other straight lines touching the circle, in two points lying between the given tangents, has the least perimeter, when those two straight lines are bisected, each in its point of contact. SCHOLIUM. The problem, solved in the above example, is evidently reduced to the dividing of a given cir cular arch into two such parts, that the aggre gate of their tangents shall be a minimum ; it has appeared that the given arch must, in that case, be bisected : and if it were shewn that when a circle is divided into 2 n equal parts, the aggre gate of the tangents of the parts is greater, than when the circle is divided into 2n parts which are not all equal ; it might thence be concluded, that a regular polygon, described about a given circle, has a less perimeter than any other polygon of THE ALGEBRAIC INVESTIGATION, &C. the same number of sides, described about the same circle, which is not equilateral and equi angular. It is manifest, at once, from the above conclusion, that when the circle is divided into pairs of equal arches, the aggregate of all the tangents of the parts may be made less, by bisect ing the aggregate of any two contiguous arches, which are not equal : and, by following the method of reasoning indicated in Art. 28. Part I. it may be shewn, that the several divisions which must be made, in order to render the aggregate of the tangents less and less, will bring the arches nearer and nearer to a state of equality. A proof of the proposition, which asserts that a regular polygon, inscribed in a circle, has a greater perimeter than another polygon of the same number of sides, inscribed in the same circle, which is not regular, may be deduced, in a similar manner, from the conclusion, that when a given circular arch is bisected, the aggregate of the sines of its part? is a maximum. ON MAXIMA AND MINIMA PART III. ON THE STRUCTURE OF THE CELLS OF BEES. XT has been asserted, that to have read the celebrated work of Cervantes, in the original, is an ample compensation for the labour of learning the Spanish tongue ; and the peculiar excellence, in its kind, of that inimitable performance may seem to justify the assertion* But if the pleasure arising from the perusal of a single work of fiction can be thought to counterbalance the pains of ac quiring a foreign language, it may with greater justice and seriousness be maintained, that the time and study consumed in acquiring a competent knowledge of the Mathematics, are more than re paid by the satisfaction to be derived from the speculations upon which we are now to enter. They terminate in conclusions so astonishing, so curious, 8 274 ON THE STRUCTURE OF THE CELLS OF BEES. and so strongly evincing the operative wisdom of the Creator of the world, that even the Grecian mathematician, when he introduces the subject of the (Economy of the Bee, seems to forget the grave and simple style of his science in his admi ration of the geometry of Providence. But the light, with which we have been in dulged by a gracious Revelation, whilst it enables us more clearly and more certainly to assign an effect like this to its true Author, has not rendered useless the contemplation of such an instance of divine contrivance. On the contrary, we shall do well, from the same source with PAPPUS, to im prove our reverence for the Being, who has endued an insect with the power of working such wonders, and to share the gratitude, which he expresses, for having been enabled to comprehend them. Natural religion, when it is rightly cultivated, will be found most powerfully to administer to our faith and to our devotion. May it not, indeed, be questioned, whether an unlettered, person, who is unacquainted with such instances of the divine contrivance as are exhibited in the mechanism of an eye, the structure of a honey-comb, or the laws of gravitation, and the planetary motions, can have a veneration for the Deity, of the same kind and degree, as that of the Christian philo sopher who has diligently studied Nature ? There is, undoubtedly, enough that is obvious, in the visible world, to inspire the least cultivated, and ON THE STRUCTURE OF THE CELLS OF BEES. 275 the least acute, of understandings, with sufficient awe of our common Maker ; nor can it be disputed, that the smallest portion of genuine religious prin ciple is infinitely more valuable than the most accurate and extensive knowledge is, without it. Still, if we mistake not, there is a peculiar sense of the