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FRONTISPIECE, 
 
 Tee CIRCEE, SOUARE, AND CRIANCLE® 
 PRIMITIVE 
 GEOMETRICAL FICURES. 
 
 ze : ae 
 London: John Bennett 4 Three Tm Passage Ivy Lane,Patemoster Row. 
 
ORIGINAL 
 
 GEOMETRICAL ILLUSTRATIONS; 
 
 OR 
 
 THE BOOK OF LINES, SQUARES, CIRCLES, 
 TRIANGLES, POLYGONS, &c. 
 
 SHOWING 
 
 AN EASY AND SCIENTI- GON, &c. TO ANY OTHER 
 
 FIC ANALYSIS FOR IN- FIGURE CONTAINING 
 
 CREASING, DECREAS- 
 THE SAME AREA, BY 
 
 ING, AND ALTERING ANY 
 PLAIN AND SIMPLE ME- 
 
 GIVEN CIRCLE, SQUARE, 
 
 TRIANGLE, ELLIPSIS, PA- TOS grace, Sane 
 
 RALLELOGRAM, POLY- AGREEABLY TO 
 
 MATHEMATICAL DEMONSTRATION, 
 
 INDISPENSABLE TO 
 
 ~“ ARCHITECTS, ARTISTS, ARTIFICERS, BUILDERS, CABINET MAKERS, . 
 CARPENTERS, ENGINEERS, (MILITARY AND CIVIL), ENGRAVERS, GLASS ' 
 CUTTERS, JEWELLERS, MACHINISTS, PAINTERS, SCULPTORS, 
 STATUARIES, &c. 
 
 Containing algo a baviety of Useful tnformation, 
 
 INTENDED AS 
 
 A COMPLETE INSTRUCTOR 
 
 TO THE 
 
 MOST USEFUL SCIENCE OF GEOMETRY AND MENSURATION. 
 
 By JOHN BENNETT, ENGINEER, 
 
 AUTHOR OF THE ARTIFICER’S COMPLETE LEXICON, LABOUR PRICES FOR BUILDERS, &c. 
 
 LONDON : JOHN BENNETT, 4, THREE-TUN PASSAGE, IVY LANE, PATERNOSTER ROW;.5 
 SIMPKIN, MARSHALL, AND CO., STATIONERS’ HALL COURT; SHERWOOD, GIL- 
 BERT AND PIPER, PATERNOSTER ROW; MR. WILLIAMS (Library of Arts), CHARLES 
 STREET, SOHO SQUARE; MR. WEALE (Architectural Library), HOLBORN; AND 
 MR. TAYLOR, WELLINGTON STREET, STRAND. 
 
 MDCCCXXXVII. 
 
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PREFACE. 
 
 GEOMETRY originally signified no more than the art of measuring the earth, 
 or any distance or dimension within it ; but, at present, it denotes the science 
 of magnitude generally, comprehending the doctrines and relations of whatever 
 is susceptible of augmentation, or diminution. Hence to Geometry may be 
 referred the consideration not only of Lines, Surfaces, and Solids, but also of 
 Time, Velocity, Number, Weight, &c. 
 
 Plato thought the word Geometry an improper name, and accordingly substi- 
 tuted in its place the more extensive one of Mensuration ; and after him others 
 gave it the title of Pantometry, as demonstrating not only the quantities of all 
 manner of magnitudes, but also their qualities, ratios, positions, transformations, 
 relations, &c. And Proclus describes it as the knowledge of magnitudes, figures, 
 &c., and their limitations, also of their motions and affections, of whatsoever 
 kind or description. 
 
 This science had its origin amongst the Egyptians, who were in a manner 
 compelled to invent it to remedy the confusion which generally happened in 
 their lands from the inundations of the river Nile, which carried away all the 
 boundaries and effaced all the limits of their possessions. This invention at first 
 consisted in measuring land, that every person might be in possession of the 
 just quantity which belonged to him. It was then termed Geometry, or the art 
 of measuring land: it is therefore probable that the draughts and schemes, which 
 they were compelled to make, caused them to discover many valuable properties 
 in the science, which subsequent speculations gradually improved, and which 
 still continue to this day. 
 
 The utility of Geometry extends to almost every art and science. It is by the 
 help of it that astronomers turn their observations to advantage, regulate the 
 duration of time, seasons, years, cycles, and epochas, and measure the distances, 
 
lv PREFACE. 
 
 motions, and magnitudes of the Heavenly Bodies. It is by it that geographers 
 determine the figure and magnitude of the whole earth, and delineate the extent 
 and bearings of kingdoms, provinces, harbours, &c. It is from this science, too, 
 that architects derive their just dimensions and proportions in the construction 
 of public edifices as well as private houses. It is by the assistance of Geometry 
 that engineers conduct their work for machines of all denominations, as also the 
 situation and plan of towns, and distances of places, and the measure of such 
 things as are only accessible to the sight. It is not only an introduction to For- 
 tification, but highly requisite to every branch of mechanism generally. 
 
 On Geometry likewise depends the theory of Music, Optics, Perspective, 
 Drawing, Hydraulics, Pneumatics, &c. &c. | 
 
 In this work, it must be observed, the various geometrical figures are laid 
 down with simplicity and perspicuity, suitable alike for the artisan, manufac- 
 turer, and gentleman ; and, being principally drawn to the full size or scale of the 
 common two-foot rule, are thereby rendered plain to the understanding and easy 
 of imitation. The student will therefore find in this work the best and readiest 
 information in the science of Geometry and Mensuration, which, combined 
 together, are calculated to lead to every relative object, however intricate or 
 remote. For by the methods shown in the divisions of the different figures, the 
 author was enabled to trace out the solutions to the intricate questions of the 
 ancients ; and it is made evidently to appear, that there is no other device 
 existing which would perform ‘Similar operations in the same space of time (or, 
 indeed at all), so perfectly correct, both in quantity and formation. 
 
 May, 1837, 
 
3% 
 
 GEOMETRICAL ILLUSTRATIONS 
 
 MATHEMATICALLY DEMONSTRATED. 
 
 oi 
 _ ABSCISSE, in conic sections, the part AP of the diameter of a curve line, intercepted 
 between the vertex A of that diameter and the point P, where any ordinate or semi-ordinate, 
 
 MP, to that diameter falls. — 
 
 From this definition it isevident that there are an infinite number of variable si 
 abscisses in the same curve, as well as an infinite number of ordinates. : 
 
 In the parabola one ordinate has but one abscisse, in an ellipsis it has two, 
 and in curves of the second and third order it may have three or four. ~ mt 
 
 ACCELERATION denotes the augmentation or increase of motion in accelerated bodies. 
 The term acceleration is chiefly used in speaking of falling bodies, or the tendency of heavy 
 bodies towards the centre of the earth, produced by the power of gravity, which, acting con- 
 stantly and uniformly upon them, must necessarily cause them to acquire, every instant, a new 
 increase of motion. 
 
 Thus, if 1 a represent the velocity acquired whilst a body falls 
 through A 1, suppose one minute, then 26 willexpress the velocity 
 acquired in two minutes, represented by A 2, 3 c¢ the velocity ac- 
 quired in three minutes, or A 3, and BC the velocity acquired whilst 
 the body falls through AB. Now the triangles A 1 a, A 20, A 3c, 
 and ABC represent the spaces described by the falling body in the 
 respective times, A 1, A 2, A 3, and AB, by reason of the uniform 
 action of gravity; but these triangles, being similar, are to each other 
 as the square’s homologous sides, Al, A 2, A3, and AB, that is, the 
 spaces are to each other as the squares of the times in which they are 
 described. Hence also follows the great law of acceleration, viz. that 
 a falling body, uniformly accelerated, describes in the whole time of = 
 its descent just one-half of the space it would have described in the same time with the motion 
 it has acquired at the end of its fall. From what has been said, it is evident that the space 
 described by a falling body, ina series of equal portions of time, will be as the odd numbers 1, 3, 
 5,7, &c. See the figure above referred to, where the space described in the time A 1 is re- 
 presented by the triangle A 1 a; whereas the space described in the second portion of time con- 
 tains three such triangles ; that, described in the third portion, five such triangles; and so on. 
 Again, that the spaces described by falling bodies in different times are as the squares of the 
 velocity acquired at the end of its fall. As the spaces represented by the odd numbers 1, 3, 5, 7, 
 
 / (7 } 
 eel = 
 
2 GEOMETRICAL ILLUSTRATIONS 
 
 &c., still approach nearer and nearer to equality, so the accelerated motion likewise approaches 
 nearer and nearer toa uniform motion ; and, if the body moves in a resisting medium, the mo- 
 tion will actually become uniform, at a certain distance. 
 
 ALTITUDE, one of the three dimensions of a body, being the same with what is otherwise 
 called height. 
 
 ALTITUDE of a figure is the nearest distance of its vertex from its base, or the length of a 
 perpendicular let fall from the vertex to the base. 
 
 ANALYSIS, the art of discovering the truth or falsehood of a proposition, or its possibility 
 or impossibility. This is done by supposing the proposition, such as it is, true; and examining 
 what follows from thence until we arrive at some evident truth, or some impossibility, of which 
 the first proposition is a neccessary consequence; and from thence establish the truth or im- 
 possibility of that proposition. The analysis of the ancient geometricians consisted in the ap- 
 plication of the propositions of Euclid, &c. until they arrived at the truth required. That of the 
 moderns, though not so elegant, must be allowed to be more ready and general. By this last 
 geometrical demonstrations are wonderfully abridged. A number of truths are frequently ex- 
 pressed by a single line, and whole sciences may sometimes be learned in a few hours which 
 otherwise would be scarcely attained in many years. 
 
 ANAMORPHOSIS, in perspective and painting, &c. a monstrous projection, or representa- 
 tion of an image on a plane or curved surface, which, beheld , 
 at a proper distance, shall appear regular andin proportion. 
 To delineate an anamorphosis upon a plane, draw the square 
 ABCD, and subdivide it into a number of little squares ; in 
 this square, let the image, to be represented deformed, be 
 drawn. Then draw the line ab equal to AB, and divide it 
 into the same number of equal parts as the side of prototype, 
 AB. Erect the perpendicular EV in the middle of a 8b, so 
 much the longer as the deformity of the image is to be 
 
 ereater. Draw VS perpendicular to EV, so much the shorter 
 as you would have the image appear more deformed. From 
 each point of division draw straight lines to V, and join the points a and S by the right line 
 aS. Through the points defg draw right lines parallel to a 6, then will abc d be the space in 
 which the monstrous projection is to be delineated: in every small trapezium of thespace a b cd 
 draw what appears delineated in the correspondent areola of the square ABCD; and thus you 
 will obtain a deformed image, which will appear in just proportion to an eye distant from it the 
 length FV, and raised above its height VS. 
 
 A 
 ANGLE, the inclination of two lines meeting each other in a point, and 
 called the legs of the angle. Thus ABC is the angle made by the two lines 
 c 
 bD 
 FEF 
 
 AB, BC, meeting in the point B, which is the vertex of the angle. - 
 
 Angles are rectilineal, or right lined, as ABC, above referred to; or they are © 
 
 Pe 
 curvilineal as DEF; or, lastly, formed of a straight line and a curved one, and decntada 
 
 thence called mixed, as HIG. tty itll a 
 
MATHEMATICALLY DEMONSTRATED. ; 3 
 
 Rectilineal angles, according to the greater or less degree of inclination, are right 
 acute, or obtuse. ; 
 
 Right angle is that formed between two lines, one of which stands 
 upright, or perpendicularly on the other, inclining no more one way than 
 it does the other. Suchis the angle EBC ; for, if BC be produced to 
 D, EB will be found to stand upright on DC, or to incline neither 
 way. A right angle is said to be an angle of ninety degrees, because 
 
 measured by a quadrant of acircle, or *"=90; so that a right angle, 
 
 or an angle of ninety degrees, is the same thing. 
 Acute angle, one whose vertex is acute or sharp, being always less than a right angle ; such 
 is the angle ABC. 
 Obtuse angle, one with a blunt or obtuse vertex, as ABD, which is always greater than a 
 right angle. Angles likewise receive other denominations from their different positions, and the 
 relations they bear to the figures they are in and to the lines which form them. Hence, 
 
 Angles in a semicircle, those subtended by the diameter of that 
 circle; as, DFC, DGC, which are always right angles. 
 
 Angle at the centre, that formed by two radii, or semi-diameters 
 of a circle as, OCN. 
 
 Angle at the circumference, or in a segment, that formed by two 
 chords of a circle meeting at the circumference; such is OPN, 
 which is only half the angle at the centre OCN subtended by the 
 same chord ON, or, which comes’ to the same thing, it is equal to 
 
 half the are OSN. Moreover, all angles in the same segment, and consequently subtended by 
 the same chord ON, as OQN, OPN, ORN, are equal to one another. 
 
 Angle of a semicircle, that formed by a diameter and the circumfer- pee tp en 
 
 ence of a circle, as BAO, which is less than a right angle, and yet 
 greater than any rectilineal-acute one. 
 
 Angle of a segment, that whic h a chord in a circle makes with the 
 tangent at the point of contact; such are the angles EDC, FDC; 
 the former being the angle of the greater segment, and the latter of the 
 lesser segment. 
 
 Angle of contact, that which the tangent of a circle forms with its circumference; as, EDA, 
 which is less than any right-lined angle. 
 
 Angles are said to be adjacent, or contiguous, which have one 
 leg common to both; as, DGI and DGE, which taken both 
 together are equal to two right angles. 
 
 Opposite, or vertical angles, those formed by two lines crossing ™ F 
 each other, as the angles DGI, EGF, which are always equal. Aa 
 An angle is also said to be opposite to the side that subtends 
 it; thus GHF is to the side oppositeGF. Again, when one of the sides of a triangle is pro- 
 duced, as from F to K, the external angle GFK is equal to the two internal angles FGH and 
 FHG, which are said to be opposite to it. 
 
4 GEOMETRICAL ILLUSTRATIONS 
 
 Alternate angles, the internal pairs of acute or obtuse angles, formed by a right line DL 
 cutting two parallel right lines IE, HK: such are EGF, and GFH, both acute and equal; also 
 the obtuse ones IGF and GFK, likewise equal. 
 
 Spherical angle, that formed by the intersection of two great circles of the sphere. 
 
 Solid angle, that formed by the meeting of three or more plane angles, not being in the same 
 plane, in one point: such is the angle of a die, of a square box, or the like. In regard to solid 
 angles, it has been demonstrated that the plane angles forming them are always less than three 
 hundred and sixty degrees, or four right angles. 
 
 ANGULAR, ina general sense, denotes something relating to or having angles. 
 
 ARCH, any part of the circumference of a circle, or D 
 curved line, lying from one point to another, by which the es 
 quantity of the whole circle or line, or some other thing B 
 
 sought after, may be obtained. If the arch BC contains the ‘ 
 
 same number of degrees as the arch DE, or, if the radius AB : \ 
 is to the radius ADas the arch BC to thearch DE, then — / 
 these two arches are similar. ie! i} 
 
 ARCHES EQUAL, those which contain the same number of degrees, and whose radii are equal. 
 
 ARCH, in architecture, a concave building, with a mould bent in the form of a curve, 
 erected to support some structure. Arches are either circular, elliptical, or strait, as they are 
 improperly called by workmen. Circular arches are of three kinds : First, Semicircular, which 
 have their centre in the middle of a line drawn between the feet of the arch. Secondly, Scheme, 
 or Skene, which are less than a semicircle, containing some ninety, and some seventy degrees. 
 Thirdly. Arches of the third and fourth point, consisting of two arches of a circle meeting in 
 an angle at the top, being drawn from the division of a chord into three or more parts at 
 pleasure. 
 
 Elliptical arches consist of a semi-ellipsis, and have commonly a key stone and imposts: they 
 are usually described on three centres. Straight arches are those used over doors and windows, 
 having plain straight edges, both upper and under, which are parallel, but both the ends and 
 joints point towards a centre. - 
 
 Arch is also used to denote the interval between two piers of a bridge. 
 
 AREA, denotes the superficial content of any figure: thus, if we suppose a parallelogram six 
 inches long, and four broad, its area will be 6 x 4—924 square inches. 
 
 ART, a system of rules serving to facilitate the performance of certain actions, in which 
 sense it stands opposed to science, or a system of merely speculative principles. Arts are 
 commonly divided into liberal and mechanical, the former comprehending poetry, painting, 
 sculpture, architecture, &c., and the latter the whole body of mechanical trades; as, carpenters’, 
 engineers, masons’ work, &c. 
 
 ARTICLE, sometimes called decade, denotes a number justly divisible into ten parts, as the 
 numbers 10, 20, 30, &c. 
 _ ASSYMPTOTE, a line which continually approaches nearer to another line, but which, though 
 continued infinitely, will never meet it: of these there are many kinds. In strictness, 
 
 however, the term assymptotes is appropriated to right lines which approach nearer and 
 nearer. 
 
MATHEMATICALLY DEMONSTRATED. 5 
 
 nearer to some curve of which they are said to be assymptotes ; but, if they and their curve 
 are indefinitely continued, they will never meet. 
 
 The nature of an assymptote will be easily conceived 
 from considering the assymptote of the chonchoid ; for if C 
 DE be a part of the curve of a conchoid, and A its pole, and 
 the right lines MN be so drawn that the parts, BC, GD, 
 FE, of right lines drawn from the pole A be equal to each 
 other, then the line MN will be the assymptote of the curve ; 
 because the perpendicular D p is shorter than BC, and EP 
 
 shorter than D p, and so on; and the points E and p can 
 never coincide, 
 
 A. 
 
 AXIOM is an established principle in some art or science. Thus it is an established axiom 
 in physics, that nature does nothing in vain ; so it is in geometry that if te two equal things you 
 add equals the sums will be equal. It is an axiom in optics that the angle of incidence is 
 equal to the angle of reflection, &c. In which sense, too, the general laws of motion are called 
 axioms. Whence it may be observed, that these particular axioms are but deductions from 
 - certain hypothesis. 
 
 AXIS, in conic sections, a right line dividing the section into two equal parts, and cutting 
 all its ordinates at right angles. Thus if A P be drawn so 
 
 A 
 as to cut the ordinate MN, at right angles, and divide the ina - 
 section into two equal parts, then is the line AP the axis of _ Ms 
 the section. The transverse, first, or principal axis of an 4, i iB 
 ellipsis or hyperbola is the axis AP, which, in the ellipsis, 
 is the longest, and in the hyperbola cuts the curves in the . 
 
 N N 
 
 points A and P. The conjugate, or second axis of an 
 
 ellipsis, is the line EF drawn through the centre C, parallel 
 
 to the ordinate MN, and perpendicular to the transverse | BE 
 axis AP, being the shorter of the two, and terminated by c 
 the curve. The conjugate axis of the hyperbola is the right 
 line EF, drawn through the centre C, parallel to the 
 ordinates MN, and perpendicular to the transverse axis AP. This axis, though no more than 
 infinite, is of a determinate length, and may be found by this proportion, as AMxPM: AP*:: 
 MN’: EF’. The axis of a parabola is of an indeterminable length. The axis of the ellipsis is 
 determinate. In the ellipsis and hyperbola there are two axis and no more, and in the 
 parabola only one. 
 
 AXIS, in mechanics. The axis of a balance is that about which it moves, or rather turns. 
 Axis of oscillation is a right line parallel to the horizon, passing through the centre about 
 which a pendulum vibrates. 
 
 AXIS, in peritrochio, one of the five mechanical powers, consisting of a peritrochium, or 
 wheel, concentric with the base of a cylinder, and movable together with it about its axis. 
 The power is applied at the circumference of the wheel, and the weight is raised by a rope that 
 is gathered up on the axis or cylinder while the machine turns round. 
 
 The power may be conceived as applied at the extremity of the arm of a lever equal to the 
 radius of the wheel, and the weight as applied at the extremity of a lever, equal to the radius 
 
6 GEOMETRICAL ILLUSTRATIONS 
 
 of the axis; only those arms do not meet at one centre of motion, as in the lever but in place 
 of this centre there is an axis of motion, viz. the axis of the whole machine. But, as this can 
 produce no difference, it follows that the power and weight are in equilibrio when they are to 
 each other inversely as the distance of their directions from the axis of the engine ; or when 
 the power is to the weight as the radius of the roller to the radius of the wheel, the power 
 being supposed to act in a perpendicular to this radius. 
 
 But, if the power act obliquely to the radius, 
 substitute a perpendicular from the axis on the di- 
 rection of the power, in the place of the radius. 
 Thus, if ABDE represent the cylindrical roller, 
 HPN the wheel, LM the axis, or right line, 
 upon which the whole engine turns, Q the point of 
 the surface of the roller where the weight W is 
 
 applied, P the point where the power is applied, 
 KQ the radius of the roller, CP the radius of the 
 wheel, then, if the power P act with a direction perpendicular to CP, the power and weight 
 will sustain each other, when P isto W as KQ is to CP or CH. But, if the power act in any - 
 other direction PR, let CR be perpendicular from C the centre of the wheel on that direction, 
 then P and W will sustain each other, when P is to W as KQ to CR; because, in this case, 
 as power P has the same effect as if it was applied to the point R of its direction, acting in a 
 right line perpendicular to CR. 
 
 The use of this machine is to raise weight to a greater height than the lever can do; because 
 the wheel is capable of being turned several times round, which the lever is not, and also to 
 communicate motion from one part of a machine to another. 
 
 AXIS, in opties, is that ray among all others that are sent to the eye which falls perpen- 
 dicularly upon it, and which consequently passes through the centre of the eye. Common or 
 mean axis is a right line drawn from the point of concourse of the two optic nerves through 
 the middle of the right line which joins the extremity of the same optic nerves. 
 
 AXIS of a vessel is an imaginary light line, passing through the middle of it per- 
 pendicularly to its base, and equally distant from its sides. 
 
 BASE, the lowest side of the perimeter of a figure: thus, the base of a triangle may be said 
 of any of its sides, but more properly of the lowest, or that which is parallel to the horizon. In 
 rectangled triangles, the base is properly that side opposite to the right angle. 
 
 Base of a solid figure the lowest side, or that on which it stands; and if the solid has two 
 opposite parallel plane sides, and one of them is the base, then the other is called the base also. 
 
 Base of a conic section is a right line in the hyperbola and parabola, arising from the 
 common intersection of the secant plane and the base of the cone. 
 
 Base line, in perspective, the common section of a picture, and the geometrical plane. 
 
 BIMEDIAL. If two medial lines, as AB, and BC, commensurable only in power, containing 
 a rational rectangle, are compounded, the whole line AC will be irra- B 
 tional, and is called a first bimedial line. ‘gs Seana eet 2 
 
 BISECTION, the division of a line, angle, &c. into two equal parts. 
 