character of God, which flashes upon the mind, from the consideration of such instances of his wisdom as those which have been alleged ; it is the present reward of a successful enquiry into his works, and, perhaps, the fore-taste of some of those intellectual enjoyments, which will constitute the happiness of our future state. The observation has already been made, that that part of the Collections of Pappus, in which he treats of the Cells of Bees, is now imperfect. The principal proposition, which was wanting, was demonstrated, in a synthetic form, by MACLAURIN ; it is here reprinted, nearly as he gave it in the Philosophical Transactions ; and an anlytical solu tion of it is subjoined. Several subordinate steps have, also, been supplied, which are not to be found either in Pappus, or in Maclaurin ; but which lead, by an easier and less interrupted ascent, to the great object of the investigation. The two methods of mathematical reasoning, which, in the former parts of this Work, have been carefully kept distinct, are here applied, in some measure, conjointly. But this mixt use of them may be said, if it need any apology, to be called S 2 276 ON THE STRUCTURE OF THE CELLS OF BEES. for by the subject ; and it does not at all interfere with the main design of this treatise ; which is to compare, in the instances adduced, the respective merits of Algebra and Geometry, in the investi gation of Maxima and Minima. -Hto* iriirf** -W)->^Wa*iiii ow ii l DEFINITIONS. 1 . A Prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another ; and the others parallelograms: And, if the sides of a prism be perpendicular to the plane of its base, it is called a Right Prism. 2. The Axis of a right prism, of which the base is a regular polygon, is a straight line drawn through the center of that polygon, perpendicular to the plane of its base. 3. A Parallelepiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel : and a Rectangular Parallelepiped is one which is contained by rectangles, having their planes perpendicular to each other. PEOP. I. 4. Theorem. If, from the angular points of a given polygon, equal straight lines be drawn, all ON THE STRUCTURE OF THE CELLS OF BEES. 277 of them perpendicular to its plane, and their extremities be joined, the planes bounded by them will contain a right prism, of which the given, polygon is the base. Let ABCD be the given polygon ; from the H ^ J) angular points A, B, C, D, let there be drawn (E. 12. 11.) the equal straight lines AE, BF, CG, DH, each perpendicular to the plane AC-, and let the points E, F, and F, G, and G, H, and //, E be joined by the straight lines EF 9 FG, GH 3 and HE ; the solid figure AG is a right prism. For (E. 6. 11.) AE and BF being equal and parallel straight lines, EF (E. 33. 1.) is equal and parallel to AB; therefore AEFB is a parallelo gram, and (E. 18. 11.) it is perpendicular to the plane AC. In the same manner BFCG, DHGC, and DHEA may each be shewn to be a paral lelogram, and to be perpendicular to the plane AC. Join E, G and A, C; then (E. 33. 1.) EG is parallel to AC, and the plane EFG is parallel to the plane AC (E. 15. 11.); for the same reason the plane EHG is also parallel to AC; wherefore 278 ON THE STRUCTURE OF THE CELLS OF BEES. (E. 14. and 4. 11.) EFG and EHG are in the same plane, and this plane is parallel to ABCD ; therefore (Art. 1. Part III.) AG is a prism ; and because the planes AF> BG, DG, and DE are each perpendicular to AC, it is a right prism. PROP. II. 5. Theorem. If a right prism and a rectan gular parallelepiped stand upon equal bases, and be of the same altitude, they are equal to one another. First, let the base ABC Q^ the prism be a tri angle, and let P be the equal base of the parallele- A H 2> M piped of the same altitude as the prism. The two solid figures are equal. f For, through Cdraw (E. 32. and 12. 1.) ECF parallel, and CD perpendicular, to AB. Bisect (E. 10. 1.) AB in H, and through A, H, and B draw (E. 11. 1.) AE, HG, and BF each perpen dicular to AB: then are ABFE, AHGE, and HBFG rectangular parallelograms ; and (E. 36. 1.) the parallelograms AG and HF are equal ; where fore AG is the half of AF, and (E. 41. 1.) the triangle ACB is also the half of AF\ therefore ON THE STRUCTURE OF THE CELLS OF BEES. 279 AG is equal to the triangle ACE ; but, by the hypothesis, the triangle ACE is equal to the rect angle P ; therefore AG is equal to P\ and it is manifest, (Art. 4. Part III,) that the prism standing on ACB may be divided into two prisms standing on the bases ADC and BDC respectively j and since (E. 34. 