MATHEMATICALLY DEMONSTRATED. 7 
 
 Bopy is otherwise called a solid. The regular bodies, or those which have all their angles 
 and sides similar and equal, are five, viz., the tetraehedron, octahedron, dodecahedron, icosa- 
 hedron, and the cube. 
 
 BREADTH, one of the three dimensions of bodies which multiplied into their length con- 
 stitute a surface. 
 
 CENTRE of a circle, a point in the middle of a circle, or circular figure, from which all lines 
 drawn to the circumference are equal. 
 
 Centre of a conic section, a point where the diameters intersect each other. 
 
 Centre in the ellipsis, the point is within the figure, and in the hyperbola without. 
 
 Centre of an ellipsis, the point where the transverse and conjugate diameters intersect each 
 other. 
 
 Centre of gravity, that point about which all the parts of a body, in any situation, invariably 
 balance other other. 
 
 The centre of gravity in regular, uniform, and homogeneal bodies, 
 as squares, circles, &c., is the middle point in a line connecting any 
 two opposite points, or angles. Therefore, if such line be bisected, the 
 * point of section will be the centre of gravity. To find the centre of gravity 
 of a triangle, bisect the base AC of the triangle ABC ; it will also bisect 
 every other line, DE, drawn parallel to the base ; consequently, the centre 
 
 of gravity of the triangle will be found somewhere in the line BG, 
 
 To find the centre of gravity in a parallelogram and parallelopiped, 
 draw the diagonal AD and EG, likewise CB and HF; since each «| Mgrs Win. 
 diagonal AD and CB divide the parallelogram ACDB into equal [ 
 parts, each passes through the centre of gravity; consequently the 
 point of intersection, I, must be the centre of gravity of the parallel- 
 ogram. In like manner, since both the plane CBFH and ADGE BL , 
 divide the parallelopiped into two equal parts, each passes through its centre of gravity, so that 
 the common intersection II€ is the diameter of gravity, the middle thereof, L, is the centre. 
 From similar directions may the centre of gravity be found in prisms, cylinders, &c., it being 
 the middle point of the right line that joins the centre of gravity of their opposite bases. The 
 centre of gravity of a parabola is found as in the triangle and cone. 
 
 Centre of the hyperbola, a point in the middle of the transverse axis. 
 
 Centre of magnitude of any homogeneal body, the same with the centre of gravity. 
 
 Centre of motion, that point which remains at rest, while all the other parts of a body move 
 about it. This is the same in uniform bodies of the same matter throughout, as the centre of 
 gravity. 
 
 Centre of percussion, in a moving body, that point wherein the percutient force is greatest, 
 or that point with which, if the body strikes against any obstacle, no shock shall be felt at the 
 point of suspension. The centre of percussion, when the percutient body revolves round a 
 fixed point, is the same with the centre of oscillation, and consequently may be determined by 
 the same rule, which is as follows, viz. if several bodies be fixed to an inflexible rod suspended 
 upon a point, and each body be multiplied by the square of its distance from the point of sus- 
 pension, and then each body be multiplied by its distance from the same point, and all the 
 
8 GEOMETRICAL ILLUSTRATIONS 
 
 former products when added together be divided by all the latter products added together, the 
 quotient which shall arise from thence will be the distance of the centre of oscillation of these 
 bodies from the said point. Hence a stick of a cylindrical figure, supposing the centre of 
 motion at the hand, will strike the greatest blow at a distance of two-thirds of its length from 
 the hand. The centre of percussion is the same with the centre of gravity, if all the parts of 
 the percutient body be carried with a parallel motion, or with the.same celerity. For the 
 momenta are the facta of the weights into the celerities, therefore to multiply equiponderating 
 bodies by the same velocity is the same thing as to take equimultiples of them. But the equi- 
 multiples of equiponderating bodies themselves equiponderate ; therefore equivalent mo- 
 menta are disposed about the centre of gravity, and, consequently, the centre of gravity in this 
 case will coincide with the centre of percussion, and what is shown of the one will hold of the 
 other. 
 
 Centre of conversion may be explained thus. Ifa stick be laid on stagnant water, and drawn 
 by a thread fastened to it, so that the thread always makes the same angle with the stick, viz. a 
 right angle, the stick will be found to turn on one of its points, which will be immovable, and 
 this point is called the point of conversion. This effect arises from the resistance of the fluid; 
 but the great question consists in knowing in what point the centre of conversion is found. 
 This has been calculated with a great deal of exactness, for if the stick be drawn by one 
 extremity in a straight line divided into twenty parts, the centre of conversion will be nearly on 
 the thirteenth, reckoning from the thread. If it be not a line, but a surface or solid, there will 
 be some change in the situation of the centre of conversion, according to the nature of the 
 surface or solid. 
 
 Centre of a parallelogram, or polygon, the point in which all its diagonals intersect. 
 
 Centre of a sphere, a point in the middle from which all lines drawn to the surface are equal. 
 
 CENTRAL FORCES, the powers which cause a moving body to tend towards, or recede from, 
 the centre of motion. 
 
 CENTRIFUGAL FORCE, that force by which all bodies that move round any other body in a 
 curve endeavour to fly off from the axis of their motion in a tangent to the periphery of the 
 curve, and that in every point of it. This force is always proportional to the circumference of 
 the curve in which the revolving body is carried round. The centrifugal force of any body is 
 to the centripetal as the square of the arch which a body describes in a given time, divided by 
 the diameter, to the space through which a heavy body moves in falling from a place where it 
 was at rest in the same time. If any body swim in a medium heavier than itself, the centrifugal 
 force is the difference between the specific weight of the medium and the floating body. All 
 moving bodies endeavour a rectilinear motion, because it is the easiest, shortest, and most 
 simple. Whenever, therefore, they move in any curve, there must be something that draws 
 them from the rectilinear motion, and detains them in their orbits ; and, were that force to cease, 
 the moving body would go straight off in a tangent to the curve in that very point, and would 
 get further and further from the focus, or centre of its curvilinear motion. It may be that, in 
 a curve, where the force of gravity in the describing body is continually variable, the centrifugal 
 force may also continually vary in the same manner, and so that one may also supply the defect, 
 or abate for the excess of the other, consequently the effect be everywhere equal to the 
 absolute gravity of the revolving body. . 
 
MATHEMATICALLY DEMONSTRATED. 9 
 
 CENTRIPETAL FORGE, that force by which a body is every where impelled, or any how 
 tends towards some point as a centre. Such is gravity, or that force whereby bodies tend 
 towards the centre of the earth; magnetical attraction, by which the loadstone draws iron; and 
 that force, whatever it be, whereby the planets are continually drawn back from right lined 
 motions and made to move in curves. The greater the quantity of matter in any body is the 
 greater will be its centripetal force, all things else alike. Ifa body laid upon a plane, revolve 
 at the same time and about the same centre with that plane, and so describe a circle; and if 
 the centripetal force, wherewith the body is drawn every moment towards the centre, should 
 cease to act, and the plane should continue to move with the same velocity, the body will begin 
 to recede from the centre about which the plane moved. 
 
 CENTRUM, the same with centre. 
 
 CHARACTERS in geometry and trigonometry, viz. ||, parallelism; A, triangle; 0, square; 
 O, a, rectangles; O, circle ; Y, equi-angular or similar; -L, equilateral; 7, an angle; |, a 
 right angle ; 1, perpendicular. 
 
 CHoRD oF THE CIRCLE, a right line drawn from one part of an arch of a circle to the 
 other. Hence the chord of an arch is a right line joining the extremes of 
 that arch; thus, AB is the chord of the arch AEB. It is demonstrated 
 that the radius CE, bisecting the chord BA in D, does also bisect the arch 
 in E, and is perpendicular to the chord AB. 
 
 From hence may be deduced the following problems :— 
 
 First, To make a circle pass through any three given points, placed 
 indiscriminately, and not in a right line. 
 
 Secondly, To find the centre of any circle. 
 
 Thirdly, To complete a circle from an arch given. 
 
10 GEOMETRICAL ILLUSTRATIONS 
 
 Fourthly, To describe a circle about any triangle given. 
 
 Fifthly, ‘To describe a square within a given circle. 
 
 Sixthly, To describe an octagon with a given circle. 
 
 CIPHER, one of the Arabic characters, or figures, used in computation, formed thus 0. A 
 cipher of itself signifies nothing; but when placed to the left of other characters, in whole 
 numbers, augments their value ten times ; and, when placed to the right hand in decimal arith- 
 metic, lessens the value of each figure in the same proportion. 
 
 Cipher denotes likewise certain characters disguised and varied, used in writing letters that 
 contain some secret not to be understood but by those between whom the cipher is agreed on: 
 for instance the word cut may be formed by the circle and square ; viz. take as much from the 
 circle as will unite two sides of the square, and transfix the other two sides as CUT. 
 
 CircLE, a plane figure, comprehended by a single curved line, called its circumference, to 
 which right lines, drawn from a point in the middle, called the centre, are equal to each other. 
 The area of the circle was found by multiplying the circumference by the fourth part of the 
 diameter, or half the circumference by half the diameter ; for every circle may be conceived 
 to be a polygon of an infinite number of sides, and the semi-diameter must be equal to the per- 
 pendicular of such a polygon, and the circumference of the circle equal to the periphery of the 
 polygon ; therefore, half the circumference, multiplied by half the diameter, give the area of the 
 circle. Circles, and similar figures inscribed inthem, are always as the squares of the diameters ; 
 so that they are in duplicate ratio of their diameters, and consequently of their radii. A circle is 
 equal to a triangle, the base of which is equal to the periphery, and its altitude to its radius: 
 circles, therefore, are in a ratio compounded of the peripheries and the radii. 
 
 To describe a circle through three given points, see Chord. 
 
 For the divisions of the Circle, see the plate; and for the quadrature of the circle, or the 
 manner of making a square whose surface is perfectly and geometrically equal to that of a 
 
MATHEMATICALLY DEMONSTRATED. 1] 
 
 circle, is a problem that has employed the geometricians of all ages. ‘‘ Charles the Fifth 
 offered a reward of one hundred thousand crowns to the person who should solve this celebrated 
 problem, and the States of Holland have also proposed a similar reward for the same purpose.” 
 See the plate. 
 
 Circle of curvature, a circle the curvature of which is equal to that of a certain curve at a 
 given point. 
 
 CiRCULAR, in a general sense, any thing that is described or moved in a round, as the cir- 
 cumference of a circle, or surface of a globe or sphere. The circular form is of all others the 
 best disposed for motion, and the most capacious. 
 
 CIRCUMFERENCE, ina general sense, denotes the line or lines bounding a plane figure. 
 However, it is generally used, in a more limited sense, for the curved line which bounds a circle, 
 and otherwise called a periphery, being the boundary of a right lined figure expressed by the 
 term perimeter. Any part of the circumference is called an arch, and a right line drawn from 
 one extreme of the arch to the other is called a chord. 
 
 CoMMENSURABLE, an appellation given to such quantities as are measured by one and the 
 same common measure ; thus, if the line A+++ a be equal to 8, and the line B+ 6 
 equal to 4 inches, these two lines will be commensurable, since the same common measure of 
 two measures them both. 
 
 COMPLEMENT is what remains of a quadrant of a circle, or of 90°, after any certain arch bas 
 been taken away from it. Thus, if the arch taken away be 40°, its complement is 50°, because 
 50 + 40 = 90. 
 
 COMPLEMENTS, in a parallelogram, are the two smaller parallelograms GAE, ie made 
 by drawing two right lines GE and FE through the point E inthe B 
 
 diagonal, and parallel to the sides AB BC of the parallelogram re 
 
 ABCD. In every parallelogram these complements are equal. 
 
 Cc ¥ D 
 
 CompouND Morton, that which is affected by the concurring action of several different 
 powers. Thus, if one power act in the direction of, and with a force proportional to, the end of 
 a parallelogram, and another act in the direction of, and with a force proportional to, its side, 
 the compound motion will be in the direction of, and proportional to, the diagonal of the paral- 
 lelogram. 
 
 ConcCAVE, the inner surface of hollow spherical bodies. 
 
 ConcaviTy, the whole space included within a concave surface, or the inner bend of a 
 curved line. 
 
 CoNCENTRIC is that body that has the same common centre with another, it stands in oppo- 
 sition to eccentric. Concentric is also used in speaking of round bodies and figures, or circular 
 and elliptical ones; but may likewise be used for polygons drawn parallel to each other, upon 
 
 the same centre. 
 
a2 GEOMETRICAL ILLUSTRATIONS 
 
 Cong, a solid figure, having a circle for its base, and its top terminated in a point or vertex. 
 
 A Cone may be conceived to be generated in the following manner :-— 
 
 Take an-immovable point A, elevated above the plane of a circle 
 BCDE, and suppose a straight line XZ, drawn through the point and 
 extended both ways from it to an indefinite length, to be carried round the 
 circle, continually touching its circumference and remaining fixed to the 
 immovable point, the line by this motion will describe two conic sur- 
 faces, which are vertical or opposite, having their common vertex at the 
 
 immovable point. vq A \ 
 ae ae al 
 (Zz. D 
 The solid contained within this conic surface between the immovable point A 
 
 A and the circumference of the circle BC DE isa cone. ‘The immovable point 
 A is the vertex, the circle BCDE is the base, and a straight line AF drawn 
 from the vertex to the centre of the base is the axis of the cone. All straight 
 lines drawn from the vertex to the circumference of the base, as AB, AC, AD, 
 AE, &c., are sides of the cone. If the axis of a cone be perpendicular to its 
 base it is called a right or proper cone, as in No. 2. If the axis be inclined to 
 the base, it is called a scalinous or oblique cone, such as No. 3, and a right 
 cone is always understood when the contrary is not expressed. 
 
 Euclid, in his eleventh book, gives a definition of a cone that is not as) 
 general, it being only of a right cone. He says, a cone is produced by the r\ 
 revolution of the plane of a right angled triangle, about the perpendicular W‘%, | | \ 
 axis remaining at rest. If this axis be greater than half the base, the solid A 
 produced is an acute angled cone; if less, it is an obtuse angled cone; and 
 if equal, a right angled cone. Thus the cone BAC, No. 4, is less acute 
 than the cone BDC, because the angle BDF is less than the angle BAF. 
 
 Conk. Cones and pyramids, having the same bases and altitudes, are equal to each other. 
 It is to be observed that every triangular prism may be divided into three equal pyramids, 
 and therefore a triangular pyramid is one third of a prism standing on the same base, having 
 the same altitude. Every multangular body may be resolved into triangular ones. A pyramid 
 is the third part of a prism, standing upon the same base, having the same altitude; and as a 
 cone may be esteemed an infinite angular pyramid, and a cylinder an infinite angular prism, a _. 
 a cone is a third part of a cylinder, which has the same base and altitude. The method of 
 
MATHEMATICALLY DEMONSTRATED. 138 
 
 measuring the solidity and surface of a cone and pyramid is, Find the solidity of a prism or 
 cylinder having the same base with the cone or pyramid, which found divide by three: the 
 quotient will be the solidity of the cone or pyramid. Or the solidity of any cone is equal to the 
 area of the base multiplied into one-third part of its altitude. As for the surfaces, that of a right 
 cone, not taking in the base, is equal to a triangle whose base is the periphery and altitude the 
 side of the cone; therefore, the surface of a right cone is obtained by multiplying the periphery 
 of the base into half of the side, and adding the product to that of the base. The altitudes of 
 similar cones are as the radii of the bases, and form the same angle with them. Cones are to 
 one another in a ratio compounded of their bases and altitades. Similar cones are in a triplicate 
 ratio of their homologous sides, and likewise of their altitudes. Of all cones standing upon the 
 same base, and having the same altitude, the superficies of that which is most oblique is 
 greatest, and the superficies of the right cone the least. 
 
 Conic SECTIONS, curves formed from the section of a cone by a plane. The curves that 
 generally pass under the name of conic sections are three, viz. the ellipsis, parabola, and hyper- 
 bola. Although the triangle and circle are formed from the sections of a cone, yet they are 
 not always considered in that capacity. 
 
 If a right cone be cut directly through its axis, the plane 
 or superficies of that section will be a plain isosceles triangle, 
 as HVG; for HV and VG, being the sides of the cone, 
 will be the sides of the triangle. HG, the diameter of the -—\ 
 cone’s base, will also be the base of the triangle, and its : 
 axis, VC, will be the height or altitudinal line. If the cone Bun 
 be cut by a right line parallel to its base, the plane of that 1h ea. 
 section will be a circle, because the base of the cone isa 
 circle. Such is HG. 
 
 If acone be cut bya right line that cuts both its sides, but not parallel to its base, as 
 TS, No. 2, the plane of that section will be an ellipsis, commonly called an oval; that is 
 an oblong or imperfect circle, having several diameters and two particular centres. 
 
 If a cone be cut into two parts by a right line parallel to one of its 
 sides, as SA, No. 3, the plane of that section, namely, Sd BA Q, is 
 called a parabola. 
 
 If a cone be cut by a right line parallel to its axis, as SA, No 4, in 
 such a manner that the intersecting line when continued through one side 
 of the cone, as at S, will meet with the other side if it be continued be- 
 yond the vertex V, as at T, then is the plane of that section, namely, 
 SB, AB, called an hyperbola. 
 
 These five sections, namely, the triangle, circle, ellipsis, parabola, 
 and hyperbola, are all the planes that can be produced from a cone. But 
 of them the last three are only sections, so called by ancient and modern 
 geometricians. 
 
 or 
 
 From the production of these sections, it may be observed how one 
 section degenerates into another. For an ellipsis being that plane of any 
 section of the cone which is between the circle and parabola, it will be ¢ 
 easy to conceive there may be great variety of ellipses produced from F 
 
 E 
 
14 GEOMETRICAL ILLUSTRATIONS 
 
 the same cone; and, when the section happens to be exactly parallel to one side of the cone, 
 then the ellipsis degenerates into a parabola. Now the parabola, being a section whose plane 
 is always exactly parallel to the side of the cone, cannot vary as the ellipsis; for, as soon as it 
 begins to move out of the position of being parallel to the side of the cone, it degenerates either 
 into an ellipsis or hyperbola. That is, if the section inclines towards the plane of the cone’s 
 base it becomes an ellipsis, but if it incline towards the cone’s vertex it then becomes a hyper- 
 bola, which is the plane of any section that falls between the parabola and the triangle ; there- 
 fore there may be as many varieties of hyperbolas produced from the same cone as there may 
 be ellipses. A circle may change into an ellipsis, the ellipsis into a parabola, the parabola into 
 a hyperbola, and the hyperbola into a plain isosceles triangle; and the centre of the circle, 
 which is its focus, divides itself into two focuses, as soon as the circle begins to degenerate into 
 an ellipsis ; but, when the ellipsis changes into a parabola, one end of it opens and one of its foci 
 vanishes, and the remaining go along with the parabola, when it degenerates to a hyperbola. 
 When the hyperbola degenerates into a plain isosceles triangle, this focus becomes the vertical 
 point of the triangle, namely, the vertex of the cone; so that the centre of the cone’s base may 
 be said to pass gradually through all the sections until it arrive at the vertex of the cone, for 
 the diameter of a circle being the right line that passes through its centre or focus, from which 
 all other right lines drawn within the circle are regulated. 
 
 CONJUGATE DIAMETER, or axis of an ellipsis, the shortest of the two diameters, or that 
 bisecting the transverse axis. : 
 
 CoNJUGATE HYPERBOLAS. If there be two opposite hyperbolas, AM, am, whose principal 
 axis is the line A a, and the conjugate axis the line B 6—if there be 
 
 ane ao 
 
 two other hyperbolas whose principal axis is the line -B 6, and conju- 
 gate the line A a—then those four hyperbolas are called conjugate hyper- 
 bolas, the two former being conjugates to the latter. 
 
 CoNOID, a solid body, generated by the revolution of a conic section about its axis. 
 
 Elliptical conoid, a solid formed by the revolution of an ellipsis about one of its diameters, 
 and more generally called a spheroid. 
 
 Hyperbolical conoid is generated by the revolution of a hyperbola about its axis. 
 
 Parabolical conoid is generated by the revolution of a parabola about its axis. 
 
 CONSTRUCTION is the drawing lines, &c., as are necessary for the making any demonstration 
 appear more plain and undeniable. 
 
 ConTACcrT is when one line, plane, or body is made to touch another, and the parts that do 
 thus touch are called the points or places of contact. The contact of two spherical bodies, and 
 of a tangent with the circumference of a circle, is only in one point. 
 
 CONTENT, the area or quantity of matter or space included within certain bounds. The 
 content of a ton of round timber is 43 feet. A load of hewn timber is 50 feet. Ina foot of 
 timber are contained 1728 cubic or square inches, and as often as 1728 are contained in a piece 
 of timber, be it round or square, so many feet of timber are contained in the piece. 
 
 CoONTIGUITY is when the surface of one body touches another. 
 
MATHEMATICALLY DEMONSTRATED. 15 
 
 CONTIGUOUS ANGLES are such as have one leg common to each angle, and are not unfre- 
 quently called adjoining angles, in contradistinction to those produced by continuing their legs 
 through the point of contact, which are called opposite or vertical angles. 
 
 The sum of any two contiguous angles is always equal to two right angles. 
 
 CONTINGENT lines are those which are casual and uncertain. 
 
 GONTRACTILE FORCE, that property or power inherent in certain bodies after being 
 extended they are enabled to draw themselves up again to their original dimensions. 
 
 CONVERGING or CONVERGENT LINES are such as continually approach nearer to one 
 another, or whose distance become still less and less. These are opposed to divergent lines, 
 the distance of which becomes continually greater. Those lines which converge one way diverge 
 the other. 
 
 CONVERGING HYPERBOLA is one whose concave legs bend in towards one another, and 
 continue both the same way. 
 
 CoNVERSE. One proposition is called the converse of another, when, after a conclusion is 
 drawn from something supposed in the converse proposition, that conclusion is supposed ; and 
 then that which in the other was supposed is now drawn as a conclusion from it. Thus, when 
 two sides of a triangle are equal, the angles under these*sides are equal; and on the converse, 
 if these angles are equal, the two sides are equal. 
 
 CoNVEX, an appellation given to the exterior surface of globular bodies, in opposition to 
 the hollow inner surface of such bodies as are called concave. 
 
 CoROLLARY is a useful consequence drawn from something already advanced or demon- 
 strated: thus, it being demonstrated that a triangle which has two equal sides has also two 
 angles equal, this corollary will follow, that a triangle which has three sides equal has all its 
 three angles equal. | 
 
 Co-SINE, the sine of an arch, which is the complement of another to 90 degrees. 
 