1.) the triangle AEC is equal to the triangle ADC, and the triangle CDB to the triangle CFB, therefore the two prisms standing on ADC and BDC will be equal to two right prisms, of the same altitude, standing on AEC and BFC, each to each : wherefore the prism standing on ACB will be the half of the prism, or parallelepi ped, of which AEFB is the base, and, therefore, will be equal to the parallelepiped standing on AHGE; but (E. 31. 11.) the parallelepiped standing on AHGE is equal to that of the same altitude, of which P is the base ; therefore the prism standing on ACB is equal to the parallelepiped standing on P. But if the base ABCDE of the prism be a G F K M polygon, let FGHI be the equal base of the parallelepiped. Join A, C and A, D ; to the 280 ON THE STRUCTURE OF THE CELLS OF BEES. straight line FG apply (E. 44. 1.) the rectangle FOLK equal to the triangle ABC; and to LK apply the rectangle KLNM equal to the triangle ACD ; then, since the two whole figures are equal, and the part FGNM, of the one, has been made equal to the part ABCD, of the other, the remainder MNHI must be equal to the remainder ADE. Again, (Art. 4. Part III.) the prism standing on ABCDE may be divided into prisms, of which the triangles ABC, ACD 9 ADE, are the respective bases ; and the parallelepiped standing on FGHI may be likewise divided into similar solid figures, of which FL, KN, and MH, are the bases ; and it has been proved that the prisms standing on ABC, ACD, and ADE are respectively equal to the parallelepipeds standing on FL, KN, and MH; therefore the whole prism standing on ABCDE is equal to the parallelepiped standing on FGHL j . w rr t *t KJ y r f j to 3. Cl . > ft K y&> d si I J _ i I i u fl PROP. III. 6. Problem. To determine the regular poly gons, which, by juxta-position, may fill space, about a given point ; all of them being situated in the same plane. Let x represent the number of sides of any of the regular polygons, which, placed in contact, may fill space, about a given point; then, since ON THE STRUCTURE OF THE CELLS OF BEES. 281 (Art. 22. Part I.) the angles of a regular polygon are all equal, if A be put for a right angle, one of the angles of that figure will (E.32. 1. Cor. 1.) be equal to 2x.A - 4. A , x - 2 . or to 2 A . x x which quantity, in order that the polygons may fill space about a given point, must be a divisor 2 x of 4 A 5 that is, - must be a whole number : X 2 wherefore - must, also, be a whole number; or f> ,__ Q r tODffjiJui onifig Oiij to c eui: .i iq -J. igrc # 2 must be a divisor of 4. But the only divisors of 4 are 1, 2, and 4 ; therefore x 2 is equal to 1, or 2, or 4; and a? must, therefore, be equal to 3, 4, or 6 ; i. e. the regular polygons, which may fill space, about a given point, must either have three, four, or six sides ; they must, therefore, be either equilateral triangles, squares, or regular hexagons. frn^filijjp3 i^rftia DVfiff as PROP. IV. r D * 7. Theorem. If two right prisms of the same altitude have for bases two equal regular polygons, that, of which the base has the greater number of sides, will have less superficies. 282 ON THE STRUCTURE OF THE CELLS OF BEES. For, the lateral surface of a right prism standing upon a regular polygon, consists (Art. 4. Part III.) of as many equal rectangles, each of the same altitude with the prism, as its base has sides ; it is, therefore, (E. 1. 2.) equal to a rectangle of that same altitude, and the base of which is equal to the perimeter of the base of the prism ; but (Art. 39. Part I.) of two equal regular polygons, the peri meter of that is the less, which has the greater number of sides ; wherefore, the lateral surface of the right prism, which stands upon the regular polygon having the greater number of sides, is the less : and the surface of the two ends is, by the hypothesis, equal in both ; therefore, if two right prisms, of the same altitude, stand upon equal regular polygons, that, of which the base has the greater number of sides, will have the less surface. 8. COR. Hence, of all equal prismatic cells, of the same depth, which may fill space, about a given straight line, that of which the base is a regular hexagon, has the least surface. For (Art. 6. Part III.) the only kinds of cells, which may fill space, about a given straight line, are such as have either equilateral triangles, squares, or regular hexagons for their bases ; and (Art. 5. Part III.) if their bases and altitudes be equal, each to each, the cell will be equally capacious ; but (Art. 7- Part III.) that, which is hexagonal, will have the least surface. ON THE STRUCTURE OF THE CELLS OF BEES. 283 PROP. V. 9. Theorem. The diameters of a rhombus bisect each other at right angles. Let ACBK be a rhombus, and AB, CK its diameters ; AB and CK are each bisected at right angles, in their point of intersection /. For (E. Def. 32. 1.) AK is equal to AC, AB is common to the two triangles CAB, KAB, and the base KB is equal to the base CB ; wherefore (E. 8. 1.) the angle KAB is equal to the angle CAB. Again, because AK is equal to AC, and AI common to the two triangles A1K, AIC, and that the angle KA1 has been shewn to be equal to the angle CAT, therefore (E. 4. l.) KI is equal to 1C, and the angle AIK to the adjacent angle AIC; wherefore, KC is bisected at right angles 284 ON THE STRUCTURE OF THE CELLS OF BEES. in /: And in the same manner AB may be shewn to be bisected at right angles in /. Therefore the two diameters AB and KC bisect each other at right angles. 1 0. COR. 1 . If BKC be an equilateral triangle, that is, (E. 15. 4.) if the angle ACB be one of the angles of a regular hexagon, CB is the double of CL For, KC (Art 9. Part III.) is the double of C7, and CB is, by the hypothesis, equal to KC. 11. COR. 2. The same supposition being made, as in the preceding corollary, if IS be made equal to the side of the square of which 1C is the dia meter, and if C, be joined, IS : SC :: CB : BA. For, (E. 47. 1.) the square of CS is equal to three times the square of IS, and the square of IB is equal to three times the square of /C; therefore the square of IS is to the square of SC as the square of 1C to the square of IB ; and, therefore, (E. 22. 6.) IS : SC :: 1C : IB X,. :: CB : BA (E. 15. and 11. 5.) ; for CB is the double of 1C (Art. 10. Part III.), and BA (Art. 9. Part III.) is the double of IB. 12. COR. 3. The same supposition and con struction being made, as in the preceding corol laries, if CL be drawn (E. 11. 1.) perpendicular to BC, and made equal to IS, that is, if LC be to ON THE STRUCTURE OF THE CELLS OF BEES. 285 1C as the side of a square is to its diameter, then, if B 3 L be joined, EL : CL :: 3 : 1. For, by the hypothesis, the square of 1C is the double of the square of IS or CL ; wherefore the square of CB, which (Art. 10. Part III. and E. 4. 2.) is four times as great as the square of 1C, is eight times as great as the square of LC ; therefore (E. 47. 1.) the square of BL is equal to nine times the square of CL; and (E. 5. Def. 5.) the square of BL is to the square of CL as 9 to 1 ; but (E. 20. 6. Cor 1.) this is the duplicate ratio of the respective sides of the two squares ; therefore, BL : CL :: 3 : 1. PROP. VI. 13. Theorem. If a right prism, bounded by A regular hexagon at each end, be cut by three planes, forming a solid angle at any point of its axis, and each of them passing through two alter nate angles of the hexagon, the capacity of the solid figure, thus formed, shall be equal to that of the prism. Let AC 9 CB be any two adjacent sides of the hexagon, which bounds the right prism AH to ward one of its ends ; let K be the center of that hexagon, and KM the axis of the figure ; let P be any point in the axis, and PAQB one of the 286 ON THE STRUCTURE OF THE CELLS OF BEES. planes^ which form a solid angle at P ; and let the plane PAQ.B pass through A and B. Join A, E and C 9 K, and let AB, CK cut x^ \ each other in /. Then (E. 15. 4.) ACBK is a rhombus, and (Art. 9. Part III.) the diameters AB, CK bisect each other at right angles in /. In the plane PQ, join P, I, and Q, 1. Then, be cause AK is equal to KB, and (E. Def. 3. 1 1.) PK is perpendicular to AK and KB, AP (E. 4. 1.) is equal to PB. Again, because AI is equal (Art. 9. Part III.) to IB, and AP has been proved to be equal to PB, and that P/is common to the two tri- ON THE STRUCTURE OF THE CELLS OF BEES. 287 angles AIP and BIP, the angle AIP (E. 8. 1.) is equal to the angle PIB, and, therefore, each of them is a right angle. In the same manner AQ, may be shewn to be equal to QB, and QIA, QIB to be right angles ; wherefore (E. 14. 1.) PIQ is a straight line. And, because in the two triangles PIK 9 C7Q, the angle PIK is equal (E. 15. 1.) to the angle C /Q, and the angles at K and C are right angles, and that CI is equal to IK, therefore (E. 26. 