 CROWN is a plane ring included between two concentric perimeters, and is generated by the 
 motion of some part of aright line round a centre, the moving part not being contiguous to it. 
 
 CuBE, a solid body consisting of six equal square sides. ‘The solidity of any cube is found 
 by multiplying the superficial area of one of the sides by the height. 
 
 CURVATURE OF A LINE is the peculiar manner of its bending or flexure by which it 
 becomes a curve of such and such peculiar properties. 
 
 Any two arches of curved lines touch each other when the same right line is the tangent of 
 both at the same point; but, when they are applied upon each other in this manner, they never 
 perfectly coincide, unless they are similar arches of equal and similar figures. And the curva- 
 ture of lines admit of indefinite variety; because the curvature is uniform in a given circle, 
 and may be varied at pleasure in them, by increasing or decreasing their diameters. ‘The curva- 
 ture of circles serve for measuring other lines. Of all the circles that touch a curve in any 
 given point, that is said to have the same curvature which touches it so closely that no circle 
 can be drawn through the point of contact between them ; and their circles are called the circle 
 of curvature, its centre the centre of curvature, and its semi-diameter the ray of curvature 
 belonging to the point of contact. As in all figures, rectilinear ones excepted, the position of the 
 tangent is continually varying, so the curvature is continually varying in all curvilinear figures, 
 the circle only excepted. As the curve is separated from its tangent by its curvature, so is it 
 separated from the circle curvature in consequence of the increase or decrease of its curvature ; 
 
16 GEOMETRICAL ILLUSTRATIONS 
 
 and as its curvature is greater or less, according as it is more or less inflected from the tangent, 
 so the variation of curvature is greater or less, according as it is more or less separated from 
 the circle of curvature. When any two lines touch each other in such a manner that no circle 
 can pass between them, they must have the same curvature ; for the circle that touches one so 
 closely that no circle can pass between them must touch the other in the same manner. And 
 it can be made appear that circles may touch curved lines in this manner, that there may be 
 indefinite degrees of more or less intimate contact between the curve and the circle of curva- 
 ture, and that a conic section may be described that shall have the same curvature with a given 
 line at a given point, and the same variation of a curvature, or a contact of the same kind with 
 the circle of curvature. The rays of curvature of similar arches, in similar figures, are in the 
 same ratio as any homologous lines of these figures, and the variation of curvature the same. 
 
 CuRVE, a line which, running on continually in all directions, may be cut by one right line 
 in a variety of points. Curves are divided into geometrical and transcendental. Geometrical 
 curves are those whose ordinates and abscisses being right lines the nature thereof can be 
 expressed by a finite equation having those ordinates and abscisses in it. ‘Transcendental curve 
 is such as when expressed by an equation, one of the terms thereof is a variable quantity. 
 Geometrical lines or curves are divided into orders, according to the number of dimensions of 
 the equation expressing the relation between the ordinates and abscisses, or according to the 
 number of points with which they may be cut by a right line. A curve of the first kind (for a 
 right line is not to be reckoned among curves) is the same with a line of the second order, and 
 a curve of the second order the same asa line of the third, and a line of an infinite order is 
 that which a right line can cut into an infinite number of points, such as a spiral, quadratrix, 
 cycloid, the figures of the sines, tangents, secants, and every line which is generated by the 
 infinite revolutions of a circle or wheel. 
 
 The curves of the second order have parts and properties similar to the first. As the conic 
 sections have diameters and axes, the lines cut by these are called ordinates, and the intersec- 
 tion of the curve and diameter, the vertex ; so in curves of the second order any two parallel 
 lines being drawn so as to meet the curve in three points, a right line cutting these parallels so 
 that the sum of the two parts between the secant and the curve on one side is equal to the third 
 part terminated by the curve on the other side, will cut in the same manner all other lines 
 parallel to these, and meet the curve in three parts, so that the sum of the two parts on one 
 side will be still equal to the third part on the other side. 
 
 These three parts, therefore, thus equal, may be called ordinates or applicates, the secant 
 may be styled the diameter, the intersection of the diameter and the curve the vertex, and 
 the point of concourse of any two diameters the centre. And, if the diameter be vertical to the 
 ordinates, it may be called axis, and that point where all the diameters terminate the general 
 centre. Again, as a hyperbola of the first order has two assymptotes, that of the second three, 
 that of the third, four, &c., and as the parts of any right line lying between the conic hyper- 
 bola and its two assymptotes are every where equal, so in the hyperbola of the second order, if 
 any right line be drawn cutting both the curve and its three assymptotes in three points, the sum 
 of the two parts of that right line being drawn the same way from any two assymptotes to two 
 points of the curve will be equal to a third part drawn a contrary way from the third assymptote 
 to a third point of the curve. As in conic sections not parabolical the square of the ordinate ; 
 that is, the rectangle under the ordinates drawn to contrary sides of the diameter, is to the 
 
MATHEMATICALLY DEMONSTRATED. 17 
 
 rectangle of the parts of the diameter which are terminated at the vertices of the ellipsis or by 
 hyperbola, so, in non-parabolic curves of the second order, a parallelopiped under the three or- 
 dinates is to a parallelopiped under the parts of the diameter, terminated at the ordinates, and 
 the three vertices of the figure in a certain given ratio, in which ratio, if you take three right 
 lines situated at the three parts of the diameter between the vertices of the figure, one answer- 
 ing to another, then these three right lines may be called the right lines of the figure, and the 
 parts of the diameter between the vertices the transverse lines. And as in the conic parabola, 
 having to one and the same diameter but only one vertex, the rectangle under the ordinates is 
 equal to that under the part of the diameter cut off between the ordinates and the vertex and 
 the right line, so in curves of the second order, which have but two vertices to the same 
 diameter, the parallelopiped under three ordinates is equal to the parallelopiped under two 
 parts of the diameter cut off between the ordinates and those two vertices and a given right 
 line. For as, in conic sections, when two parallels terminated on each side of the curve are cut 
 by two other parallels terminated on each by the curve, the first by the third and the second 
 by the fourth, as the rectangle under the parts of the first is to the rectangle under the 
 parts of the third, as the rectangle under the parts of the second is to that under the 
 parts of the fourth, so when four such right lines occur in a curve of the second kind, each in 
 three points, then shall the parallelopiped under the’parts of the first line be to that under the 
 parts of the third, as the parallelopiped under the parts of the second line to that under the 
 parts of the fourth. Lastly, the legs of curves, both of the first, second, and higher kinds, 
 are either of the parabolic or hyperbolic kind, a hyperbolic leg being that which approaches in- 
 finitely towards some assymptote, a parabolic that which has no assymptote. These legs are 
 best distinguished by their tangents; for, if the point of contact goes off to an infinite distance, 
 the tangent of the hyperbolic leg will coincide with the assymptote, and the parabolic leg recede 
 infinitely and vanish. The assymptote, therefore, of any leg is found by seeking the tangent 
 of it to a point infinitely distant ; and the bearing of an infinite leg is found by seeking the 
 position of a right line parallel to the tangent when the point of contact is infinitely remote, 
 for this line tends the same way towards which the infinite leg is directed. 
 
 CyYcLoIpD, a curve of the transcendental kind. Upon this curve the doctrine of pendulums 
 and time measuring instruments, in a great measure, depends. 
 
 CYLINDER, a solid body, supposed to be generated by the rotation of a parallelogram, as 
 CBEF, about one of its sides CF. If the generating parallelogram be rectan- 
 gular, as CBEF, the cylinder it produces will be a right cylinder, that is, it will 
 have its axis perpendicular to its base. If the parallelogram be a rhombus or 
 rhomboides, the cylinder will be oblique or scalenous. The section of every 
 cylinder by a plane oblique to its base is an ellipsis. The superficies of a 
 cylinder is equal to the periphery of the base multiplied into the length of 
 its side. The solidity of acylinder is equal to the area of its base, multiplied 
 into its altitude. Cylinders of the same base and standing between the same 
 parallels are equal. Every cylinder is to a spheroid inscribed in it as three to 
 two. If the altitudes of two cylinders be equal to the diameters of their bases, those cylinders 
 are to one another as the cubes of the diameters of their bases. To find a circle equal to the 
 surface of acylinder we have this theorem :—The surface of a cylinder is equal to a circle whose 
 
 F 
 
18 GEOMETRICAL ILLUSTRATIONS 
 
 radius is a mean proportional between the diameter and height of the cylinder. The diameter 
 of a sphere and altitude of a cylinder equal thereto being given to find the diameter of the 
 cylinder, the theorem is the square of the diameter of the sphere is to the square of the 
 
 diameter of the cylinder equal to it, nearly, as triple the altitude of the cylinder to double the 
 diameter of the sphere. 
 
 CYLINDRICAL, that body which is in the form of or resembling a cylinder. 
 
 CYLINDROID, a solid body approaching to the figure of a cylinder, but differing from it in 
 some respects, as having the bases elliptical, but parallel and equal. 
 
 Dara, a term for such things or quantities as are given or known, in order to find other 
 things thereby that are unknown. The word data is used for such spaces, lines, and angles 
 as are given in magnitude, or to which others can be assigned equal. 
 
 DECAGON, a plane figure with ten sides and ten angles: it is called a regular decagon when 
 all the sides and angles are equal. 
 
 DEDUCTION, a subtracting or retrenching a smaller quantity from a greater. 
 
 DEFINITION, the elucidating one method by several other not synonymous methods. 
 
 DEFINITIVE, a term applied to the termination of a question. 
 
 DEGREE, a division of a circle, including the three hundred and sixtieth part of the cir- 
 cumference. 
 
 DEMONSTRATION, evident truths or propositions already established. 
 
 DeEptH, the same as altitude, although we only use the term depth to denote how much one 
 body or part of a body is below another. ; 
 
 DESCENT is the tendency of a body from a higher to a lower place. All bodies, unless other- 
 wise determined by a force superior to their gravity, descend towards the centre of the earth. 
 
 Heavy bodies, meeting with no resistance, descend with a uniformly accelerated motion, for the 
 laws of which see Acceleration. 
 
 All bodies near the surface of the earth descend perpendicularly at the rate of sixteen feet 
 one inch in a second of time. The velocity of a body descending in an inclined plane at the end 
 of any given time is to the velocity that it would acquire by descending perpendicularly in the 
 same time as the altitude of the inclined plane is to its length. The last velocity acquired by 
 the direct descent is to the last velocity acquired in the same time by the oblique descent as the 
 absolute gravity is to the relative gravity of the descending body. The line described by the 
 direct descent is to the line described in the same time by the oblique descent as the length of 
 the oblique plane is to its height. Ifthe line described by the direct descent be to the line 
 ~ described by the oblique descent as the height of the inclined plane to its length, then the times 
 of descent will also be in the same proportion, and the last velocities equal. The last velocities 
 acquired upon several inclined planes of the same height, however different in length, are equal. 
 The time of any oblique descent through any chord of a circle drawn from its lowest point is 
 equal to the time of a direct descent through the diameter of that circle. The last acquired 
 velocities of a body, descending to the lowest point of a given circle through different chords, 
 
MATHEMATICALLY DEMONSTRATED. 19 
 
 are as those chords. The time of the descent of a body in any arch of a semi-cycloid is equal to 
 the time of its descent through any other arch, whether longer or shorter, of the same curve. A 
 body will descend sooner along an arch of a cycloid than along any other curve drawn between 
 the same points. If water runs out through a hole made in the bottom of a parabolic conoid, 
 the surface of the water will descend equal spaces in equal times. A body descends in a resist- 
 ing medium with a force only equal to the excess of its gravity above that of an equal bulk of 
 the medium. Ifa body be thrown downwards in a resisting medium with such force as shall 
 make the resistance of the medium equal to the acceleration of gravity, it will afterwards descend 
 with a uniform motion. If a body descends through any number of inclined planes, it will 
 acquire the same velocity at the end of its fall as though it had fallen through a plane equal in 
 height to the whole, and of the same inclination with the last of them. 
 
 DESCRIBENT, a line or surface which, by moving parallel to itself, describes a surface or 
 
 solid. 
 
 DETERMINATE PROBLEM, that which has but one, or at least a limited number of answers, 
 as the following problem, which has but one solution, viz. To describe an isosceles triangle upon 
 a given line whose angles at the base shall be double that at the vertex. But the following has 
 two solutions, viz. To find an isosceles triangle whose area and perimeter are given. A de- 
 terminate problem may be either simple or linear, plain, solid, or sursolid. 
 
 DEXTER, an appellation given to whatever belongs to the right side. 
 
 DIAGONAL, a right line drawn across a quadrilateral figure from one angle to the other. 
 
 Thus a 6 is called a diagonal. It is demonstrable that every diagonal - 
 
 14, 
 te 
 te, 
 “1, 
 
 divides a parallelogram into two equal parts,—that two diagonals 
 drawn in any parallelogram bisect each other. 
 
 The line fg, passing through the middle point of the diagonal of a ; 
 parallelogram, divides the figure into two equal parts. The diagonal dt \ 
 of a square is incommensurable with one of its sides. 
 
 That the sum of the squares of the two diagonals of every parallelogram is equal to the sum 
 of the squares of the four sides. This proposition is of great use in the theory of compound 
 motions; for in an oblique angled parallelogram, the greater diagonal being the subtuse of 
 an obtuse and the less of an acute angle which is the complement of the former, if the obtuse 
 angle be conceived to grow until it be infinitely great with regard to the acute one, the great 
 diagonal will be the sum of the two sides, and the less one nothing. Now two contiguous sides 
 of a parallelogram being known, together with the angle they include, it is easy to find one of 
 the diagonals in numbers, and then the foregoing proposition gives the other. This second 
 diagonal is the line which would be described by a body impelled at the same time by two forces 
 which should have the same ratio to each other as the contiguous sides have, and act in those 
 two directions; the body would describe this diagonal in the same time as it would have described 
 either of the contiguous sides in, if only impelled by the force corresponding thereto. 
 
 D1AGRAM, the explanation and demonstration of the properties of any figure, whether circle, 
 square, or triangle, &c. 
 
 DIAMETER, a right line passing through the centre of a circle, and terminated at each side 
 
20 GEOMETRICAL ILLUSTRATIONS 
 
 by the circumference thereof. The properties of the diameter are, that it divides the circum- 
 ference of a circle into two equal parts; hence we have a method of i 
 
 describing a semicircle upon any line, assuming its middle point for the yo th uM 
 centre, The diameter of a circle is the greatest of all chords. 
 
 Diameter of a curve is a right line AC, bisecting the right lines 
 DE, DE, drawn parallel to one another, and are either of finite or 
 infinite lengths. Although a right line, bisecting all parallel lines 
 drawn from one point of a curve to another, is taken in a strict sense 
 only for the diameter of a curved line, yet it may not be amiss more 
 generally to define a diameter, in saying that it is that line, whether right or curved, which 
 bisects all parallels drawn from one point of a curve to another. 
 
 Diameter conjugate. See Conjugate. 
 
 Diameter conjugate of a sphere is the diameter of the semicircle by whose rotation the 
 sphere is generated, in which sense it stands the same as axis. 
 
 Diameter of gravity, in any surface or solid, is that line in which the centre of gravity is 
 found to be placed. 
 
 DIMENSION is either length, breadth, or thickness: hence a line has one dimension, viz. 
 length ; a superficies two, viz. length and breadth; and a body or solid three, length, breadth, 
 and thickness. 
 
 DIRECTRIX or DIRIGENT signifies the line of motion along which the describent line or 
 surface is carried in the formation of any plane or solid figure. 
 Thus, if the line AB move along the line AC so that the point A 
 always keeps in the line AC, a parallelogram, as ABCD, will be 
 produced, of which the side AB is the describent, and the line 
 AC the dirigent; and also, if the surface ABCD is supposed to be 
 carried along CE in a position always parallel to itself in its first 
 situation, the solid ADFH will be formed, where the surface AD is the describent, and the line 
 CE the dirigent. 
 
 DISTANCES, ACCESSIBLE, are such as can be readily measured. 
 
 Distances, inaccessible, are such as cannot be measured by the chain, &c., im consequence of 
 
 rivers or other obstructions, and are measured by angles from stations. 
 
 DIVERGENT, or DIVERGING LINES, are those which constantly recede from each other. 
 
 DOoDECAGON, a regular polygon, consisting of twelve equal sides and angles. 
 
 DoDECAHEDRON, a solid contained under twelve equal and regular pentagons. Its solidity 
 is found by multiplying the area of one of the pentagons by twelve, and then this product by. 
 one third of the distance of the face from the centre of the dodecahedron, which is the same with 
 the centre of the circumscribing sphere. The side of a dodecahedron, inscribed in a sphere, is 
 the greater part of the side of a cube, inscribed in the same sphere, cut into extreme and mean 
 proportion. 
 
 If the diameter of the sphere be 1,0000, the side of the dodecahedron inscribed on it will be 
 ,30682 nearly. All dodecahedrons are similar, and are to one another as the cubes of their sides ; 
 their surfaces are also similar, and therefore they are as the squares of their sides. Whence, as 
 09282 is to 10,51462 so is the square of the side of any dodecahedron to the superficies 
 
MATHEMATICALLY DEMONSTRATED. 21 
 
 thereof; and as ,3637 is to 2,78516, so is the cube of the side of any dodecahedron to its 
 solidity. 
 
 DUPLICATION signifies the doubling of any thing, or multiplying by two. The duplication 
 of a cube consists in finding the side of a cube that shall be double in solidity to a given cube. 
 
 EccENTRIC, a term applied to circles and spheres which have not the same centre, and con- 
 sequently are not parallel, in opposition to concentric, where they are parallel, having one common 
 centre. 
 
 ELLiPsiIs, a curved line returning into itself, and produced from the section of a cone, by 
 a plane cutting both its sides, but not parallel to the base. An 
 easy way of describing this curve, when the transverse and 
 conjugate axes AB, ED, are given, is this :—Take the points 
 F f in the transverse axis AB, so that the distances CF, Cf, 
 from the centre C, be equal; or that the lines FD, f D, be 
 equal to AC. Then, having fixed two pins in the points F, f, 
 which are called the foci of the ellipsis, take a thread equal in 
 length to the transverse axis AB, and fastening the two ends, 
 one to the pin F and the other to f, with another pin M stretch 
 the thread tight: then, if the pin M be moved round until it 
 returns to the place from whence it first set out, keeping the 
 
 thread always extended so as to form the triangle FM f, it will describe an ellipsis whose axes 
 are AB, DE. The greater axis, AB, passing through the two foci F, f, is called the transverse 
 axis; and the lesser one, DF, the conjugate or second axis: these two always bisect each other 
 at right angles, and the centre of the ellipsis is the point C, where they intersect. Any right 
 line passing through the centre, and terminating by the curve of the ellipsis on each side, is 
 called a diameter; and two diameters which mutually bisect all the parallels to each other 
 bounded by the ellipsis are called conjugate diameters. Any right line, not passing through 
 the centre, but terminated by the ellipsis, and bisected by a diameter, is called the ordinate, or 
 ordinate applicate to that diameter. And a third proportional to two conjugate diameters is 
 called the parameter of that diameter, which is the first of the three proportionals. 
 
 The reason of the name is this: let BA, ED, be any two conjugate diameters of an ellipsis 
 (as No. 2, where there are two axes); at the end A, of the diameter AB, raise the perpendicular 
 AF, equal to the parameter, being a third proportional to AB, ED, and draw the right line 
 BF: then if any point P be taken in BA, and an ordinate PM he drawn, cutting BF in N, 
 the rectangle under the absciss AP and the line PN will be equal to the square of the ordi- 
 nate PN. For, in drawing NO parallel to AB, it appears that this rectangle, or the square of 
 the ordinate, is less than that under the absciss AP and the parameter AF, by the rectangle 
 under AP and OF or NO and OF, on account of which deficiency the name of an ellipsis 
 was given to this curve. 
 
 In every ellipsis, as AEBD, No. 2, the squares of the semiordinates MP, mp, are as the 
 rectangles under the sequents of the transverse axis APx PB, ApxpB, made by these ordi- 
 nates respectively, which holds equally true with the circle, where the squares of the ordinates 
 are equal to such rectangles, as being mean proportionals between the sequents of the diameter. 
 In the same manner, the ordinates to any diameter whatever are as the rectangles under the 
 
 G 
 
22 GEOMETRICAL ILLUSTRATIONS 
 
 sequents of that diameter. As to the other principal properties of the ellipsis, they may be 
 reduced to the following propositions :—If from any point M in an ellipsis two right lines, 
 MF, Mf, No. 1, be drawn to the foci F, f, the sum of these two lines will be equal to the 
 transverse axis AB: this is evident from the manner of describing an ellipsis. The square of 
 half the lesser axis is equal to the rectangle, under the sequents of the greater axis, contained 
 between the foci and its vertices. Every diameter is bisected in the centre C. The transverse 
 axis is the greatest, and the conjugate axis the least of all diameters. Two diameters, one of 
 which is parallel to the tangent is the least of all diameters. Two diameters, one of which is 
 parallel to the tangent in the vertex of the other, are conjugate diameters ; and, on the contrary, 
 a right line drawn through the vertex of any diameter parallel to its conjugate diameter touches 
 the ellipsis in that vertex. If four tangents be drawn through the vertices of two conjugate 
 diameters, the parallelogram contained under them will be equal to the parallelogram con- 
 tained under tangents drawn through the vertices of any other two conjugate diameters. If a 
 right line, touching an ellipsis, meet two conjugate diameters produced, the rectangle under the 
 segments of the tangent, between the point of contact and the diameters, will be equal to the 
 square of the same diameter, which is conjugate to that passing through the point of contact. 
 In every ellipsis the sum of the squares of any two conjugate diameters is equal to the sum of 
 the squares of the two‘axes. In every ellipsis the angles FGI, fGH, No.1, made by the tan- 
 gent HI, and the lines FG, fG, drawn from the foci to the point of contact, are equal to each 
 other. The area of an ellipsis is to the area of the circumscribed circle as the lesser axis is to 
 the greater, and on the contrary, with respect to an inscribed circle, so that it is a mean pro- 
 portional between two circles, having the transverse and conjugate axes for their diameters, 
 This holds equally true of all the corresponding parts belonging to an ellipsis. 
 
 Exuretic. The elliptical space is the area contained within the curve of the ellipsis, which 
 is to that of a circle described on the transverse axis as the conjugate diameter is to the trans- 
 yerse axis; or it is a mean proportional between two circles described on the conjugate and 
 transverse axes. 
 
 ENNEAGON, a polygon with nine sides. 
 