1.) PK is equal to CQ, and PI is equal to /Q; wherefore, also, (E. 4. 1.) AQ is equal to AP\ and the figure APBQ is a rhombus, where- ever P be taken in the axis KM. Lastly, because the triangle BKA is equal (E. 34. 1.) to the triangle BCA, and P^has been proved to be equal to CQ, therefore (E. 5. 12.) the solid figure QABC f which is cut off from the prism, is equal to the solid figure PABK; and, therefore, the whole of what is cut off from the prism, by the three rhombs, which form the solid angle at P, is equal to the solid space, which they add by their junction ; and the capacity of the solid so formed is equal to that of the prism. PROP. VII. 14. Problem. To determine when the solid, described in the sixth proposition, has the least superficies. Let AH be the original prism, and, the con- 288 ON THE STRUCTURE OF THE CELLS OF BEES. struction remaining as it is described in the pre- ceding article, in CF, the common section of any at the rectangles which bound the prism, take CL to CI as the side of a square is to its diameter; and let AMBL be a rhomb, passing through A and B, and cutting CFin L; then, AMBL is one. of the rhombs which contain the solid angle, when the surface of the figure is a minimum. For, join L, /; let APBQ be any other rhomb passing through A and B ; and, first, let it meet CF, in a point Q, which lies between C and L. From Q draw (E. 12. 1.) QO perpendicular to ON THE STRUCTURE OF THE CELLS OF BEES. 289 IL. The triangles LOQ and LCI having a right angle in each, and another angle common, are similar ; therefore (E. 4. 6.) LO : LQ :: LC : LI, but, by the construction, and (Art. 11. Part III.) LC : LI ,: CB : BA ; therefore (E. 11. 5.) LO : LQ :: CB : BA, and (E. 16. 6.) the rectangle contained by AB and LO is equal to that contained by LQ and CB. But (E. 34. and41.-l. and Art. 9. Part III.) the rhomb ML is equal to the rectangle contained by AB, LI, and the rhomb PQ is equal to the rectangle AB, Q/ ; and (E. 17. 1.) the angle CQI is less than a right angle ; wherefore (E. 13. 1.) the angle IQL is greater than a right angle, and (E. 19. 1.) IL is greater than IQ ; and, therefore, the rhomb ML is greater than the rhomb PQ, and exceeds it by the difference of the two rect angles AB, LI, and AB, IQ ; i. e. by a rectangle which has AB for its altitude, and the excess of IL above IQ for its base; and, because (E. 19. 1.) IQ is greater than IO, LO is greater than the excess of IL above IQ ; wherefore, the rhomb ML exceeds the rhomb PQ, by a rectangle which is less than that contained by AB and LO. Again, the excess of the two trapeziums AEFQ, FQBG above the two trapeziums AEFL, FLBG }s equal to the aggregate of the two equal tri- 290 ON THE STRUCTURE OF THE CELLS OF BEES. angles SQL, AQL, or to the rectangle (E. 41. 1.) contained by CB and LQ, which has been shewn to be equal to the rectangle AB 9 LO ; therefore the excess of the two trapeziums AEFQ, FQBO above the two AEFL, FLBG is greater than that of the rhomb ML above the rhomb PQ ; therefore the rhomb ML, together with the two trapeziums LE, LG, is less than the rhomb PQ, together with the two trapeziums QE, QG. If, now, a similar construction be made at D y and at the angle opposite to C, the surface of the solid figure thus formed, may be shewn to be a mini mum. And the same mode of proof is applicable, if the point Q lie between L and F. 15. COR. 1. In the solid figure, thus deter mined, bounded by three equal rhombs and six equal trapeziums, the obtuse angle of the rhomb is equal to the obtuse angle of the trapezium ; the acute angle, also, of the one, is equal to the acute angle of the other; and the three plane angles^ which form any of the solid angles of the .figure, are equal to each other. For, let ALBM be one of the equal rhombs j bisect MI in R, and join AR ; wherefore (Art. 9. Part III.) LI is the double of IR; also, from the construction, LC : CB :: LI : AB :: IR : IA (E. 15. 5.) because LI is the double of IR, and AB is the double of IA (Art. 9. Part III.) j wherefore ** ON THE STRUCTURE OF THE CELLS OF BEES. LC : CB :: IR : IA, and the angles LCB, AIR, are right angles; ^_ *M s* therefore (E. 6. 6.) the triangles BCL, ARI are similar; and (E. 4. 6.) AR : RI :: EL : CL ; but (Art. 12. Part III.) BL : CL :: 3 : 1 ; therefore AR is equal to three times jR7; and RL is also equal to three times RI; therefore AR is equal to RL, and (E. 5. 1.) the angle RAL is equal to the angle RLA ; and (E. 32. 