 EPICYCLOID, a curve generated by the revolution of the periphery of a circle, ACE, along 
 the convex or concave side of the periphery of another circle, DGB. C 
 The length of any part of the curve that any given point in the 
 revolving circle has described, from the time it touched the circle 
 it revolved upon, shall be double the versed sine of half the arch 
 which all the time touched the circle at rest; as the sum of the dia- 
 meters of the circles to the semidiameter of the resting circle, if the 
 revolving circle moves upon the convex side of the resting circle; but, if 
 upon the concave side, as the difference of the diameters to the semi- 
 diameter of the resting circle. The area of any cycloid or epicycloid is 
 to the area of the generating circle as the sum of double the velocity of 
 the centre, and circular motion to the velocity of the circular motion: and in the same propor- 
 tion are the areas of segments of those curves to those of analogous segments of the generating 
 circle. 
 
MATHEMATICALLY DEMONSTRATED, 23 
 
 EQUAL, a term of relation between two or more things of the same magnitude, quantity, or 
 quality. 
 
 EQUALITY, that agreement between two or more things whereby they are denominated 
 equal. The equality of two quantities is denoted by two parallel lines placed between them : 
 thus 4+2=6, that is, 4 added to 2 are equal to 6. 
 
 EQUATION of a curve, an equation expressing the nature of a curve, the relation between 
 an absciss and a corresponding ordinate. 
 
 EQUIANGULAR, a term given to those figures whose angles are equal, such as the square, 
 equilateral triangle, &c. 
 
 EQUICRURAL, the same as isosceles. 
 
 EQUIDISTANT, an appellation given to things placed at an equal distance from some fixed 
 point or place to which they are referred. 
 
 EQUILATERAL, in general, something that has equal sides, as an equilateral angle. 
 
 Equilateral hyperbola, one whose transverse diameter is equal to its parameter, and so all 
 the other diameters equal to their parameters. In such a hyperbola, the assympotes always 
 
 cut each other at right angles in the centre. 
 ERECTION, the art of raising or elevating, as the erection of a perpendicular. 
 
 ESSENTIAL, something necessarily belonging to the essence or nature of a thing, from 
 which it cannot be conceived distinct: thus the primary qualities of bodies, as extension, 
 figure, number, &c., are essential or inseparable from them in all their changes and altera- 
 tions. 
 
 EVOLUTE, a curve which, by being gradually opened, describes another curve. Such is 
 the curve BCF; if a thread, FCM, be wrapped about or applied 
 to the curve and then unwound again, the point M thereof will de- 
 scribe another curve, AMM. The part of the thread MC is called 
 the radius of the evolute, or of the osculatory circle, described on the 
 centre C with the radius MC. When the point B falls in A, the 
 radius of the evolute MC is equal to the arch BC; but, if not, to AB 
 
 and the arch BC. The radius of the evolute CM is perpendicular to \ 
 the curve AM, because the radius MC continually touches it; it is \ 
 evident from its generation that it may be described through innu- on 
 
 merable points, if the tangents in the parts of the evolute are produced, 
 until they become equal to their corresponding arches. 
 
 EXTERNAL, or EXTERIOR, a term of relation applied to the surface or outside of a body ; 
 or that part which appears or presents itself to the eye, touch, &c., in contradistinction to 
 internal. 
 
 EXTERNAL ANGLES, the angles on the outside of any right-lined figure, when all the sides 
 are severally produced and taken together, are equal to four right angles. 
 
24 GEOMETRICAL ILLUSTRATIONS 
 
 EXTREME AND MEAN PROPORTION is when a line AB is so divided in F that the rec- 
 tangle under the whole line AB, and the lesser segment FB, are equal to the: 
 square of the greater segment AF. Let a square be formed upon the line 
 AB, and one of its sides AC be equally divided in the point D; draw DB 
 and take the line DG, equal to the line BD;«then the square AGHF will 
 be equal to the rectangle FE. As the line AC is equally divided in the point 
 D and lengthened by the line AG, the rectangle CH, together with the 
 square of the line AD, will be equal to the square of the line DG or DB. 
 But the square AE, with the square of the line AD, is also equal to the 
 square of the line DB. Therefore the square AE is equal to the rectangle 
 CH. Taking away from both the rectangle CF, the pay FE will be equal to the 
 square FG. 
 
 FIGURE, in conic sections, is the rectangle made under the latus rectum and transversum of 
 the ellipsis and hyperbola. 
 
 Figure, of the diameter, is the rectangle under any diameter, and its proper parameter is 
 in the ellipsis and hyperbola, called the figure of that diameter. 
 
 Figure, of the’ superficies, included between one or more lines, and is denominated either 
 
 rectilinear, curvilinear, or mixed, according as their extremities are bounded by right lines, 
 curved lines, or both. 
 
 FINITE, something bounded or limited, in contradistinction to infinite. 
 
 Focus, in conic sections, is applied to certain points in the parabola, ellipsis, and hyperbola, 
 where the rays reflected from all the parts of these curves concur and meet. 
 
 Foci of an ellipsis are two points in the longest axis, on which, as centres, the figure. is 
 described. If from the foci two right lines are drawn, meeting each other in the periphery of 
 the ellipsis, their sum will be always equal to the longest axis; and therefore when an ellipsis 
 and its two axes are given, and the foci are required, take half the longest axis in the compasses, 
 and, setting one foot in the end of the shorter, the other will cut the longer in the focus sought. 
 
 Focus of a hyperbola is that point in the axis through which the latus rectum passes, from 
 whence if any two right lines are drawn, meeting in either of the opposite hyperbolas, their 
 difference will be equal to the principal axis. 
 
 Focus of a parabola, a point in the axis within the figure, distant from the vertex one-fourth 
 part of the latus rectum. 
 
 Foor SQUARE is the same measure in breadth and length, containing 144 square or super- 
 ficial inches ; the solid foot, or foot cube, is the same in all its three dimensions, length, breadth, 
 and depth, and contains 1728 solid inches. 
 
 FRUSTUM, a part of some solid body separated from the rest. The frustum of a cone is 
 the part that remains when the top is cut off by a plane parallel to the base, and is otherwise 
 called a truncated cone. 
 
 The frustum of a pyramid is also that part which remains after the top is cut off by a plane 
 parallel to its base. The frustum of a globe, or sphere, is any part cut off by a plane the solid 
 content of which may be found in this manner :—To three times the square of the semidiameter 
 of the base add the square of the height; multiply that sum by the height; and the product 
 multiplied by ,5286 will give the solidity of the frustum. 
 
MATHEMATICALLY DEMONSTRATED. 15) 
 
 A frustum or portion of any solid, generated by the revolution of any conic section upon its 
 axis, and terminated by any two parallel planes, may be thus compared to a cylinder of the 
 same altitude, whose base is equal to the middle section of the frustum made by a parallel 
 plane. The difference between such frustum and cylinder is always the same in different 
 parts of similar solids, when the inclination of the planes to the axis and the altitude of the 
 frustum are given. In the parabolic conoid this difference vanishes, the frustum being always 
 equal to a cylinder of the same height, upon the section of the conoid that bisects the altitude 
 of the frustum, and is parallel to its bases. In the sphere, the frustum is always less than the 
 cylinder by one-fourth part of a right angled cone of the same height as the frustum, or by 
 one-half of a sphere of a diameter equal to that height; and this difference is always the same 
 in all spheres whatever, when the altitude of the frustum is given. In the cone the frustum 
 always exceeds the cylinder by one-fourth part of the content of a similar cone that has the 
 same height. 
 
 GENERATING LINE or FIGURE is that which, by its motion, produces any other plane or 
 solid figure. A right line moved any way parallel to itself generates a parallelogram. Round a 
 point in the same plane, with one end fastened in that point, it generates a circle. One entire 
 revolution of a circle, in the same plane, generates the cycloid; and the revolution of a semi- 
 circle round its diameter generates a sphere. 
 
 GLOBE, the same as sphere. 
 
 Gnomon. [If in a!parallelogram ABCD the diameter AC be drawn, also two lines EF, 
 HI, parallel to the sides of the parallelogram, and cutting the diameter A, E RB 
 in the same point G, so that the parallelogram is by these parallels 
 
 divided into four parallelograms, then are the two parallelograms 
 DG,BG, through which the diameter does not pass, called comple- 
 ments, those through which the diameters pass, EH,FI, are called the 
 parallelograms about the diameter; and a gnomon consists of the two D 
 complements and either of the parallelograms about the diameter, viz. GD+HE+EI, or 
 GD+FI+GB. 
 
 HEIGHT is a perpendicular from the vertex or top of any right lined figure, upon the base 
 
 or side subtending it. It is likewise the perpendicular height of any object above the horizon. 
 
 HEutx, the spiral. 
 
 HEMISPHERE is the half of a globe or sphere, when it is supposed to be cut through its 
 centre in the plane of one of its great circles. The centre of gravity of a hemisphere is five 
 eighths of the radius distant from the vertex. 
 
 HEMISPHEROIDAL, an appeilation given to whatever approaches to the figure of a hemi- 
 sphere, but is not exactly so. 
 
 HEPTAGON, a figure consisting of seven sides and as many angles. 
 
 HEPTANGULAR, an appellation given to figures which have seven angles. 
 
 HETEROGENEOUS, or HETEROGENEAL, that which consists of parts of dissimilar kinds, in 
 opposition to homogeneous. . 
 
 Heterogeneous, such bodies whose density is unequal in different parts of their bulk ; or they 
 are such whose gravities in different parts are not proportionable to the bulks thereof: whereas, 
 bodies equally dense or solid in every part, or whose gravity is proportionable to their bulk, are 
 
 said to be homogeneous. 
 H 
 
26 GEOMETRICAL ILLUSTRATIONS 
 
 HEXAGON, a figure of six sides and angles; if these sides and angles be equal, it is called 
 a regular hexagon. The side of every regular hexagon, inscribed in + F 
 a circle, is equal in length to the radius of that circle ; and, by laying 
 off the radius six times upon the circumference, it will inscribe the 
 hexagon in a circle. To describe a regular hexagon on a given right D G 
 line AB draw an equilateral triangle ACB, and the vertex C will be 
 the centre of the circle, which will circumscribe the hexagon required, 
 
 ABDEFG. As 1 is to 1,672, so is the square of the side of any regu- “\ 
 lar hexagon to the area thereof nearly. 
 
 HEXAHEDRON, one of the five Platonic bodies, or regular solids, being the same as the 
 cube. 
 
 HoMocENTRIC, the same as Concentric. 
 
 HoMocGENEoUS, or HOMOGENEAL, an appellation given to things, the parts of which are 
 similar, or of the same nature and properties. | 
 
 HoMoLOGOUS, an appellation given to the corresponding sides and angles of similar figures, 
 as being proportional to each other. 
 
 Thus, in two similar triangles, ABC, DEF, the sides AB and Ez B 
 DE, BC and EF, and AC and DF are homologous. And these 
 triangles are to each other as the squares of their homologous sides. D MN c 
 
 HoRIzONTAL, relating to the horizon, or that is taken in, or on a 
 level with the horizon: thus we say a horizontal plane, being that which is perfectly parallel to 
 the horizon, and in no manner inclined thereto. 
 
 HYPERBOLA, the section GEH of a cone ABC, No. 1, made by a plane, so that the axis 
 EF of the section inclines to the opposite leg of the cone BC, which in | 
 the parabola is parallel to it, and in the ellipsis intersects it, The axis te 
 of the hyperbolical section will also meet with the opposite side of the F 
 cone, when produced above the vertex at D.—No. I. \ NB 
 
 Definitions, No.1. If at the point E, No. 2, in any plane, the end of 4. 
 the rule EH is so fixed that it may be freely carried round as about a E 
 centre, and at the other end of the rule H there is fixed the end of a 
 thread shorter than the rule, let the other end of the thread be placed at 
 the point F in the same plane (but the distance of the points E F must be £:7" \ fe ga 
 greater than the excess of the rule above the length of the thread), then | Nay Sana 
 let the thread be applied to the side of the rule EH by the pin G and 
 be stretched along it, afterwards let the rule be carried round, and in 
 the mean time let the thread, kept stretched by the pin, be con- 
 stantly applied to the rule—a line will be described by the motion of 
 the pin which is called the hyperbola. But if the extremity of the 
 same rule which was fixed in the point E is fixed in the point F, 
 and the end of the thread fixed in the point E, and the same per- 
 formed as before, there will be described another line opposite to the 
 former, which is likewise called a hyperbola, and both together called 
 opposite hyperbolas. These lines may be extended to any distance 
 from the points EF, viz., if a thread be taken of a length greater 
 than that distance. 2.The points E and F are called the foci. 3. The point C, which 
 
MATHEMATICALLY DEMONSTRATED. 27 
 
 bisects the right line between the two focuses, is called the centre of the hyperbola, or of the 
 opposite hyperbolas. 4. Any right line passing through the centre, and , 
 meeting the hyperbolas, is called a transverse diameter, and the point s | 
 in which it meets them their vertices; but the right line, which passes — 
 through the centre and bisects any right line terminated by the opposite 
 hyperbolas, and not passing through the centre, is called a right dia- 
 meter. 5. The diameter which passes through the foci is called the 
 transverse axis. 6. If from A or a, the extremities of the transverse 
 axis, there is put a right line AD equal to the distance of the centre C 
 from either focus, and with A as a centre, and the distance A1), there is 
 a circle described meeting the right line which is drawn through the 
 centre of the hyperbola at right angles to the transverse axis in B d, the 
 line B @ is called the second axis. 7. Two diameters, either of which bisects, 
 all the right lines parallel to the other, and which are terminated both ways | 4 
 by the hyperbola or opposite hyperbolas, are called conjugate dia- 
 meters. 8. Any right line not passing through the centre, but 
 terminated both ways by the hyperbola or opposite hyperbolas, 
 and bisected by a diameter is called an ordinate applied, or simply 
 an ordinate to that diameter: the diameter likewise, which is 
 parallel to the other right line ordinately applied to the other 
 diameter, is said to be ordinately applied to it. 9. The right 
 line which meets the hyperbola in one point only, but produced 
 both ways falls without the opposite hyperbolas, is said to touch it 
 at that point, or is a tangent to it. 10. If, through the vertex of 
 the transverse axis, a right line be drawn equal and parallel to the | 
 second axis, and is bisected by the transverse axis, the right 
 lines drawn through the centre and the extremities of the parallel line are called assymptotes. 
 11. The right line drawn through the centre of the hyperbola, parallel to the tangent, and 
 equal to the sequent of the tangent between the assymptotes, and which is bisected in the centre; 
 is called the second diameter of that which is drawn through the point of contact. 12. A third 
 proportional to the two diameters, one of which is the transverse, the other second to it is called 
 the right line or parameter of that diameter, which is the first of the three proportionals 
 And, 13. lastly, No. 5. If upon two right lines A a, 
 Bb, mutually bisecting each other at right angles, the - 
 opposite hyperbolas AG, ag are described, and if upon K 
 the same right lines there are described two other opposite 
 hyperbolas BK, bk, of which the transverse axis Bd is the _ 
 second axis of the two first, and the second axis of the 
 two last, Aa, is the transverse axis of the two first—these 
 four are called conjugate hyperbolas, and their assymptotes — 
 shall be common. Proposition I. No. 2. The square of 14 
 half the second axis is equal to the rectangle contained by 
 the right lines between the foci and the vertex of the transverse axis. Let A a be the transverse 
 
28 GEOMETRICAL ILLUSTRATIONS 
 
 axis, C the centre, E and F the foci, and B @ the second axis, which is evidently bisected in 
 the centre C from this definition; let AB be joined: then, as by definition 6, AB and CF are 
 equal, the squares of AC and CB together will be equal to the square CF, that is, to the 
 square of AC and the rectangle AF a together; therefore, taking away the square of AC, 
 which is common, the square of CB will be equal to the rectangle AF a. Proposition II. If 
 from any point G, Nos. 3 and 4, of the hyperbola, a right line GD be drawn at right angles to 
 the transverse axis A a, and if from the same point there is drawn the right line GF to the focus 
 nearest to that point, the half of the transverse axis CA will be to the distance of the focus from 
 the centre, viz. CF, as the distance of the perpendicular CD is to the sum of half of the trans- 
 verse axis, and the right line drawn to the focus. Let GE be drawn to the other focus, and, on 
 the axis A a produced, let there be set off AH equal to GF; then B 
 with the centre G and the distance GF describe a circle cutting the N°6. 
 axis aA in K and F, and the right line EG in the points LandM: K& @ 
 then, as EF is double CF, and FK double FD, EK shall also be 
 double CD, and as EL or Aa is double CA, and LM double GF or 
 AH, EM shall also be double CH: but because the circle EL or 
 Aa: EF: : EK: EM: and taking their halves, it will be as CA: 
 CF:: CD: CH. Proposition III. The same being proposed, if F OM: 
 from A, the extremity of the transverse axis nearest to the point G, there is set off a right line 
 AH on the axis produced equal to the distance of the point G from the focus F nearest to the 
 said extremity, the square of the perpendicular GD shall be equal to the excess of the rect- 
 angle EHF, contained under the segments between H (the extremity of the right angle AH) 
 and the foci, above the rectangle AD a, contained under the segments va. 
 
 cut off between the perpendicular and the extremities of the axis. For 
 the right line CH is in any manner cut in A, the squares of CA and CH 
 together will be equal to twice the rectangle ACH, and the square of 
 AH (7. 2.), that is, because CA, CF, CD, CH, are proportionals to twice 
 the rectangle FCD and to the square of AH or GF, that is, to twice the Kr 
 rectangle FC D and the squares of FD and DG, that is, to the squares of fr G 
 
 | 
 
 FC, CD, and DG (7. 2): therefore the two squares of CA and CH are 
 equal to the three squares of FC, CD, and DG; and, taking away the 
 squares of CA and CF from both sides, the remaining rectangle EHF will be equal to the 
 remaining rectangle AD q@ and to the square of DG (6. 2). Proposition IV. If from any 
 point G of the hyperbola there be drawn a right line parallel to the second axis B 6 meeting 
 the transverse axis A ain D, the square of the transverse axis shall be to the square of the 
 second axis as the rectangle contained under the segments of the transverse axis between the 
 parallels and its extremes to the square of the parallel. Proposition V. No. 4. If from any 
 point G of the hyperbola there is drawn a right line parallel to the transverse axis A a meeting 
 the second axis in N, the square of the second axis shall be to the square of the transverse as 
 the sum of the squares of half of the second axis and its segment between the centre and the 
 right line to the square of the line itself. Proposition VI. It is another property of the hyper- 
 bola that the assymptotes, D d, E ¢, never absolutely meet with the curve. Proposition VII. 
 If through any point F, No. 5, of the hyperbola, there is drawn a line IFL parallel to the 
 
MATHEMATICALLY DEMONSTRATED. 29 
 
 second axis and meeting the assymptotes in I and L, the rectangle contained under the right 
 lines which are intercepted between the assymptotes and the hyperbola are equal to the square 
 of half the second axis. Proposition VIII. If from any point, F, of the hyperbola there is drawn 
 to the transverse diameter, AB, a right line ordinately applied to it, FG, and from the extre™ 
 mity of the diameter there is drawn AH perpendicular to it, and equal to the latus rectum, 
 the square of the ordinate shall be equal to the rectangle applied to the latus rectum, being of 
 the breadth of the abscissa between the ordinate and the vertex, and which exceeds it by a figure 
 alike situated to that which is contained under the diameter and the latus rectum. Join BH, 
 and from the point G let there be drawn GM parallel to AH and meeting BH in M, and 
 through M let there be drawn MN parallel to AB, meeting AH in N, and let the rectangles 
 MNHO, BAHP, be completed. For the rectangle AGB is to the square of GF as AB is to 
 AH and as GB is to GM, the rectangle AGB is to the rectangle AGM, AGB shall be to the 
 square of GF as the same AGB to the rectangle AGM: therefore the square of GF is equal 
 to the rectangle AGM, which is applied to the latus rectum AH, having the breadth AG, and 
 exceeding the rectangle HAGO by the rectangle MNHO, as BAHP, from which excess the 
 
 name of hyperbola was given to this curve. 
 
 An easy method to describe the hyperbola, having the transverse diameter DE, and the foci 
 N n, given.—From N at any distance, as NF, strike an arch; and with the same opening of the 
 compasses, with one foot in E the vertex, set off EG equal to NF in the axis continued; 
 then with the distance GD, and one foot in nm, the other focus, cross the former arch in F, 
 So F becomes a point in the hyperbola: and by this method repeated may be found any other 
 point f further on, and as many more. 
 
 Ambigenal hyperbola is that which has one of its infinite legs inscribed and the other circum- 
 scribed. Equilateral hyperbola is that wherein the conjugate axes are equal. 
 
 HYPERBOLIC CYLINDROID is a solid figure, viz. two opposite hyperbolas being joined by 
 the transverse axis, and through the centre a right line drawn at right angles to the axis, and 
 about that as an axis the hyperbolas being supposed to revolve, by such revolution a body will 
 be generated which is called the hyperbolic cylindroid, whose bases and all sections parallel to 
 them will be circles. 
 
 Hyperbolic line is sometimes used for the hyperbola itself. In this sense, the plane surface 
 terminated by the curved line is called the hyperbola, or hyperbolic space, and the curved line 
 that terminates it the hyperbolic line. 
 
 HyPorTuHENUSE, the longest side AC of a right-angled triangle ABC, or it is that side 
 which subtends the right angle B. Cc 
 
 A 
 
 HYPOTHESIS denotes something supposed to be true, or taken for granted, in order to prove 
 or illustrate a point in question. 
 
 I 
 
30 GEOMETRICAL ILLUSTRATIONS 
 
 ICOSAHEDRON, a regular solid, consisting of twenty triangular pyramids, whose vertexes 
 meet in the centre of a sphere, supposed to circumscribe it, and have Fig.2 
 their height and bases equal: therefore the solidity of one of those pyramids 
 multiplied by twenty, the number of bases, gives the solid content of the 
 
 AN 
 icosahedron. /\ 4 \ 
 a 
 
 If figure 1 be drawn on pasteboard, and cut about Hal 
 half way through, and afterwards folded together, it as 
 will represent an icosahedron, as fig. 2. 
 
 INCLINATION signifies the approach, tendency, or leaning of two lines or two planes towards 
 each other, so as to make an angle. [Inclination of a right line to a plane is the acute angle 
 which that line makes with another right line drawn in the plane, through the point where the 
 inclined line intersects it, and through the point where it is also cut by a perpendicular drawn 
 from any point of the inclined plane. 
 
 INCLINED PLANE, that which makes an oblique angle with the horizon. 
 
 INCOMMENSURABLE is used where two lines, when compared to each other, have no com- 
 mon measure, however small, that will exactly measure them both. And, in general, two 
 quantities are said to be incommensurable when no third quantity can be found that is an aliquot 
 part of both. Such are the diagonal and side of a square; although each of these lines have 
 infinite aliquot parts, as the half, third, &c., yet not any part of the one, let it be ever so little, 
 can possibly measure the other. As two surfaces, which cannot be measured by a common sur- 
 face, are said to be incommensurable in power. 
 
 INDEFINITE, that which has no certain boundary. 
 
 INDIVISIBLES, the elements or principles into which any body or figure may be ultimately 
 resolved, which elements are supposed to be infinitely small: a line may be said to consist of 
 points, a surface of parallel lines, and a solid of parallel and similar surfaces, because each 
 of these elements is supposed indivisible; if in any figure a line be drawn through the ele- 
 ments perpendicularly, the number of points in that line will be the same as the number of the 
 elements; from whence we may see that a parallelogram, prism, or cylinder, is resolvable into 
 elements, or indivisibles, and all equal to each other, parallel and like to the base ; a triangle 
 into lines parallel to the base, but decreasing in arithmetical proportion, and so are the circles 
 which constitute the parabolic conoid, and the plane of a circle or surface of an isosceles cone. 
 The method of indivisibles is found of great use sometimes in short- 
 ening mathematical demonstrations ; for instance, suppose a cylinder, 
 a hemisphere, and an inverted cone to have the same base and alti- 
 tude, and cut by infinite planes parallel to the base, of which dg is 
 one. It is plain the square of dh will be every where equal to the- 
 square of kc (the radius of the sphere), the square 4c=eh square; and consequently such 
 
MATHEMATICALLY DEMONSTRATED. 3l 
 
 circles are to one another as the squares of the radii; all the circles of the hemisphere will be 
 equal to all those of the cylinder, deducting all those of the cone; therefore the cylinder, de- 
 ducting the cone, is equal to the hemisphere: but it is understood that the cone is one-third of 
 the cylinder, consequently the sphere must be two-thirds of it. 
 
 INFINITE, or infinitely great line, denotes an indefinite or indeterminate line, to which no 
 certain bounds or limits are prescribed. 
 
 INSCRIBED. A figure is said to be inscribed in another when all its angles touch the sides 
 or planes of the other figure. 
 
 INTERNAL signifies whatever is within another. 
 
 ISOPERIMETRICAL FIGURES are such as have equal perimeters or circumferences. Of iso- 
 perimetrical figures, that is the greatest which contains the greatest number of sides, or the 
 most angles ; consequently the circle is the greatest of all figures that have the same peri- 
 meter it has. Of two isoperimetrical triangles having the same base, whereof two sides of one 
 are equal and of the other unequal, that is the greater whose two sides are equal. Of those figures 
 whose sides are equal in number, that is the greatest which is equilateral an@ equiangular. 
 
 ISOSCELES TRIANGLE, one that has two equal sides, as AB, AC, where the side AB is equal 
 to AC. In every isosceles triangle the angles ABC, ACB, sub- A 
 tended by the equal sides, are equal; and a line, AD, bisecting 
 the base BC in D, is perpendicular to it. 
 
 LATUS RECTUM, in conic sections, a constant line, as parameter. 
 
 LATUS TRANSVERSUM, in the hyperbola, that part of the tranverse diameter intercepted 
 between the vertices of the two opposite sections. 
 
 LATUS PRIMARIUM, aright line belonging to a conic section drawn through the vertex of 
 the section and within it. 
 
 LEVELLING, the art of finding a line parallel to the horizon at one or more stations, in order 
 to determine the height of one place with regard to another. A truly level surface is a segment 
 of a spherical surface, which is concentric to the globe D B A H 
 of the earth. A true line of level is an arch of a great 
 circle, which is imagined to be described upon a truly 
 level surface. The apparent level is a straight line drawn 
 tangent to an arch, or line of true level. Every point of P 
 the apparent level, except the point of contact, is higher than the true level: thus, let HAG 
 be an arch of a great circle drawn upon the earth; to a person who stands upon the earth at A, 
 the line HD is the apparent level parallel to his rational horizon RR; but this line, the further 
 it is extended from its station A, the further it recedes from the centre ; for BC is longer than 
 AC, and DC is longer than BC, &c. 
 
 LEVER, an inflexible right line, rod, or beam, supported in a single point on a fulcrum, used 
 
32 GEOMETRICAL ILLUSTRATIONS 
 
 for raising weights, being either void of weight itself, or at least having such weight as may be 
 counterbalanced. Ina lever there are three things to be con- D B 
 sidered: the weight to be sustained, as O, the power by which it 
 
 is to be raised, as B, and the fulcrum D, whereon the lever is sus- 
 
 tained. Ina lever, the moving force or power that has to the 
 
 weight the same ratio which the distance of the line of direction of the weight from the fulcrum 
 has to the distance of the direction of the power from the fulcrum sustains the weight; and 
 therefore, if it be but little increased, it will raise the weight. It is manifest from this theorem 
 that the spaces which are described by a power and weight, according or contrary to their pro- 
 pensions, are proportionable to the distances of the lines of direction from the fulcrum; but 
 the velocities are proportionable to these spaces, and consequently will also be proportionable to 
 the distance. 
 
 LIMITED PROBLEM, a problem that admits of but one solution: for instance, to make a 
 circle pass through three given points not lying in a right line. 
 
 LINE, a quantity extended in length only, without any breadth or thickness. There are 
 two kinds of lines, viz., right lines and curved lines. If the pointA a - 
 move towards B, by its motion it describes a line which, if the point go $ 
 the nearest way towards B, will be a right or straight line, whose distine- 
 tion therefore is the nearest or shortest distance between any two 
 points, or a line all whose points tend the same way. If the points F 
 go about as one of the lines ABC, or A c B, it will trace out a crooked 
 line, as in the upper, Ac B, or else two more straight ones, as in the 
 lower, ACB. Right lines are all of the same, but curves are of an in- 
 finite number of species. 
 
 Lines in perspective are, first, the geometrical line, which is a right 
 line drawn in any manner on the geometrical plane ; secondly, the ter- 
 restrial or fundamental line, being a right line wherein the geometrical 
 plane and that of the draught intersect each other; such is the line 
 NI, formed by the intersection of the geometrical plane LM, and the perspective plane HK; 
 thirdly, the line of the front is any right line parallel to the terrestrial line ; fourthly, the vertical 
 line, the common section of the vertical and the draught; fifthly, the visual line, the line or 
 ray imagined to pass from the object to the eye; sixthly, the line of station, the common 
 section of the vertical and geometrical planes; seventhly, the line of an object whence the 
 
 appearance is sought for in the draught. 
 
 Line of direction, that line wherein a body actually moves, or would move were it not pre- 
 vented. It also denotes the line that passes through the centre of gravity of the heavy body to 
 the centre of the earth, which must also pass through the fulerum,’or support, of the heavy body, 
 without which it would fall. 
 
 LINEAR PROBLEM, that which may be solved geometrically by the intersection of two 
 right lines. This is called a simple problem, and is capable of but one solution. 
 
 Lozenck, the rhombus, a quadrilateral figure, consisting of four equal and parallel sides, 
 two of whose opposite angles are acute and the other two obtuse; the distance between the 
 
MATHEMATICALLY DEMONSTRATED. 33 
 
 two obtuse ones, being always equal to the length of one side: when the sides are unequal, 
 this figure is called a rhomboides. 
 
 MEAN, in general, denotes the middle between two extremes: as mean distance, mean 
 proportion, &c. 
 
 MENSURATION denotes the art of measuring lines, superficies, solids, &c. 
 
 MINUTE, the sixtieth part of a degree of a circle. 
 
 OBLIQUE, a deviation from the perpendicular. An oblique angle is either an acute or an 
 obtuse one, or any angle except a right one. 
 
 Oblique line, that which falling on another line makes oblique angles with it, viz., one acute, 
 the other obtuse. 
 
 OBLONG denotes a figure that is longer taan broad; such is a parallelogram, an ellipsis, &c. 
 
 OBTUSE, in opposition to acute, sharp, &c. 
 
 OCCULT LINES are lines made scarcely perceptible, or those made by a fine pencil, which 
 can be removed when the work is completed. 
 
 OcTAGON, a figure of eight sides and angles; and, when all the sides and angles are equal, 
 it is a regular octagon. 
 
 OCTAHEDRON, one of the five regular bodies, consisting of eight equal and equilateral 
 triangles. The square of the side of the octahedron is to the square of the diameter of the 
 circumscribing sphere as one to two. If the diameter of the sphere be 2, the solidity of the 
 octahedron inscribed in it will be 1.83333 nearly. The octahedron is two pyramids put together 
 at their bases, therefore its solidity may be found by multiplying the quadrangular base of 
 either by one-third of the perpendicular height of one of them, and doubling the product. 
 
 OPPOSITE SECTIONS are two hyperbolas made by cutting two opposite cones by the same 
 plane. 
 
 OPPOSITION, the relation of two things between which a line may be drawn perpendicular 
 to both. 
 
 ORDINATES, or ordinate applicates, are parallel lines terminating in a curve, and bisected 
 by adiameter,as AD. The half of these, as MP, mp, is properly the semi- 
 ordinate, though commonly called ordinate. 
 
 ORTHOGRAPHY, the art of drawing or delineating the plan of any object, and of expressing 
 the heights or elevations of each part. It is called orthography from its determining objects by 
 perpendicular lines falling on a geometrical plane. 
 
 In perspective, it is the side or plane that lies parallel to a straight line that may be imagined 
 to pass through the outward convex points of the eye, continued to a proper or convenient 
 length. 
 
 OVAL, an oblong curvilinear figure, otherwise called ellipsis. See ELLIPSIS. 
 
 K 
 
54 GEOMETRICAL ILLUSTRATIONS 
 
 PARABOLA, a figure arising from the section of a cone when cut by a plane parallel to one of 
 its sides. ‘To describe a parabola, draw a right line ,y 5 zB LM 
 AB, and assume a point C, without it; then, in the 
 same plane, with the line and point, place a square 
 rule DEF so that the side DE may be applied to 
 the right line AB, and the other EF turned to the 
 side in which the point C is situate. This done, and 
 the thread FGC, exactly of the length of the side of 
 the rule EF, being fixed at one end to the extremity 
 of the rule F, and the other to the point C, slide the 
 side of the rule DE along the right line AB, and, by 
 the means of a pin G, continually apply the thread to the side of the rule EF, to keep it always — 
 stretched as the rule is moved along: the point of the pin will describe a parabola, GHO. The 
 right line, AB, is called the directrix. The point C is the focus of the parabola. All perpen- 
 diculars to the directrix, as LK, MO, &c., are called diameters; the points where these cut the 
 parabola are called its vertices; the diameter, BI, which passes through the focus C, is called 
 the axis; and its vertex, H, the principal vertex; a right line terminated on each side by the 
 parabola and bisected by a diameter is called the ordinate applicate, or simply the ordinate to 
 that diameter. A line equal to four times the segment of any diameter, intercepted between 
 the directrix and the vertex where it cuts the parabola, is called the latus rectum, or parameter, 
 of that diameter. A right line which touches the parabola in one point only, and, being produeed 
 on each side, falls without it, is a tangent to it in that point. Any right line, as GE, drawn 
 from any point of the parabola, G, perpendicular to AB, is equal to a line GC drawn from the 
 same point to the focus. This is evident from the description, for the length of the thread, 
 FGC, being equal to the side of the rule, EF, if the part, FG, common to both, be taken away, 
 there remains EG=GC. The reverse of this proposition is equally evident, viz., if the dis- 
 tance of any point from the focus of a parabola be equal to the perpendicular drawn from it to 
 the directrix, that point will fall in the curve of the parabola. 
 
 PARABOLIC CONOID, a solid generated by the rotation of a parabola about its axis: its 
 solidity is equal to half of its circumscribing cylinder. The circles 
 conceived to be the elements of this figure are in arithmetical propor- 
 tion, decreasing towards the vertex. A parabolic conoid is to a cylinder 
 of the same base and height as one to two, and to acone of the same 
 base and height as one and a half to one. 
 
 Parabolic cuneus, a solid figure formed by multiplying all the DB’s 
 into the DS’s. Or, which amounts to the same, on the base APB erect 
 a prism whose altitude is AS; this will be a parabolic cuneus, which 
 will be equal to the parabolic pyramidoid, as the compound rectangles in one are severally 
 equal to all the component squares in the other. 
 
 Parabolic pyramidoid, a solid figure generated by supposing all the squares of the ordi- 
 nate applicates in the parabola so placed that the axis passes through all the centres at 
 right angles, in which case the aggregate of the planes will form the parabolic pyramidoid. The 
 
MATHEMATICALLY DEMONSTRATED. 35 
 
 solidity thereof is obtained by multiplying the base by half the altitude, the reason of which is 
 quite obvious ; for the component planes being series of arithmetical proportionals, beginning 
 from the cypher, their sum will be equal to the extremes multiplied by half the number o¢ 
 terms. 
 
 Parabolic space. The segment of a parabolic space is that space included between two 
 ordinates. 
 
 Parabolic spindle is eight-fifteenths of its circumscribing cylinder. 
 
 PARALLEL, an appellation given to lines, surfaces, and bodies every where equidistant from 
 each other, and which, though infinitely produced, would never "4 : 
 meet: thus, the line OP is parallel to QR. If two parallels, OP and P Seite oO 
 QR, be cut by a transverse line, ST in A and B, the alternate 
 angles, w and y, are equal. The external angle, u, is equal to the 
 
 internal opposite one, y. The two internal opposite ones, «x and s, 
 are also equal to the two right angles. 
 
 It is shown in the principles of optics that if the eye be placed between two parallel lines 
 they will appear to converge towards a point opposite the eye, and, if they run to such length 
 that the distance between them be but as a point thereto, they will then appear to coincide: 
 Parallel lines are described by letting fall equal perpendiculars, and drawing lines through the 
 extremes. 
 
 Parallel planes are such planes as have all their perpendiculars drawn between them equal to 
 each other. 
 
 PARALLELOPIPED, a regular solid comprehended under six parallelograms, the opposite 
 being similar, parallel, and equal. All parallelopipeds, prisms, cylinders, 
 &c., whose bases and heights are equal, are themselves equal. 
 
 A diagonal plane divides a parallelopiped into two equal prisms, so 
 that a triangular prism is half a parallelopiped upon the same base and 
 of the same altitude. All parallelopipeds, prisms, cylinders, &c., are 
 in a ratio compounded of their bases and altitudes: therefore, if their 
 
 bases be equal, they are in proportion to their altitudes. All parallelopipeds, cylinders, cones, 
 &c., are in a triplicate ratio of their homologous sides, and also of their altitudes. Equal 
 parallelopipeds, prisms, cones, cylinders, &c., reciprocate their bases and altitudes. To measure 
 the surface and solidity of a parallelopiped, find the areas of the parallelogram, ILMK, LMON, 
 and OMKP;; add these into one sum and multiply it by two; the factum will be the surface of 
 the parallelopiped. Or if the base, ILMK, be multiplied by the altitude, MO, the product 
 will be the solidity. 
 
 PaRALLELISM, the situation or quality whereby any thing is denominated parallel. 
 
 PARALLELOGRAM, a quadrilateral right-lined figure, whose opposite sides are parallel and 
 equal to each other. It is generated by an equal motion of a right line always parallel to 
 itself. When it has all its four angles right, and only its opposite sides equal, it is called a 
 rectangle. When the angles are all right, and the sides equal, it is called a square. If all the 
 sides are equal, and the angles unequal, it is called a rhombus or lozenge: and, if the sides and 
 angles be unequal, it is called a rhomboides. 
 
36 GEOMETRICAL ILLUSTRATIONS 
 
 In every parallelogram, as ABCD, a diagonal, DA, divides it into two equal parts ; the 
 angles diagonally opposite BC and AD are equal ; the opposite angles .- E A 
 of the same side CD and AB, &c., are together equal to two right \ 
 angles, and the two sides together greater than the diagonal. ‘Two 
 parallelograms, ABCD and ECDF, on the same or equal base, CD, 
 and the same height, AC, or between the same parallels, AF,CD, are 
 equal: and the two triangles, CDA and CDF, on the same base, and of 
 the same height, are also equal. i \ 
 
 Every triangle, CFD, is half a parallelogram, ACDB, upon the same bad + 
 or an equal base, CD, and of the same altitude, or between the same parallels. A triangle is 
 equal to a parallelogram having the same base and half the altitude, or half the base and the 
 same altitude. Parallelograms are, in a given ratio, compounded of their bases and altitudes. 
 
 If the altitudes be equal, they are as the bases, and conversely. In similar parallelograms and 
 triangles, the altitudes are proportional to the homologous sides, and the bases are cut propor- 
 tionably thereby. Similar parallelograms and triangles are in duplicate ratio to their homologous 
 sides, as also of their altitudes and the segments of their bases; they are therefore as the 
 squares of the sides, altitudes, and homologous segments of the bases. In every parallelogram 
 the sum of the squares of the two diagonals is equal to the sum of the squares of the four sides. 
 For, if the parallelogram be rectangular, it follows that the two diagonals are equal ; and, con- 
 sequently, the square of a diagonal, or, which comes to the same, the square of a hypothenuse 
 of a right angle, is equal to the squares of the sides. If the parallelogram be not rectangular, 
 the diagonals will be unequal, which is the general case. The proposition then becomes of vast 
 extent ; for instance, in the whole theory of compound motions, &c. To find the area of the 
 rectangled parallelogram, ABCD, find the length of the sides, AB and AC; multiply AC, sup- 
 posed=345 into AB=33; the product will be the area of the parallelogram, namely, 11,385. 
 Rectangles are in a ratio compounded of their sides AC and AB. If, therefore, three lines be 
 continually proportional, the square of the middle one is equal to the rectangle of the two 
 extremes ; and, if four lines be proportional, the rectangle under the two extremes will be equal 
 to that under the two middle terms. Other parallelograms, not rectangular, have their areas 
 found by resolving them by diagonals into two triangles, and adding the areas of the separate 
 triangles into one sum. 
 
 PARAMETER, a constant line, otherwise called latus rectum. The parameter is said to be 
 constant, because, in the parabola, the rectangle under it and any absciss is always equal to the 
 square of the corresponding semi-ordinate ; in the ellipsis and hyperbola, it is a third propor- 
 tional to the conjugate and transverse axis. 
 
 PELECOIDES, a figure contained under the two inverted quadrantal ares, AB and AD, 
 
 and the semicircle BCD. The area of the pelecoides is demonstrated to be ce 
 
 equal to the square AC, and that again to the parallelogram EB. It is 
 
 equal to the square AC, because it wants of the square on the left the two 3B D 
 
 segments AB and AD, which are equal to the two segments BC and CD, 
 
 by which it exceeds on the right. io 5 
 A 
 
 PENTAGON, a figure of five sides and five angles. If the five sides be equal, the angles are 
 
MATHEMATICALLY DEMONSTRATED. 37 
 
 also equal ; and the figure is called a regular pentagon. Such is ABCDE, inscribed in the 
 circle. The most considerable property of a pentagon is, that one of B 
 
 its sides, DE, is equal in power to the sides of a hexagon and a deca- pic aes 
 
 gon inscribed in the same circle, ABCDE, that is, the square of the as 
 
 side DE is equal to the sum of the squares of the sides d E, and Eb. **\ . \~ 
 The area of the pentagon, like any other polygon, may be obtained \\ 4 
 
 by resolving it into triangles. The dodecahedron, which is the fourth  \ \ A y) 
 regular solid, consists of twelve pentagons. x <——-— 
 
 PERIMETER, the bounds or limits of any figure or body. The perimeter of surfaces or 
 figures are lines, those of bodies are surfaces. In circular figures, instead of perimeter, we say 
 circumference, or periphery. 
 
 PERIPHERY, the circumference of a circle, ellipsis, or any other regular curvilinear figure. 
 
 PERPENDICULAR, a line falling directly upon another, so as to make equal angles on each 
 side, called also a normal line: thus, the line AB is perpendicular B 
 to the line CD; that is, it makes right angles therewith. 
 
 From the nature of a perpendicular it follows, That the perpendicu- 
 larity is natural, that is, if-a Ime, as AB, be perpendicular to another 
 CD, that is also perpendicular to the first. That only one perpendicu- 
 lar can be drawn from one point in the same place. That if a perpen . 
 dicular be continued through the line it was drawn perpendicularly to, ” 
 the continuation will also be perpendicular to the same. That if there D- 
 be two points of a right line, each of which is of an equal distance from 
 two points of another right line, that line will be perpendicular to the 
 other. That line which is perpendicular to another, is also perpendicu- 
 lar to all the parallels of the other. A perpendicular line is the shortest 
 of all those which can be drawn from the same point to the same right 
 line. The shortest distance of a point from a line isa right line drawn & 4 ¢ 
 from the point perpendicular to the line or plane, the altitude of a figure ¢ 
 is a perpendicular let fall from the vertex to the base. 
 
 To raise from the point A a line perpendicular to the line BC, make AB=AC, and from the 
 points B and C as centres, with the same opening of the dividers, describe two arches cutting 
 each other in the point D, and the line DA shall be the perpendicular described ; that is, the 
 angles, DAB, DAG, shall be equal, because all the sides of the triangle 
 DAB will be equal to all the sides of the triangle DAC. 
 
 To let fall a perpendicular upon a line, BC, from a point given without 
 it, A, on the point A describe an arch which shall cut the line in the 
 points Band C: then making the equilateral triangle BEC, the line 
 AE will be perpendicular to the line BC. The triangles, ABD and ACD, 
 have the side AD common, and AB is equal to AC, and the angle BAD 
 is equal to the angle CAD, they are equal in every respect, and the angle 
 D equal on each side: therefore, by this definition, the line AD is per- 
 pendicular to BC. | 
 
 To erect a perpendicular at the end of a given line, suppose at R, open the dividers to any 
 
 L 
 
38 , GEOMETRICAL ILLUSTRATIONS 
 
 convenient distance, and setting one foot in ©, draw the semicircle PRS, Lay a ruler from 
 
 S through C, and it will find the point P in the circumference, then draw = 
 PR, which is the perpendicular required. For the angle PRS, being in a : on® 
 semicircle, must be a right one; consequently, PR must be perpendicular bias . 
 
 to SR. A line is said to be perpehdicular to a plane when it is perpendicu- 
 lar to more than two lines drawn in that plane ; and a plane is said to be per- 
 pendicular to another plane, when a line in one plane is perpendicular to the ‘ ahs es 
 ial yb: 
 
 Perpendicular to a parabola is a right 1 line cutting the parabola in the point in which any 
 other right line touches it, and is also itself perpendicular to that tangent. 
 