1.) the angle ARM is equal to the double of ALM, that is, to the obtuse angle ALB of the rhomb ; but the angle ARI is equal to the angle BLC, the two tri angles ARI, BLC having been shewn to be similar; also, since (E. 8. 1.) the two triangles ALC, BLC, are equal, the angle ALC is equal to BLC t and is, therefore, equal to ARI-, wherefore 292 ON THE STRUCTURE OF THE CELLS Of BEES. (E. 13. 1.) the angle BLF is equal to the angle ARM, and consequently to its equal, the obtuse angle ALB of the rhomb ; therefore, also, (E. 13. and 29. 1.) the acute angle BLC is equal to the acute angle of the rhomb. Again, since the angle ALC is equal to the angle BLC, therefore (E. 13. 1.) the angle ALF is equal to the angle BLF, and the three obtuse angles, which contain the solid angle at Z/, are equal to each other. It is manifest, also, that the three acute angles, which form the solid angle at B 9 may, in the same manner, be shewn to be equal, as well as the three angles at the summit M of the figure. l6. COR. 2. The surface of the solid figure thus determined, in the seventh proposition, is less than the surface of a right prism of the same capacity^ which has a hexagon for its base. For the demonstration of the seventh propo sition may be applied to the case, in which the rhomb APBQ, instead of cutting the hexagonal prism, coincides with the rhomb AKBC; so as to shew, that the surface of the solid figure, deter mined in that proposition, is less than that of the hexagonal prism : and (Art. 13.) the capacities of the two solids are equal to one another *. * The surface of the cell, when it is terminated by a feexagor>, will be found to exceed its surface when terminated by ON THE STRUCTURE OF THE CELLS OF BEES. 293 PROP. VIII. 17. Problem. To determine, algebraically, when the solid described in the sixth proposition has the least surface ; and to compute the angle, at which each of the equal rhombs is, in that case, inclined to the axis of the solid, and also the angles of the rhomb. It is manifest, from the description of the figure in Art. 13, that the aggregate of the rhomb ALBM and the two trapeziums AEFL, LFGB, is to be a minimum. For C7, or (Art. 10.) f CB, put a ; for CF t b ; and for CL, 2; therefore LI = (LC* 4- 67")* = (z* + and by a pyramid of the form above described, by SAB* (1C IN), that is, by -(2)2 ^ or 55 xAB.IC nearly : and the whole surface of the three terminating rhombs is 3AB X /= 3^. AB.IC, or 3. 668. x AB x 1C nearly; (2)4 therefore the excess of the former surface, above the latter, is nearly one-sixth part of the whole surface of the three termi nating rhombs. 294 ON THE STRUCTURE OF THE CELLS OF BEES. And the two trapeziums = 2 (CO - BCL) = 2(2ab az) is to be a minimum, or ^"3 . (** + *)5 4- 26- * is to be a minimum ; *.* " \/ 3 /l - ^1 - * = 0; * + a (* .-. 3z* = ft + a 2 ; = a a , and s*=~ a i = /=* ) and : a :: 1 : v 2; v 2 i. e. CL : C/ :: 1 as the side of a square is to its diameter, according to the construction in Prop. 6. ,-^-- To compute the angle ILC, which is equal to the angle, at which the plane of the rhomb is in clined to the axis of the figure : In the right-angled triangle Z/C/, CL : CI :: rad. : tan z /LC; i. e. 1 : \/~2 :: rad. : tan z ILC; .*. log. rad. -flog. \/*2 = 10. 1505150 = log. tan 54. 44 . 8". ON THE STRUCTURE OF THE CELLS OP BEES. 295 .-. the angle ILC is an angle of 54. 44 . 6". To compute the angle of the rhomb. In the right-angled triangle BCL, EL : LC :; rad. : cos L BLC; i.e. (Art. 12. Part III.) 3:1:: rad. : cos L BLC-, .-. log. rad,- log. 3 = 10 .47712 12 = 9.5228788 ** * = log. co, 7 0. 31 . 44". Therefore, the acute angle LBM of the rhomb, which (Art. 15. Part III.) has been shewn to be equal to the angle BLC, is an angle of 70. 31 . 44". Therefore, the obtuse angle of the rhomb is an angle of 109. 