 PLANE denotes a plain surface, or one that lies evenly between its bounding lines: and as 
 a right line is the shortest extension from one point to another, so a plain surface is the shortest 
 extension from one line to another. 
 
 PLANIMETRY, that part of geometry which considers lines and plain figures, without con- 
 sidering their height or depth. 
 
 PoINT is a quantity which has no parts, or which is indivisible. Points are the ends and 
 extremities of lines. Ifa point was supposed to be moved, it would, by its own motion, de- 
 scribe a line. ' 
 
 POLYGON, a figure with many sides, or whose perimeter consists of more than four sides; 
 such are the pentagon, hexagon, heptagon, &c. Every polygon may be divided into as many 
 triangles as it has sides, for if you assume a point, as A, any where within the polygon, and from 
 thence draw lines to every angle ab, ac, ad, &c., they will make as 
 many triangles as the figure hath sides. Then, if the polygon has six 
 sides, the double of that.is twelve, from whence take four, and there 
 remain eight, therefore all the angles, bcdefg, of that polygon taken 
 together, are equal to eight right angles. For the polygon, having 7 
 six sides, is divided into six triangles, and the three angles of each 
 are equal to two right ones; so that all the angles together make 
 twelve right ones: but each of these triangles has one angle in the f 
 
 point a, and by it they complete the space round the same point; N22. : 
 YY > 
 Ze | > 
 Bt 
 
 the right angles of the hexagon. It is therefore plain, the figure / 
 
 and all the angles about a point are known to be equal to four right 
 Every polygon circumscribed about a circle is equal to a rectangled Bp aw 
 
 Nel. rg 
 
 ones, therefore the four taken from twelve leave eight, the sum of 
 
 has twice as many right angles as it has sides, excepting four 
 
 triangle, one of whose legs shall be the radius of the circle, and the 
 
 other the perimeter (or sum of all the sides) of the polygon. Every regular polygon is equal 
 to a rectangled triangle, one of whose legs is the perimeter of the polygon, and the other a per- 
 pendicular drawn from the centre to one of the sides of the polygon. And every polygon cir- 
 cumscribed about a circle is larger than it; and every polygon inserted is less than the circle. 
 The perimeter of every polygon circumscribed about a circle is greater than the circumference 
 of the circle, and the perimeter of every polygon inscribed is less. A circle is equal to a vight- 
 angled triangle, whose base ig the circumference of the circle, and its height the radius of it. 
 
- 
 
 MATHEMATICALLY DEMONSTRATED. 39 
 
 For this triangle will be less than any polygon circumscribed, and greater than any inscribed ; 
 because the circumference of the circle, which is the base of the triangle, is greater than the 
 compass of any inscribed, therefore it will be equal to the circle. For ifthe triangle be greater 
 than any thing that is less than the circle, and smaller than any thing that is greater than the 
 circle, it follows that it must be equal to the circle. This is called the quadrature or squaring 
 of the circle ; that is, to find a right-lined figure equal to a circle, upon a supposition that the 
 basis given is equal to the circumference of the circle: but actually to find a right line equal to 
 the circumference of a circle is not yet discovered geometrically. Ona regular polygon to cir- 
 cumscribe a circle, or to circumscribe a regular polygon upon a circle, bisect two of the angles 
 of the given polygon, Aand B (No. 2), by the right lines AF, BF, and on the point F where 
 they meet, with the radius AF, describe a circle which will circumscribe the polygon. . Next, 
 to circumscribe a polygon, divide 360 by the number of sides required, to find e Fd; which set 
 off from the centre F, and draw the line de, on which construct the polygon as in the following 
 problem :—On a given line to describe any given regular polygon, find.the angle of the poly- 
 gon in the table, and in E set off an angle equal thereto ; then drawing EA=ED through the 
 points EAD, describe a circle, and in this applying the given right line as often as you can, 
 the polygon will be described. To find the sum of all the angles in any given regular polygon, 
 multiply the number of sides by 180°, from the product subtract 360°, and the remainder will 
 be the sum required: thus, in a pentagon, 180 x5 = 900, and 900 — 360= 540 = the sum of all 
 the angles in a pentagon. ‘To find the area of a regular polygon, multiply one side of the 
 polygon by half the number of sides; and then multiply this product by a perpendicular let fall 
 from the centre of the circumscribing circle, and the product will be the area required: thus, 
 if AB (the side of a pentagon, =54x23,)=135, and 185x29 (the perpendicular)=3915, the 
 area required. To find the area of an irregular polygon, let it be resolved into triangles, and 
 the sum of the areas of these will be the area of the polygon. 
 
 POLYHEDRON denotes a body, or solid, comprehended under many sides or planes. 
 
 PRISM, an oblong solid contained under more than four planes, whose bases are equal, 
 parallel and alike situated. The prism is generated by the motion of a rectilinear : 
 figure, as ABC, descending always parallel to itself along the right line AE. Perl 
 If the describent be a triangle, the body is said to be a triangular prism; if square, 
 a quadrangular one. From the principle of the prism, it is evident it has two 
 equal and opposite bases, ABC and EDF; and it is terminated by as many 
 _parallelograms as the base consists of sides; and all the sections of a prism, 
 parallel to its base, are equal. Every triangular prism may be divided into three F D 
 equal pyramids. To measure the surface of any prism, find the area of each side, Eb 
 whether a triangle, parallelogram, or other rectilinear figure, as directed under these articles, 
 and the sum of all these taken together make the whole superficies of the prism. 
 
 The solid content of a given prism may be found thus: let the area of the base of the prism 
 be measured, as directed under the article TRIANGLE; and this area multiplied by the height, 
 and the product will give the solid content of the prism. 
 
 PRISMOID, a solid figure, bounded by several planes whose bases are right-angled parallel- 
 ograms, parallel and alike situate. 
 
40 GEOMETRICAL ILLUSTRATIONS 
 
 PROBLEM is a proposition, wherein some operation or construction is required, as to divide — 
 a line or angle, erect or let fall perpendiculars, &c. A problem consists of three parts; the 
 proposition, which expresses what is to be done; the solution, wherein the several steps whereby 
 the matter required is to be effected, are rehearsed in order ; and, lastly, the demonstration, 
 wherein is shown, that by doing the several things prescribed in the solution the object required 
 
 is obtained. 
 
 Propuct, the factum of two or more numbers, or lines, &c., into one another: thus 
 5x4=20, the product required. 
 
 PROPOSITION is either some truth advanced and shown to be such by demonstration, or 
 some operation proposed and its solution shown. If the proposition be deduced from several 
 theoretical definitions compared together, it is called a theorem ; if from a praxis, or series of 
 operations, it is called a problem. | 
 _ Puncrum. The punctum formatum, or generatum in conics, is a point determined by the 
 intersection of a right line drawn through the vertex of the cone, to a point in the plane of the 
 base that constitutes the conic section. 
 
 PyRAMID, a solid standing on a triangular, square, or polygonal basis, and terminating in a 
 point at the top: or, it is a solid figure, consisting of several 
 triangles, whose bases are all in the same plane and have one 
 common vertex. The superficies of a given pyramid_ are 
 readily formed by measuring their triangles separately ; for 
 their sum added to the area of the base is the surface of the 
 pyramid required, ‘To find the solid content of a pyramid, 
 the area of the base being found, let it be multiplied by the 
 third part of its height, or the third part of the base by the 
 height, and the product will give the solid content. If the 
 solid content of a frustrum of a pyramid is required, let the 
 solid content of the whole pyramid be found; from which 
 subtract the solid content of the part wanting, and the solid content of the frustrum 
 or broken pyramid will remain. Every pyramid is equal to one-third of its circumscrib- 
 ing prism, or that has the same base and height; that is, the solid content of the prism, * 
 ABD, No. 2, is equal to one third of the prism, ABFE. Suppose the base Aa, Bd, a square,, 
 then does the pyramid consist of an infinite number of such squares, whose sides are continually 
 increasing in arithmetical progression, beginning at the vertex or point D; its base, Aa, Bd, 
 being the greatest term, and its perpendicular height, CD, the number of all the terms: but 
 the last term multiplied into the number of terms will be triple the sum of all the series, or 
 “'=S=the solid content of the pyramid. All pyramids are in a ratio compounded of their 
 bases and altitudes; so that, if their bases be equal, they are in proportion to their altitudes, &c. 
 Equal pyramids reciprocate their bases and altitudes; that is, the altitude of one is to that of 
 the other as the base of the one is to the base of the other. 
 
 QUADRANGLE, a quadrilateral figure, or one consisting of four sides and four angles. 
 
 ~ QUADRANT, an arch of a circle containing ninety degrees, or the fourth part of the entire 
 periphery. Sometimes the space, or area, included between this arch and two radii drawn from 
 
 . ~ 
 
\ 
 
 MATHEMATICALLY DEMONSTRATED. Al 
 
 the centre to each extremity thereof is called a quadrant, or more properly a quadrantal space, 
 being the quarter of an entire circle. 
 
 QUADRANTAL TRIANGLE, a spherical triangle, one of whose sides at least is the quadrant 
 of a circle, and one of its angles a right angle. 
 
 ‘QUADRATURE denotes the squaring or reducing a figure to a square! The finding of a 
 square, which shall contain just as much surface or area as a circle, ellipsis, or triangle, &c., is 
 the quadrature of a circle, ellipsis, &c. 
 
 QUADRILATERAL, a figure whose perimeter consists of four right lines, making four angles; 
 whence it is also called a quadrangular figure. The quadrilateral figures are either a parallelo- 
 gram, trapezium, rectangle, square, rhombus, or rhomboides, 
 
 QUINDECAGON, a plane figure with fifteen sides and fifteen angles, which, if the sides be 
 all equal, is termed a regular quindecagon, and an irregular when otherwise. The side of a 
 regular quindecagon inscribed in a circle is equal in power to the half difference between the 
 side of the equilateral triangle and the side of a pentagon inscribed in the same circle; also the 
 difference of the perpendiculars let fall on both sides taken together. | 
 
 RADIAL CURVES are curves of the spiral kind, whose ordinates (if they may be so called) 
 all terminate in the centre of the including circle, appearing like radii of that circle. 
 
 Ravivs, the semidiameter of a circle, or a right line drawn from the centre to the circum- 
 ference. 
 
 RATIO is that relation of homogeneous things which determines the quantity of one from 
 another without the intervention of a third. 
 
 RECIPROCAL FIGURES, those which have the antecedents and consequents of the same 
 ratio in both figures. Thus the side A: B:: ©: D; or 
 12:4::9:3; that is, as much as the side A in the 
 first rectangle is longer than B, so much deeper is * 
 the side C in the second rectangle than the side D 
 in the first, consequently the greater length of the 
 one is compensated by the greater breadth or depth 
 of the other; for as the side A is one-fourth longer 
 than C, so B is one-fourth longer than D, and the 
 rectangles of course equal, that is, Ax D=BxC, or 
 12x3=4x9=36. This is the foundation of the 
 theorem, viz., that the rectangle of the extremes is always equal to the means ; and, conse- 
 quently, the reason of the rule of three. Therefore it follows, that if any two triangles, paral- 
 
 _lelograms, prisms, parallelopipeds, pyramids, cones, or cylinders, have their bases and altitudes 
 
 reciprocally proportional, those two figures or solids are equal to each other ; and, on the, other 
 
 hand, if they are equal, then their bases and altitudes are reciprocally proportional. 
 RECTANGLE, the same as a right-angled parallelogram. 
 
 RECTANGLED, RECTANGULAR, or RIGHT-ANGLED, appellations given to figures and solids 
 which have one or more right angles: a triangle with one right angle is termed a rectangled 
 triangle; also parallelograms with right angles, squares, cubes, &c., are rectangular. Solids, 
 
 M 
 
42 GEOMETRICAL ILLUSTRATIONS 
 
 as coues, cylinders, &c., are also said to be rectangular, with respect to their situation, when 
 their axes are perpendicular to the plane of the horizon. 
 
 RECTIFICATION is the finding of a.right line equal in length to a curve. 
 
 ReEcTILINEAR, right-lined, are figures whose perimeter consists of right lines. 
 
 REGULAR denotes any thing agreeable to the rules of art: as a regular figure in geometry 
 whose sides, and consequently the angles, are equal, and a regular figure with three or four 
 sides is commonly termed a triangle, or square, as all others with more sides are called regular 
 polygons. All regular figures may be inscribed in a circle. A regular solid, called also a 
 platonie body, is that which is terminated on all sides by regular and equal planes, and whose 
 solid angles are all equal. The regular bodies are the five following: first, the tetrahedron, 
 which is a pyramid comprehended under four equal and equilateral triangles: second, the hexa- 
 hedron, or cube, whose surface is composed of six equal squares: third, the octahedron, which 
 is bounded by eight equal and equilateral triangles: fourth, the dodecahedron, whieh is con- 
 tained under twelve equal and equilateral pentagons: fifth, the icosihedron, consisting of twenty 
 equal and equilateral triangles. The proportion of the five regular bodies, the diameter of the 
 sphere being 2. The circumference of the greatest circle is 6.28318. Superficies of the greatest 
 circle 3.14159. Superficies of the sphere 12.56637. Solidity 
 of the sphere 4.18859. Side of the tetrahedron 1.62299. Su- g 
 perficies of a tetrahedron 4.6188. Solidity of a tetrahedron . A 
 0.15132. Side of a cube or hexahedron 1.1547. Superficies of the Ge 
 hexahedron 8.000. Solidity of the hexahedron 1.5896. Side 
 of an octahedron 1.41421. Superficies of the octahedron 6.9282. 
 Solidity of the octahedron 1.33333. Side of the dodecahedron 
 0.71364. Superficies of the dodecahedron 10.51462. Solidity 
 of the dedeeahedron 2.78516. Side of the icosihedron 1.05146. 
 Superficies of the icosihedron 9.57454. Solidity of the icosihedron 
 2.53615. If one of these five regular bodies were required to 
 be cut out of the sphere of any other diameter, it will be as the 
 diameter of the sphere two is to the side of any one solid inscribed in the same (suppose the cube 
 1.1547), so is the diameter of any sphere (suppose 8) to 9.2376, the side’ of the cube inscribed 
 in this latter sphere. Let dr be the diameter of any sphere, and da one-third of it equal to ad 
 and br. Erect the perpendiculars ae, cf, and bg, and draw de, df, er, fr, and gr. Then will 
 re-be as the side of the tetrahedron, df the side of the hexahedron, de the side of the octa- 
 hedron. Cut de in extreme and mean proposition in hf, and dh will be the side of the dode- 
 cahedron. Set the diameter dr up perpendicularly at 7, and from the centre c to its-top draw 
 the line cg, cutting the circle ing. Let fall the perpendicular gd; then is dr the side of the 
 icosihedron. . 
 
 RESIDUAL FIGURE, the figure remaining after taking a lesser from a greater. 
 RHOMBOIDES, a quadrilateral figure whose opposite sides and angles are equal, but neither 
 equilateral nor equiangular ; as the figure NOPQ. Pp 
 
 So 
 
 Q NK 
 
MATHEMATICALLY DEMONSTRATED. 43 
 
 RHoMBUS, an oblique-angled parallelogram, or a quadrilateral figure whose sides are equal 
 and parallel, but.the angles unequal, two of the opposite ones being x 
 obtuse, and the other two acute, as ABCD. To find the area of a £] 
 rhombus, upon CD, assumed as a base, let fall the perpendicular Ae, 
 which is the altitude of the figure ; then multiply the base by the alti- 
 tude, the product will be the area. 
 
 RIGHT signifies the same as strait: therefore a strait line is a right line. 
 
 RorTATION, a term applied to the circumrotation of any surface round a fixed and immovable 
 line, which is called the axis of its rotation; and by such rotations it is that solids are conceived 
 
 to be generated. 
 
 SCALENCE, or SCALENOUS TRIANGLE, a triangle whose sides and angles are unequal. 
 
 SCHOLIUM, a note, annotation, or remark, occasionally made upon some passage, proposi- 
 tion, &c. This term is used in geometry and the mathematics, where, after demonstrating a 
 proposition, it is customary to point out how it might be done some other way; or give some 
 advice or precaution, in order to prevent mistakes, or add some particular use upon application 
 thereof. 
 
 SCREW, ascrew is a cylinder cut into several concave surfaces, or rather a channel or 
 groove made in a cylinder, by carrying on two spiral planes the whole length of the screw, that 
 they may be always inclined equally to the axis of the cylinder in their whole progress, and also 
 always inclined to the base of it inthe same angle. The screw may also be considered as a 
 wedge carried round a cylinder, which is called the arbor of the screw; the wedge so carried 
 
 onwards making the thread of the screw. 
 SECANT is aline that cuts another, or divides it into two parts. 
 
 SECTION denotes a side or surface appearing of a body or figure cut by another; or the 
 parts where lines, planes, &c., cut each other. The common section of two planes is always a 
 right line, being the line supposed to be drawn on one plane by the section of another, or by its 
 
 entrance into it. 
 
 SECTOR is part of a circle, comprehended between two radii and the arch; or it is a mixed 
 triangle formed by two radii and the arch of a circle. 
 
 SEGMENT of a circle, that part of the circle contained between a chord and an arch of 
 the same circle. The portion AFB, and the chord AB, is a segment of 2 
 the circle ABFD. It is evident every segment of a circle must either ,, 
 be greater or less than a semicircle, the greater part of the circle cut off / 
 by a chord; thatis, the part greater than a semicircle is called the greater 
 segment, as ABED; and the lesser part, or part less than a semicircle, 
 the lesser segment, as AFB. It appears that the area of the sector D 
 ABCD, No. 2, is produced by multiplying half of the arch into the 
 
44 GEOMETRICAL ILLUSTRATIONS 
 
 radius; and likewise the area of the segment ABC is found by subtracting from the area of 
 the sector the area of the triangle ABC. } 
 Segment of a sphere is a part of a sphere terminated by a portion of its 
 surface, and a plane which cuts it off, passing somewhere out of the 
 centre, being more properly called the section of a sphere: the base of 
 
 such segment, it is evident, must always be a circle. : 
 
 AP eG 
 D 
 
 SEMI-CIRCLE, half a circle, or that figure comprehended between the diameter of a circle 
 and half the circumference. 
 
 SEMI-DIAMETER, half the diameter, or a right line drawn from the centre of a circle, or 
 sphere, to its circumference, being the same with what is otherwise called the radius. 
 SEMI-ORDINATE, the half of an ordinate. 
 
 SEXANGLE, a figure having six sides, consequently six angles, 
 
 SIMILAR FIGURES, such as have their angles respectively equal, and the sides about the 
 equal angles proportional. 
 
 Souip. A solid is defined to be the third species of magnitude, or that which has three dimen- 
 sions, viz., length, breadth, depth or thickness. A solid may be conceived to be formed by 
 the revolution or direct motion of a superficies of any figure whatever, and is always terminated’ 
 or contained under one or more planes or surfaces, as a surface is under one or more lines. 
 Solids are divided into regular and irregular. The regular solids are those terminated by regular 
 and equal planes, and are-only five in number, viz., the tetrahedron, which consists of four 
 equal triangles; the cube, or hexahedron, of six equal squares ; the octahedron, of eight equal 
 triangles ; the dodecahedron, of twelve; and the icosihedron, of twenty equal triangles. : 
 
 The irregular solids are almost infinite, comprehending~all such as do not come under the 
 definition of regular solids; as the sphere, cylinder, cone, parallelogram, prism, parallelo- 
 piped, &c. : 
 
 SOLID ANGLE is that formed by three or more planes meeting in a point. 
 
 SOLID PROBLEM is one which cannot be geometrically solved unless by the intersection of 
 a circle and a conic section: or, by the intersection of two other conic sections besides the 
 circle: as to describe an isosceles triangle on a given right line, whose angle at the base shall 
 be triple to that at the vertex. This will help to inscribe a regular heptagon in a given circle, ~ 
 and may be resolved by the intersection of a parabola and a circle. This problem also helps 
 to inscribe a nonagon ina circle, and may be solved by the intersection of a parabola and an 
 hyperbola between its assymptotes, viz., ‘To describe an isosceles’ triangle, whose angle at the 
 base shall be quadruple of that at the vertex. And such a problem as’this has four solutions, 
 because two conic sections can cut one another but in four points. 
 
 Souipiry. The solidity of a body denotes the quantity or space contained in it, and is 
 called also its solid content. 
 
 SOLUTION. The answering a question, or resolving any problem proposed. 
 
 SPACE denotes the area of any figure, or that which fills the interval or distance between the 
 lines that terminate it. . 
 
MATHEMATICALLY DEMONSTRATED. 45 
 
 SPECIFIC GRAVITY is that by which one body is heavier than another of the same dimen- 
 sion, and is always as the quantity of matter under that dimension. 
 
 SPHERE is_a solid contained under one uniform round surface, such as would be formed by 
 the revolution of a circle about a diameter thereof, as an axis. The circle revolving about the 
 diameter will generate a sphere, whose surface will be formed by the circumference of the circle. 
 The centre and axis of a sphere are the same as the centre and diameter of the generating circle: 
 and as a circle has an indefinite number of diameters, so a sphere may be considered as having 
 also an indefinite number of diameters, round any one of which the sphere may be conceived to 
 be generated. All spheres are to one another as the cubes of their diameters. The surface of 
 the sphere is equal to four times the area of one of its great circles. To find the superficies of any 
 sphere, let the area of a great circle be multiplied by four, and the product will be the superficies. 
 The solidity of a sphere is equal to the surface multiplied into one-third of the radius: or, a 
 sphere is equal to two-thirds of its circumscribing cylinder, having its base equal to a great 
 circle of the sphere. A sphere is equal to a cone whose height is equal to the semi-diameter of 
 the sphere, and its base equal to the superficies of the sphere, or to the area of four great circles 
 of the sphere, or to a circle whose radius is equal to the diameter of the sphere. 
 
 SPHERICS is that part which treats of the position and mensuration of arches of circles 
 described on the.surface of the sphere. 
 
 SPHEROID, a solid approaching to the figure of a sphere. The spheroid is generated by the 
 entire revolution of a semi-ellipsis about its axis. 
 
 SPIRAL, a curved line of the circular kind, which, in its progress, recedes from its centre. 
 A spiral is generated by having one end fixed in B upon the line 
 ABD, and equally moved round, so as with the other end A to 
 describe the periphery of a.circle ; and, at the same time, a point 
 be conceived to move forward equally from B towards A in the 
 right line BA, so as that the point describes such line, while the 
 line generates the circle: then will the point, with its two motions, 
 describe the curved line B1, 2,3, 4,5, &c., which is called the 
 helix or spiral line ; and the plane space, contained between the 
 spiral line and the right line BA, is called the spiral space. If 
 the point B was to be conceived to move twice as slow as the line AB, so that it shall get but 
 half-way along the line BA, when that line shall have formed the circle; and if then you 
 imagine a new revolution to be made of the line carrying the point, so that they shall end their 
 motion at last together, there will be formed a double spiral line, and the two spiral spaces, as 
 you see in the figure. From the principle of this curve, the following corollaries may be drawn. 
 First, the lines B 12, B 11, B 10, &c., making equal angles with the first and second spiral 
 (as B 12, B10, B8, &c.), are in arithmetical proportion. Second, the lines B 7, B 10, &c., 
 drawn any how to the first spiral, are to one another as the arches of the circle intercepted 
 betwéen BA and those lines. Third, any lines drawn from B to the second spiral, as B 18, 
 B 22, &c., are to each other as the aforesaid arches, together with the whole periphery added 
 on both sides. Fourth, the first spiral space is to the first circle as one to three.. And fifth, 
 the first spiral line is equal to half the peripbery of the first circle; for the radii of the sectors, 
 and consequently the arches, are in a simple arithmetic progression, whilst the periphery of the 
 
 N 
 
46 | | GEOMETRICAL ILLUSTRATIONS 
 
 circle contains as many arches equal to the greatest; therefore, the periphery to all those 
 arches is to the spiral lines as two to one. 
 
 SQUARE, a quadrilateral figure, both equilateral and equiangular. To find the area of a 
 square, obtain the length of one side; multiply this by itself, and the product is the area of 
 the square. : 
 
 STEREOMETRY, that part of geometry which teaches how to measure solid bodies, that is, 
 to find the solidity of solid contents of bodies, as globes, cylinders, cubes, &c. 
 
 STEREOTOMY, the art of cutting solids, or making sections. thereof. 
 
 SUBCONTRARY POSITION is when two similar triangles are so placed as to have one com- 
 mon angle V at the vertex, and yet their bases not parallel. If the 
 scalenous cone, BVD, be so cut by the plane CA, as that the angle at 
 C = the angle at D, the cone is then said to be cut contrarily to its 
 base BD. 
 
 ’ Cc’ 
 SUBTEUSE, the same with the chord of an arch. The subteuse of an angle is a right line, 
 supposed to be drawn between the two extremities of the arch that measures the angle. 
 
 SUPERFICIES, or SURFACE, a magnitude considered as having two dimensions, or extended 
 in length and breadth, but without thickness or depth. In bodies, the superficies is all that 
 presents itself to the eye. A superficies is chiefly considered as the external part of a solid. 
 When we speak of a surface simply, and without any regard to body, we usually call it figure. 
 The several kinds of superficies are as follow. ectilinear superficies that are comprehended 
 between curved lines; plane superficies are those which have no inequality, but lie evenly 
 between their boundary lines ; convex superficies is the exterior part of a spherical or spheroidal 
 body. The measure or quantity of a superficies, or surface, is called the area thereof. 
 
 TANGENT is defined in general to be a right line, ET, which touches any arch of a curve, 
 HE in E, in such a manner that no right line can be drawn through E between the 3 
 right line ET and the arch EH, or within the angle HET that is formed by them. 
 
 The tangent of an arch is aright line drawn perpendicularly from the end ofa 
 diameter, passing to one extremity of the arch, and terminated by a right line 
 drawn from the centre through the other end of the arch, and called the secant. 
 The cotangent of an arch is a tangent of the complement of the arch. The tangent - 
 of a curve is a right line which only touches the curve in one point, but does not 
 
 cut it. 
 
 TETRAHEDRON, one of the five regular or Platonic bodies or solids, com- # iE 
 prehended under four equilateral and equal triangles. : 
 
 TETRAGON, a general name for any four-sided figure, as a square, a parallelogram, rhombus, 
 or trapezium. . 
 
- 
 
 MATHEMATICALLY DEMONSTRATED. AT 
 
 THEOREM, a speculative proposition, demonstrating the properties of any subject. Theo- 
 rems are either universal, which extends to any quantity without restriction, universally; as 
 this, that the rectangle of the sum and difference of any two quantities, is equal to the dif- 
 ference of their squares: or particularly, which extends only to a given quantity, as in an equi- 
 lateral right-lined triangle, each of the angles is sixty degrees. A negative theorem is that 
 which expresses the impossibility of any assertion; as, the sum of two biquadrate numbers cannot 
 make a square number. A local theorem is that which relates to a surface ; as, the triangles of 
 the same base and altitude are equal. A plane theorem is that which either relates to a rec- 
 tilinear surface, or to one terminated by the circumference of a circle; as, that all angles in the 
 same segment of a circle are equal. And a solid theorem is that which considers a space ter- 
 minated by a solid line; that is, by any of the three conic sections. 
 
 . THEOREM RECIPROCAL is one whose converse is true, viz., if a triangle have two equal 
 sides, it must have two equal angles; the converse of which is likewise true; if it has two 
 equal angles, it must have two equal sides. 
 
 TRAPEZIUM, a plane figure, contained under four unequal right lines. First, Any three 
 sides of a trapezium taken together are greater than the third. Second, The two diagonals of 
 a trapezium divide it into four proportional triangles. Third, If two sides of a trapezium be 
 parallel, the rectangle under the aggregate of the parallel sides and one half their distance is 
 equal toit, Fourth, Ifa parallelogram circumscribes a trapezium, so that one of the sides of 
 the parallelogram be parallel to a diagonal of the trapezium, that parallelogram will be double 
 the trapezium. Fifth, If any trapezium has two of its opposite angles, each aright angle, and 
 a diagonal be drawn joining these angles; and if from the other two angles be drawn two per- 
 pendiculars to that diagonal, the distances from the feet of these perpendiculars to those right 
 angles, respectively taken, will be equal. Sixth, If the sides of a trapezium be each bisected, 
 and the points of bisection be joined by four right lines, these lines will form a parallelogram, 
 which will be half of the trapezium. Seventh, If the diagonals of a-trapezium be bisected, 
 and a right line join the points, the aggregate of the squares of the sides are equal to the aggre- 
 gate of the squares of the diagonals, together with four times the square of the right line joining 
 the point of bisection. Eighth, In any trapezium, the aggregate of the diagonals is less than 
 the aggregate of the four right lines drawn from any point (except the intersection of the dia- 
 gonals) within the figure. 
 
 TRIANGLE, a figure of three sides and three angles. ‘Triangles are either plane or sphe- 
 rical, A plane triangle is contained under three right lines; and a spherical one A 
 is a triangle contained under three arches of great circles of the sphere. Tri- 
 angles are denominated from their angles, right, obtuse, and acute. A right- 
 angled triangle is that which has one right angle, as ABC, No.1. An obtuse- 
 angled triangle is such as has one obtuse angle, as DEF, No.2. An acute- 
 angled triangle is that which has all its angles acute, as ghi, GHI, No. 3 and 4. 
 Any triangle that is not right-angled, is called oblique-angled or amblygonial. % ° 
 
48 GEOMETRICAL ILLUSTRATIONS, &¢e 
 
 An equilateral triangle is that which has all its sides equal; as ghi, No.3. An 
 isosceles triangle is one, that has only two sides equal; as GHI, No. 4. And 
 a scalenous triangle has no two sides equal; as DEF, No. 2. Triangles — 
 in the same base, and having the same height or place between the “same 
 parallels, are equal: also triangles on equal bases and between the same paral- 
 lels are equal. Ifa perpendicular be let fall upon the base of an oblique-angled 
 triangle, the difference of the squares of the sides is equal to the double rec- 
 tangle under the base, and the distance of the perpendicular from the middle of 
 the base. The side of an equilateral triangle, described in a circle, is in power 
 triple the radius. The sides of a triangle are cut proportionably by a line drawn 
 parallel to its base. A whole triangle is to a triangle cut off by 
 
 a right line drawn parallel to its base, as the rectangle under 
 the cut side is tothe rectangle of the two other sides. In a 
 right-angled triangle, a line drawn from the right angle at the §& 
 top, perpendicular to the hypotheneuse, divides the triangle 
 into two other right-angled triangles, which are similar to the 
 first triangle, and to one another. In every right-angled - 
 triangle, the square of the hypotheneuse is equal to the sum of 
 
 the squares of the other two sides. If any angle of a triangle be bisected, the bisecting line 
 will divide the opposite side in the same proportion as the legs of the angle are to one another. 
 Every triangle is half of a parallelogram of the same base and height. 
 
 TRISECTION, the division of a figure into three, or the division of an angle into three equal 
 parts. The trisection of an angle geometrically is one of those great problems whose solution 
 has been so much sought by mathematicians for these two thousand years, being in this respect 
 on a footing with the quadrature of the circle, and the duplicature of the cube. 
 
 TRUNCATED is an appellation given to such figures as have, or seem to have, the points cut 
 off, as a truncated cone, pyramid, &c. 
 
 VERTEX, the upper part or top, as the vertex of a cone or pyramid, &c., being the top of those 
 
 figures. The vertex of an angle is the angular point, and the angles 
 
 which, being opposite and touch only in the angular point, are called 
 vertical angles: such are the angles ABC, and DBE, wherein the sides 
 AB and CB are only continuations of the legs of the other, BE and 
 BD; such angles are demonstrated to be equal. The vertex of any 
 plane figure is the angle opposite to the base; and the vertex of a 
 curve is the point from which the diameter is drawn, or the intersection of the diameter and 
 curve. 
 
 VERTICAL PLANE is a plane passing through the vertex of the cone, and parallel to any 
 conic section. And a vertical line is a right line drawn on the vertical plane, and passing 
 through the vertex of the cone. 
 
 VOLUTE, the spiral—See SPIRAL. 
 
 WEDGE, one of the mechanical powers. The wedge is a triangular prism, whose bases 
 are equilateral acute-angled triangles. 
 
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DESCRIPTION OF THE PLATES. 
 
 PLATE I. 
 
 —_—_ 
 
 Tue Circle, Square, and Triangle, are the primitive or leading figures 
 in the science of geometry, unto which all others are connected or re- 
 lated throughout the whole of their various transmutations. They are 
 the only emblems that may be termed regular ; for, from their nature, 
 they can imply no other sensible representation. In them also are gene- 
 rated and established the extensive and comprehensive system from 
 which arises the production of every other geometrical appearance, what- 
 ever be the qualified external formation. 
 
 PLATE IL. 
 
 The Sphere and Cube. The axis of the sphere being equally divided 
 into sixteen parts, thirteen of those parts compose the side of the cube 
 whose solidity or admeasurement will be the same, and consequently of 
 
 equal gravitation, provided the two bodies be fabricated from the same 
 material. 
 
 PLATE IIf. 
 
 — 
 
 The square is here represented divided into aliquot parts, viz. the 
 fourth, half, and three-fourths, by equal diagonal lines. No figure, or 
 
2 DESCRIPTION OF THE PLATES. 
 
 representation, has ever yet been discovered to perform this operation 
 exactly: as this method demonstrates. ~The dimension of the square 
 in the division is not positively requisite, and may be or not dispensed 
 with, as the case may happen ; for, in the event of fractional parts arising 
 from the divisions by common arithmetic, many figures are used, and 
 the solution less likely from that circumstance alone to be correct. Al- 
 lowing, therefore, a square to be divided whose given dimension was five 
 inches and three-quarters, or five feet nine inches and a quarter, it 
 matters not ; the method shown here is simply to lay the squared own to 
 the scale of the common two-foot rule, and, as soon as the divisional 
 parts are produced, their quantity or dimension will be ascertained. 
 
 PLATE IV. 
 
 This square discloses the method of taking half of another square, and 
 the same to be precisely square. Halving the square with parallel lines 
 will not effect squares, but rectangles. Upon this figure, A, the triangle, 
 which is one-fourth of the square, added to B below, makes half. A line 
 drawn parallel to B through the centre would form the square into two 
 parallelograms of similar areas. 
 
 PLATE V. 
 
 Corresponding areas of the square and circle. The divisions shown 
 upon this diagram would be adapted for any other, and the dimension of 
 the square or circle need not be known: it will always be found sufficient 
 to place the figure at such convenient scale as to obtain a correct solu- 
 
 tion ; and the larger the better, for the truth will be manifest upon the 
 face of it. 
 
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 Invented and drawn by John Barnett 
 
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CORRESPONDING AREAS 
 
 ORs et Aue 
 
 Sa Aree ON GS RC LIE. 
 
 The traxsverse_A of the Grele, beng divided into twenty sta equal parts; 
 twenty one of those parts are found to occupy one fourth of the cirele, as 
 shown at B; and trom the divistons C,and D, the sides of the square are found, 
 
 and drawn, containing the Area of the crcle. 
 
 1a 5 ec 
 
 Invented and drawn by John Bennett, 3 "LehY 1636. 
 
 This problem has remained altogether in obscurity ; although, rewards were 
 ottered by Charles 5” of one hundred thousand crowns: and the States of Holland 
 
 a simular sum ,to any person effecting u: but,it does not appear ever to have 
 been pertormed ; being never published 
 
 London; John Bennett, 4, Three Tum Passase, vy Lane, Pater 
 
 1836 
 
RAP E21 My. 
 
 Invented and drawn by John Bennett 
 
 ~ 5 74 r 4 1m 
 
 London; John Bennett 4,Three Ton Passage, Ivy Lane, Paternoster Row, 
 1836 
 
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THE SQUARE. 
 
 B the half of A..C the half of B,D the half of C, ke. 
 
 Again, Triangle B, the halt of A. and so on tor the other. 
 
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 Was : Or, oy shy ah Ae pein 2 
 
TRIAZBRG LE. 
 
 A..Lhe Triangle, divided into three parts B.C.D, ot equal AVES, 
 
 and the dotted line ¥. ¥, equahses. 
 
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 Une g IRE A ey Ce eee ae ee eo 5k Pee ST PETE ES meee ts tee en ew oe Sg ee eee > 
 
 The trisection ,or division of the triangular ngure ,into three equal parts. 
 ‘The trisectton. of an angle Geometrically, 71s one of those great problems whose solution has 
 been so much sought by mathematicians tor these two thousand years; being in this respect, on a 
 footing with the quadrature of the Circle; and the dupleature of the cube” New Lichonary Arts and 
 
 Sciences, Published by a society of Gentlemen, 1754. 
 
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DESCRIPTION OF THE PLATES. 3 
 
 PLATE VI. 
 
 The trapezium, being an unequal-sided figure, is shown divided in the 
 same manner as the square, figure 3. Although this figure is untrue, 
 this can make no difference in the correct quantities, . notwithstanding 
 the dissimilarity in the shape of the smaller divisions. 
 
 PLATE VII. 
 
 The square diagonally divided from centre points. _B is precisely half 
 the area of A, and C half that of B, &c. This figure shows also the com- 
 plement of the triangle with that of the square. 
 
 PLATE VIII. 
 
 The triangle divided into three parts of equal quantities this figure 
 clearly demonstrates. It is evident, from the immense sums of money 
 that were formerly offered by kingdoms for the performance of this very 
 difficult problem, that its solution was held by the ancients of infinite 
 importance. It must be particularly observed that there can be no dia- 
 gram throughout the whole science which will serve to exhibit the value of 
 this discovery so much as the triangle, in the performance of abstruse 
 questions. ‘The trapezium B, and the unequal triangles C and D, have 
 methodically the same areas ; the dotted line EF equalises the triangle C, 
 and the same with D. B, having a different figure, contains the same 
 area as the true figure C, notwithstanding. 
 
4 DESCRIPTION OF THE PLATES. 
 
 PLATE IX. 
 
 The rhombus divided into parts, &c. This figure is similar to the 
 square, but differently posited, for the greater facility in comprehending 
 its nature and effects. 
 
 PLATE X. 
 
 This blended figure, which is composed of circular and angular lines, 
 being purely speculative, has no particular qualities to enumerate ; it may 
 be considered merely as a reference for the formation of others from its 
 diversified positions. In many instances its uses may be imagined from 
 events which frequently occur in the absence of matter to fill up orna- 
 mental spaces, and in the arrangement of component parts, in order that 
 the whole should harmonise together. It may likewise be found a desira- 
 ble reference in panelling out ceilings, laying down embellished floor- 
 ings, devising pleasure-grounds to noblemen’s mansions, and inlaying 
 cabinet work. 
 
 PLATE XI. 
 
 This figure shows that the invention of dividing the square, as shown 
 by the plate 3, is not confined to regular, perfect, or true figures, but can 
 be extended to any other, however disproportionate they may happen to 
 be; for herein it is made very evident that neither disposition nor 
 quality can alter the verity of this system, for, as they are divided, the 
 result will be certain to follow ; and, although the figure itself is altogether 
 untrue, the divisions will be found mathematically exact. 
 
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 DESCRIPTION OF THE PLATES. 
 
 PLATE XII. 
 
 Emel 
 
 The hyperbola divided into various parts. This diagram may be, in 
 some instances, of considerable utility, when any alteration is required 
 for large door-ways or windows of buildings or churches. It will extend 
 also to furnace-doors; &c., for steam-engine work, the circular head falling 
 into the divisions with equal order and precision. 
 
 PLATE XIII. 
 
 From a given square produce a circle containing the same area. Draw 
 the diagonal lines as shown, and divide one portion of the same into 
 fifteen parts ; one point of the dividers placed at the centre, the other at 
 twelve, describe a circle, which will be of the same capacity as the 
 square. ‘This diagram will always be of practical utility, and shows in- 
 finitely more than can be explained in any written description. In the 
 square there are four triangles ; each of those triangles contains one-fourth 
 of the square; the quarter circle, whose figure differs materially from. 
 the other, holds a similar area notwithstanding: and it may be ascer- 
 tained from this axiom that a quarter circle and triangle, formed of the 
 dimension these are, their quantities must invariably be equal. 
 
 PLATE XIV. 
 
 The relative proportions of any square that may be produced from 
 this figure may be accurately ascertained by the diagonal line AB, which 
 in the true figure will be seventeen and the side twelve. 
 
 C 
 
6 DESCRIPTION OF THE PLATES. 
 
 PLATE XV. 
 
 The division of the circle into aliquot parts. The direction of this 
 figure follows in its nature precisely the same as the square, and the parts 
 are equally explicit. In taking the circle herewith shown as the plan for 
 the cylinder, the aliquot parts of the cylinder may be ascertained from 
 this to any extent required. 
 
 - 
 
 PLATE XVI. 
 
 It is to be perceived, from this frame, that the external and internal 
 A’s have the same area or capacity as the middle.part B. Therefore, if 
 the quantity of B were to be deducted from A, it would be exactly half. 
 
 PLATE XVII. 
 
 The right-angled triangle given, to be made square, holding the same 
 area or capacity. The triangle ABC is here introduced without refer- 
 ence to any dimension; therefore, let the diagonal line AB be divided _ 
 into thirty-four equal parts, set off ten parts from A and ten from B, 
 leaving fourteen in the middle; introduce the lines ED and DF, which 
 will produce the square, as the diagram shows, holding the same space 
 or area as the triangle. | 
 
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 Invented and Drawn by John Lenrett . 
 
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FRAME. 
 
 eframe AA AA . 
 
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 The dotted space B BBB, is half the area of ti 
 
 FIG.1G, 
 
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 Stxetnches. 
 
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 DE.F.G the Sguare required. 
 
 F1IG.17. 
 
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RHOMBOIDES. 
 
 A Rhhomboides being given, as N,B.C, VI); @ Square, the same Area, ts 
 described. as ¥.¥, GM. 
 
 Londgon: John Bennett.4, Three Tun Passace, hry Lane 1836. 
 
PE aya Pee tn 
 
DESCRIPTION OF THE PLATES. 7 
 
 PLATE XVIII. 
 
 This right-angled triangle being half of the square, is herewith intro- 
 duced, divided by the dotted lines as shown, into aliquot parts: namely,_ 
 three-fourths, half, one-fourth, and one-eighth parts; the division of which 
 will be found set out upon the base of the square ; viz., six for the eighth, 
 twelve for the fourth, seventeen for the half, and twenty-one for the three- 
 fourths parts ; from this scale of divisions any intermediate quantity of 
 
 this, or any other similar triangle, may with the greatest accuracy be 
 obtained. 
 
 PLATE XIX. 
 
 The rhomboides ABCD given, whose dimensions are four inches by 
 three and three quarters inches, is required to be made into a square, 
 containing a similar area. Let a perpendicular fall from the vertex B 
 upon the base line GD, which produces three inches and a half; let that 
 be multiplied by the content of the line AC or CD, either of which are 
 four, the result will be fourteen, the contents of the square EFGH. Or 
 divide the side of the square at EG into twenty-eight parts, set off ano- 
 ther part in addition for the square as shown by the line EF. Conse- 
 quently, the square holding the same area as the rhomboides may be 
 readily obtained, as this figure elucidates. 
 
8 DESCRIPTION OF THE PLATES. : 
 
 PLATE XX. 
 
 —_— 
 
 Shows the rhomboides divided into parts, viz., an eighth, fourth, half, 
 and three-fourths: the divisions are not only correct in their respective 
 areas, but take the precise form of the original figure in all its bearings. 
 . This diagram will serve to demonstrate the properties herein laid down, 
 that notwithstanding the nature of the figure, whether true or otherwise, 
 ' the divisions are brought out perfect. 
 
 PLATE XXI. 
 
 The equilateral triangle, divided by the same method as the figures 
 before explained ; and any fractional part of either of those divisions. can 
 be found by this scale. 
 
 PLATE XXII. 
 
 The divisions of this square are clearly exemplified in the preceding 
 plates ; but the introduction of this deviation may be extremely useful in 
 taking any intermediate equal part from the whole ; or of adding to, and’ 
 deducting from, such parts of the figure as may at:any time be required. 
 In bordering or frame-work, and many other useful dispositions in work- 
 manship, this diagram may be held in requisition, more especially where . 
 a correct attainment of the divisions is considered to be of any conse- 
 quence. 
 
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 RHOMBOIDES. 
 
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 London; John Bennett 4 Three l'un Passage Ivy Lane. 1836. ; 
 
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EQUILATERAW TRIANGLE. 
 
 ¢: 
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 F1IG.21. 
 
 Invented and drawn by John Bennett. 
 
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 SQUARE. 
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 lnvented ara draw OY JSoht SCRE 
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[RREGULAR. 
 
 FIGE23. 
 
 London; John Bennett .4,Three Tun Passage, Paternoster Row, 1836 
 
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 The Triangle A.B. ©. being given; let the same be divided itto 1 equal Triangles. Bisect. | 
 the 3 sides.to which draw the lines, as shown in the (allowing diagram. ——D. the 
 hecagon tormed of the triangle ._ 
 
 Trvented and drawn-by John hennet. 
 