28 . l6", which is the double of the angle at which each of the equal rhombs is in* clined to the axis of the solid. - SCHOLIUM. The fabrication of the system of cells which compose a honey-comb is carried on in strict con formity with the theory which has been established in the preceding articles. In order that the honey-comb may be compact and strong, and that it may not occupy more space than is absolutely necessary, it is evident that there ought to be no interstices between the cells of which it is composed; it appears, from Art. 6. Part III, that there is a choice of figures, which have this 296 ON THE STRUCTURE OF THE CELLS OF BEES. important property, of filling space about a given point, when placed in contact with each other ; and it is shewn, in Art. 7. Part III, that, of all these, the hexagonal prism is of the most cecono- mical structure. Now this is the model according to which the bee really works. But a complete hexagonal prism is not, in every respect, suited to its purpose. The safety of the grub, and the preservation of the honey in which it is bedded, and which is its nutriment, re quire that the cell should terminate in a solid angle rather than in a flat surface. Here again there is a choice of figure, and indeed an infinite variety of modes, in which such a termination might be formed; but it is proved, in Art. 14. Part III, that there is one particular mode, that is the most advantageous, as requiring the least quantity of labour and materials : and the cells of the honey-comb were found by MARALDI, upon exact measurement, to terminate in that very angle, which is mathematically ascertained to ren der the surface the least possible, in such a struc ture. Further, the surface of the cell, which from its figure thus secures the safety of the embryo, is demonstrated, in Art. l6. Part III, to be less even than that of a complete hexagonal prism of equal capacity. Another remarkable consequence of the actual construction of the cell, is the simplicity of the component parts of its termination. It may be seen, in Art. 15. Part III. that there are only two ON THE STITCTURE OF THE CELLS OF BEES. 297 different angles, and those each the supplement of the other, employed in the fabrication. This advantage is not wholly unconnected with the hexagonal form of the cells. If they had not been hexagonal, but had still, for a reason which has been given, ended in pyramids, these pyramids must have been bounded by trapeziums, and not by rhombs; and there would neither have been the same regularity, nor the same facility of con struction, A saving of a different kind, also, from that which has been pointed out, accrues from the termination of the cells in these equal solid angles, and from the rows of cells being placed back to back. For, by the combination of two bases of cells belonging to one row, with a third belonging to the same row, the bottom of a new cell is formed, belonging to the opposite row; and the work is strengthened by this junc tion of opposite cells, and their locking, as it were, into each other. The regularity with which this most ingenious structure is carried on, by so many thousands of insects, labouring together without any plan to imitate, the great delicacy of the work manship, and the extraordinary exactitude with which three sets of rhombs are continued in three planes, are truly surprizing. But if such depth of Geometry be manifested in the form, there is displayed as wonderful a regard to the principles of chemical science in the size, also, of the cell of the honey -comb. For, had its dimen- sipns been larger than they are, the honey, with 298 ON THE STRUCTURE OF THE CELLS OF BEES. which it is filled, would have fermented, and would thus have been spoiled. Many other circumstances, also, belonging to this portion of Natural History, which do not lie within the sphere of mathematical enquiry, are most worthy of observation. Such, for example, dfe the anatomy of th bee, the conformation of the different organs with which it extracts the honey and the wax, and the mecha nism of its sting; its habits and propensities, its prognostications of the Weather ; its prospective in dustry in laying up stores for winter, and hernleti- cally sealing those cells whicih are last to be opened; its precautions against cold j , which appears to be its greatest physical evil; its strong affections and hatreds ; the distribution of labour which takes place in the hive ; the modelling of a few cells of larger dimensions for the propagation of drones, and of a still less number, apart from the rest, as nurseries for the queens of future swarms ; thb singular instinct by which the queen-bee constantly deposits the proper egg in the peculiar kind of cell which is accommodated to it. All these particulars might be enlarged up6h, if this were the proper place, and many more, equally curious, might be described, which it is at once delightful and instruc tive to contemplate. fc t<- { ^^S<r U.C. BERKELEY LIBRARIES. CD37S71SSb 0097 1 5 UNIVERSITY OF CALIFORNIA LIBRARY m wt ,.,f, --S^~^ ^ W ^^\ ^^^