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 London, John Bennett .4 Three Tun Passage, Paternoster Row, 1836 . 
 
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TRIANGLE’ — 
 AND 
 
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 Pk bsvamiashdunannianqaracasaesssnesroesie ser DB Trachies pretence nenseneans. iegntsncans. 
 
 Jnverted and drawn by John Lennete. 
 The triangle &., being given,anad difficulty arising respecting tts quantity by fornang the square. 
 \B.C.D. Ew i; the contents may speedily be ascertained : tor instance , the square being 6x0-306: 
 that divided by 8, leaves 44; then 3%3,-9:halt of which , ts 442,which was tobe demonstrated . 
 
 London Jchn Bennett, 4,Three Ton Paes age, Paternoster Row, 1636. 
 
 a & 
 
DESCRIPTION OF THE PLATES. 9 
 
 PLATE XXIII. 
 
 This irregular figure, comprehended under two sides of like dimen- 
 sion, and two sides of equal, but larger dimensions ; although such is de- 
 scribed by this diagram, it is nevertheless divided as any of the foregoing 
 figures, whose dimensions are equal. The diagonal line in this figure 
 appears curved, which must be the case where the sides do not corre- 
 spond; but, where the sides are equal, the diagonal invariably will be 
 straight from point to point. 
 
 PLATE XXIV. 
 
 ed 
 
 The triangle laid out into four divisions, the whole of which are equal 
 in capacity and alike in figure: these divisions are found by bisecting 
 the sides, and drawing the dotted lines as shown. This method will per- 
 form the like operation to any three-sided figure. The hexagon, or six- 
 sided figure, as shown at D (providing the sides of the triangle are equal), 
 may be obtained within it, by dividing the three sides into three equal 
 parts, as is described. 
 
 PLATE XXV. 
 
 The square disposed or laid out into eight equal triangles. Half of 
 this square, from B to D, forms the rectangle, and composes four of the 
 D 
 
10 DESCRIPTION OF THE PLATES. 
 
 triangles : the contents of one of those triangles, multiplied by eight, will 
 introduce the area of the square, and by four, the area of the rectangle, 
 being half the area of the square; and the triangle CBD, being half the 
 contents of the square, will also be equal in quantity of area to the 
 rectangle. 
 
 PLATE XXVI. 
 
 The ellipsis, or oval, divided into equal parts, viz., an eighth, fourth, 
 half, three-fourths, &c. Upon the face of this plate will be observed a 
 reference to the rhombus, No. 2, page 7, which is incorrectly placed ; 
 it should be to the rhombus, No. 1, Fig. 9, wherein the method which 
 is adopted for the equal division of that figure will directly apply to 
 this. By the dotted lines from A to B the points are struck circular from 
 a centre, which may be found ; so that the lines should intersect each 
 other without crippling. No. 1 constitutes the eighth division; No. 2, 
 the fourth ; No. 3, the half; and No. 4, the three-fourths part of this 
 ellipsis ; and by this method any division of the ellipsis may be obtained, 
 without reference to the transverse, or conjugate. 
 
 PLATE XXVII. 
 
 ——— 
 
 The section of an irregular pyramid, or four-sided figure, whose sides 
 are equal, being six inches, as the plate demonstrates ; notwithstanding 
 such inequality, it is divided into aliquot parts: viz., one-eighth, fourth, 
 
ih Wy At eT 
 
 dn ellipsis being given ,as A,B,C,D, let the same be diaied mto parts, 
 viz’ an eighth fourth halt: three fourths, &e.asW°*1, 2,3,4, 5. 
 
 See Rhombus VN? ?, Tage a. 
 
 FIG, 26. 
 
 D. SLOT and drawn by: bhn Bennett. 
 
 + - mi = “! ads Ss rte 
 London John Bennett,4,Three Tun Passage, Paternoster Row, 183¢ 
 
Peg ey PRR SS 
 
 ey 
 
 az afl | pe ie yaa. Moa a 
 
Lwented and drawn by John Bennet. 
 
 ter Row. 1836. 
 
 Paternos 
 
 ie 
 
 London: John Bennett 4.Three Tun Pass 
 
‘ Par 
 
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 PORT, 5 
 
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aie toes alatd cabs eroisdiae Seis ead SO ther atiy onte of the ~ 
  @rangles. must be «364 part of the whole square : eye ae all the sae at ae 
 bear equal proportions to each other. 4: 
 > 
 FIG. 28, 
 
 - Lwented and trawn by Sohn Lennett. 
 
 london; John Bennett, 4,Three Tun Passage. Paternoster Row, 1836. 
 
Wee va? 
 MET GiN ti} 
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DESCRIPTION OF THE PLATES. 11 
 
 half, and three-fourths ; the areas of each respectively will be found, when 
 worked out, perfectly correct ; although, in the representation of this 
 figure, the smaller divisions bear but an indifferent resemblance of the 
 original. 
 
 PLATE XXVIII. 
 
 This figure is shown divided into nine small squares, and thirty-six 
 triangles ; therefore, one of the triangles must contain an area of the 
 thirty-sixth part of the whole square ; consequently all the squares and 
 triangles bear equal proportions to each other. In this figure much may 
 be discovered relative to triangles, rectangles, and squares : for, by taking 
 half this square diagonally, it will be found to contain eighteen small 
 triangles, as are shown dotted upon the plate ; the base line of each, or 
 either, will be two inches; nine triangles, whose areas double the fore- 
 going, the base line of which will be two and seven-eighths inches. The 
 next larger has a base of three inches and seven-eighths, and contains 
 nine of the small triangles, and doubles also the area of the last. And 
 the triangle which holds a fourth part of the whole square, contains nine 
 of the small triangles. Two-thirds of this square will form the rectangle, 
 containing twenty-four small triangles; and the remaining rectangle, 
 whose dimension will be six inches by two inches, will contain twelve 
 triangles : therefore, from the transformations this figure undergoes, much 
 may be obtained in some cases in the reduction of a figure by triangles, 
 in order to obtain some particular result or definition: for, in this it is 
 made evident, if one triangle will not come in with even numbers, the 
 next in succession may, and therefore by such means a correct solution 
 may be obtained with infinitely less trouble than by any other. 
 
12 DESCRIPTION OF THE PLATES. 
 
 PLATE XXIX. 
 
 The unequal four-sided figure, whose sides are respectively five and a 
 quarter inches, four and three quarters inches, three and a quarter inches, 
 and two and three quarters inches, is given in order that it may be made 
 into a square, of the same capacity, or to contain the same area. The 
 contents of the sides of the untrue figure, added together, as shown upon 
 the plate, make sixteen, which, being divided by four, its number of 
 sides, leaves four inches for the side of the square holding a similar area. 
 In this instance, and probably in many others, this operation may be per- 
 formed with accuracy ; but it will not in all; therefore, should this 
 method fail in producing a satisfactory result, others must be resorted to. 
 
 PLATE XXX. 
 
 The quadrangle, or rectangle, divided into equal or aliquot parts, viz., 
 one-eighth, one-fourth, half, and three-fourths; in the formation of these 
 divisions, it is not of the least consequence what dimensions the rectangle 
 may be composed of: for the disposition by which this rectangle is laid 
 out upon the plate, is certain to perfect these, or any quantity required, 
 to the greatest possible nicety. 
 
 PLATE XXXI. 
 
 a 
 
 The parallelogram ABCD being given, a square is required, contain- 
 ing its contents, as EFGH, the square of the same area. From this 
 
THE: SOUA RES NO*4. 
 
 Qu. Zhe tigure A whose stiles are all unegual vir. 34 24 34 
 and 4° wches, tts reguired that an egual sided gure 
 
 should be made, occupying the samme area oF space only ? Dhece. of Aiba G8 
 ot 
 9ay 
 Ans? 34 B the equ sided ngure whose side 7s tour urches .; Ro, Wea og B= 16. 
 V6 
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 : London: Jona Bennett, 4 tiree Dim bassage ivy Lan 
 
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 Three ig ica 
 
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 THE QUADRANGLE. 
 equal parts viz* %.#4.34.%. and one inch 
 
 7wzto 
 
 Londen; John Bemett,4,Three Ton Passa fe, Ivy Lane 1837 . 
 
 thn Bennett. 
 
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 Divided 
 
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PARAL CLO GRAM. 
 
 j 
 ~ 
 
 Q The Parallelogram A\B.C.D. being given, a Square ts required, containing 
 the sarne area only ” 
 A.E.F.GH hase 
 
 FIG.21. | ‘ 
 
 eS Se ee 
 
 le 
 o 
 
 coo 
 
 Lwedted and drawn by Joh Bermew. 
 
 London: Jchm Bennett 4. Three Tun Passage. hy Lane.l837. 
 
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 ein pS Ae is 
 Dy int Nate Se 
 OY eee NL 1) ON ee 
 
THE SQUARE, NOB. 
 
 Another method see Fig. 3. 
 
 FIG 32. 
 
 4 4 Inches 
 
 4 47 Inches 
 
 sabtellsbana-seteirabbanegharhanibertiiaiegah <o-curadhinietehisalhainkipc nanan nitnnd id streecnterererereretennenanenanenanannnsersnannnn fy Inches ——e Bn = I Se Ber ES eee I ee a Ts mieaetecs: 5 
 lLnvented and drawn by John Bermett 
 
 London; John Bemnett,4, Three Ton Passage, vy Lane, 1837 
 
 ~<O Inches 
 
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 ri : oi. may 
 me et 
 
WAU Gocttifiesrh tt AONUG LE. 
 
 Lhe Foundation angle of Stic, hight, and Ten, MBC. making the tourlh of the square . 
 
 FUS 33 
 B 
 
 oO 
 
 pig Sages 7 3 
 Drawn by Jolin Bermet. Scale % of an Inch to one toot 
 
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DESCRIPTION OF THE PLATES. 13 
 
 problem it appears to be an easy matter to square the rectangle, or paral- 
 lelogram’; and it is so whenever the quantities happen to fall into equal 
 squares, as this figure demonstrates ; but, although it will sometimes and 
 with some quantities work correct, it will be found by no means sufii- 
 ciently so for general adoption. 
 
 PLATE XXXII. 
 
 In this figure the square of six inches is halved in its area by the square 
 of four and a quarter inches, as is here represented. The halving of the 
 square may be seen in a preceding figure, but in this it is to be further 
 explained, viz., The dotted diagonal shows the square divided into two 
 triangles, a horizontal line would divide the square in the middle into two 
 rectangles ; therefore this square of four and a quarter inches would be 
 equal in area to either of them. 
 
 PLATE XXXITI, 
 
 or 
 
 The triangle ABC shows the method by which a square may be 
 obtained to the greatest truth upon any undetermined space. Let the 
 line of six inches be made as this diagram shows, and also the vertical 
 line of eight, and adjust the points of those to the diagonal of ten, as this 
 plate serves to represent ; the angle at A will be right, and true for any 
 workmanship to proceed from whatever. 
 
14 DESCRIPTION OF THE PLATES. 
 
 PLATE XXXIV. 
 
 The equilateral triangle being given, let the same be made into a 
 square, containing a similar area. Divide the side A into twenty-four 
 equal parts, let them be drawn parallel and ranging with the opposite line, 
 intersecting each of them upon the base line as shown. At part four, in- 
 troduce the vertical line B, intersecting one side of the triangle in the 
 eighth part ; and as the base line shows from figure four to twelve, the 
 distance of the square, let the rest be set out for the remainder of it, as 
 this diagram serves to elucidate. | 
 
 PLATE XXXV. 
 
 — 
 
 The crescent divided into eight sections, the pyramid is also divided 
 in like manner ; but the difference in their areas is as five to four, the 
 area of the crescent being two and a half, and the pyramid two ; it is there- 
 fore evident, from this diagram, that the curved lines of the crescent 
 gain upon the straight lines of the pyramid. 
 
 PLATE XXXVI. 
 
 The pyramid divided into aliquot parts, namely, an eighth, fourth, half, 
 and three-fourths ; by the divisional lines as shown, the manner in which 
 this is performed is similar to the preceding figures. 
 
EQUILITERAL TRIANGLE 
 AND 
 SQUARE. 
 
 AAA-Qu: An equititeral triangle being given, whose side is Six inches; required the stde of 
 
 a square, contaaang the same area 7 
 
 BBBB. 4ns” Four Irches. 
 
 ee aetceate gta . 5. ae ae eee ne : eT eee | ie, SECON AY PRD RS b- nc anal aaa NO AiR, Sarre a AC ae ee 7 RN ee See 
 ented and drawn by Jol Bennat 
 
 London; John Bennett , 4, Three Tum Fassase, vy Lage,1837 
 
4 y 3 wt . 
 CARESS. ST4 A MaRS ae eats, “Wy 
 
 See URS Ct wi bess - ai ale ¥ 
 
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CRESCENT. 
 
 . 
 
 * 
 
 FIG.35. 
 
 Swented and drawn by John Bane. 
 
 age, Ivy Lane, 1837. 
 
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 London, John Bemett ,4, Three Tum Pass 
 
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 B- a ee es 
 
 38. 
 
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 A.A The Sem? , B the outer square, C Bete the square head required , 
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RHOMBUS. eae es 
 
 _ A rhombus ABCD. being given EF.GA, ts @ Sguare, occupying the same space . 
 
 Invented. and. drawn by John Bennett, 
 
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DESCRIPTION OF THE PLATES. 15 
 
 PLATE XXXVII. 
 
 The octagon divided into aliquot parts, by the same method, which 
 has been sufficiently before explained. The area of this octagonal figure 
 amounts to nineteen ; consequently, each compartment must be two and 
 three-eighths ; and the rectangle equal thereto would be two and three- 
 eighths by one, in consequence of the base line of the octagon or the 
 length of one of its sides being two. 
 
 PLATE XXXVIIL. 
 
 The semi or half circle A being given, let it be formed with a square 
 head as shown by the plate. Draw the dotted line B at the full extent 
 of the semi, then the inner dotted line C, bisect AC as shown at D, and 
 D will form the square head required. 
 
 PLATE XXXIX. 
 
 The rhombus introduced to be made into a square, occupying a similar 
 area only. Proceed with this figure, as is explained to the fig. 19, the 
 rhomboides: the same method which performs the operation with one, 
 will perfectly describe the other ; the positions of the figures, as placed 
 on the plates, nor the difference of size, will make any alteration in its 
 solution. 
 
16 DESCRIPTION OF THE PLATES. 
 
 PLATE XL. 
 
 The rhombus, No. 2, divided into aliquot parts, viz., one-eighth, fourth, 
 half, three-fourths, &c., by the methods practised with the preceding 
 figures: the divisional parts of this are found, which are not altered in 
 appearance from the leading figure by their diminution. 
 
 PLATE XLI. 
 
 —e_ 
 
 This plate comprises three diagrams, viz., first, the cube ; secondly, 
 the surfaces, or superficies of the cube ; and, thirdly, the transformation 
 of those surfaces into another position. From these figures it must be 
 observed, that allowing the actual measurement of another similar figure 
 to consist of fractions, this method, in almost every instance, would serve 
 to consolidate them, and render their solution much easier, with greater 
 expedition and accuracy, than by taking one side only for the operation. 
 The cube, No. 1, whose side is supposed to be twelve inches, or one foot, 
 the superficies of which will amount to 864; the figure, No. 2, whose 
 dimensions are 36x24=864; and the figure, No. 3, whose dimensions 
 are 24x24=576, which is the content of the middle Square: the four 
 angles will make, viz., 12 x 12=144+144=288, which, added to 576, 
 introduces the like solution of 864. From such variable deductions as 
 these much valuable and interesting information might be gained from 
 the simplicity (which these figures are chosen for) of disarranging a figure 
 in order to obtain a true disposition of its relative quantities and com- 
 ponent parts. 
 
Pad ‘ ; , . 
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 a RHOMBUS,N°2. 
 
 H 
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 Destanedl and Zrawm by John Bernat. 
 
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 Ionden, ohn Bermett, 4, Three Tun Passage, Ivy Lane, 1837. | 
 
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 3. The same surtitces transtormed . 
 FIG.41. ee 
 
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 — STnvented and awn ty John Bennett. 
 
 London ; John Bennett.4, Three Ton Fassage, vy Lane,1837. 
 
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 QALWARSTER CIRCLE, 
 
 OR | 
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 Lrawn by John Bennett. 
 
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SQUARE and OCTAGON. 
 
 Lhe Syuare transtormed to an Octagon, containing Oued juilt’ 
 tts ares, by continuny the cant A,to A PB. 
 
 wtG. 43. 
 
 1/2 tuches 
 
 Srwented and drawn by John Bennett. 
 
 London; John Bennett 4, Three Tun Passage, Ivy Lane, 1837. 
 
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ROMAIN SG LES. 
 ~ 
 Qu: dn unequal sided triangle being given, required an equal sided one , 
 containing the same space only? ash 44, 442, 6074, sides. 
 Ans” B35 inches. 
 
 FIG.44. 
 
 The Triangles A and Bare | \\>s. Jie Gren of A= 9: $56 06 
 of equal areas, although os 
 
 their sides are of different 
 dimenstons; which will be 
 sen by the quantities of their” 
 sides being added together; vez. 
 
 The Qua ¢ B= 0.8253 
 
 SN 
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 Designed and drawn by John Bennett. 
 
 London; John Bennett, 4,Three Tun Passage, Ivy Lane, 1837. 
 
7 
 
 DESCRIPTION OF PLATES. iwi 
 
 PLATE XLII. 
 
 The quarter circle, or quadrant, divided into aliquot parts ; this figure, 
 like those preceding, is easily divided, as is shown by this method, as 
 the others ; therefore requires no further explanation to make it under- 
 stood. 
 
 1 pee 
 
 PLATE XLIItL. 
 
 The triangle A, of the square, is equal to the triangle AB of the octa- 
 gon, consequently, the four triangles must constitute half the square. 
 This diagram may be considered of great practical utility in machinery, 
 viz., the introduction of iron gudgeons to wood shafts for windmills, or 
 water wheels, &c. 
 
 PLATE XLIV. 
 
 mene 
 
 ~ This figure comprehends the unequal-sided triangle A, whose sides are 
 4 and a quarter, 4 and a half, and 6 and a quarter inches ; the same is in 
 this diagram formed into the equilateral triangle B, whose area is nearly 
 the same. The method by which this problem is performed in the 
 margin of the plate, is as to its linear capacity ; but it does not always 
 follow with triangles that they will work to the truth with that nicety that 
 may sometimes be required ; although, in many instances, they will in- 
 troduce the contents within a trifling fraction. _ 
 
 F 
 
18 DESCRIPTION OF PLATES. 
 
 PLATE XLV. 
 
 The equilateral triangle, formed into two squares, of similar areas. 
 This figure, although simple, might be so constructed as to introduce 
 several others ; and also to show where many difficult questions arise from 
 first appearances ; but, in this diagram, it may very readily be observed, 
 that the upper triangle, No. 1, being reversed, and placed at the sides of 
 A or B, will produce the quantity required for the square to either. 
 
 PLATE XLVI. 
 
 The out and inside angles. This useful problem, in taking angles of 
 buildings, &c:, may be considered of great practical utility ; for, as this 
 diagram shows, it is merely the running out of the line of front and the 
 side to equal distances ; and then measuring the diagonals, which must 
 necessarily furnish the angle, let it be whatever it may:.and, by the 
 same rule, this method can be rendered of much service in the setting 
 out of large works, where the space cannot with convenience be made 
 applicable. 
 
 PLATE XLVII. 
 
 The square, No. 1, whose area is four: and No.2 contains also the 
 same quantity of space; the dotted lines of which introduce the octagon. 
 Figures No. 3 and 4 are also fabricated from them, in which are displayed 
 much diversification of character, thereby assuming plans in the simplest 
 
— 
 
 THE TRIANGLE FORMING TWO SQUARES. 
 
 trom this triangle, the 2 sguares A. and B, are constructed ; containing the same space. 
 
 FIG.45, 
 
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 Lhvented anid drawn by John Bennett 
 
 London; John Bennett,4 Three Tun Passase. ly 
 
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OUTER ann INNER ANGLES. 
 
 - 
 
 4 sinple method tor ascertaning-any Angle. 
 
 Outer 
 
 1 ee So cpecet seen Be Og. ~/ 5) Ser ES ie ere tte ca sete ata aaa Gat ae 
 
 Invented and drawn by John Kennett. 
 
 ~= 
 
 Londcn.John Bennett.4,Three Tun Passage, Ivy Lane,1837. 
 
SQUARE 
 eee la 
 
 EP LG. 47. 
 
 Lhe Compound riqures N? 3 & 4, contain the same arcas,as VL & 2, 
 
 Snwented and drawn hy John Bennett. 
 
 Lendon;Jchn Bennett, 4, Three Tun Passage, Ivy Lane. 
 
COM POUND. 
 
 hd ‘ 
 
 This Compound fibricated trom the plan A, the two squares. 
 
 lnwented and drawn by John Dennett. 
 
 London; John Bennett. 4, Three Tun Passage Ivy Lane. 
 
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 London John Bennett.4, Three Tuu Passage Ivy Lane. 
 
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 DESCRIPTION OF PLATES. 19 
 
 manner, suitable to some of the branches of fortification, being alike 
 useful and ornamental. It is necessary to observe that the areas of No. 
 3 and No. 4 are precisely the same as No. 1 and No. 2, for such parts as 
 are removed from the internal angles are again added to the external, leav- 
 ing the original area of four for the whole of these figures. 
 
 PLATE XLVIII. 
 
 Is another plan introduced of the crossed squares, as shown by the 
 smaller plan A. This figure may form a very useful process for the student 
 in the science of fortification; and indeed a variety of other problems 
 equally useful and instructive, requiring diversity of character: this plan 
 may be altered into many more geometrical devices, and still the same 
 area always contrived to be kept, which may serve to operate materially 
 in some instances where any particular point requires more attention than 
 another: the mutations upon this figure may be so constructed as to afford 
 abundance of means requisite to fulfil any disposition required, under 
 almost any circumstances or arrangement. 
 
 PLATE XLIx. 
 
 The straight line, formed into a triangle, as at A: this diagram, al- 
 though extremely simple, would be found one of considerable difficulty to 
 be performed in some situations upon an extensive scale; but, from a 
 knowledge of the readiest methods, the mind, when once informed, is not 
 readily divested of it, but much more likely to enlarge upon it by other 
 plans of far greater intricacy. The triangle formed into a square, as at B: 
 
20 DESCRIPTION OF PLATES. 
 
 Let either of the sides be divided into sixteen parts, mark off thirteen, 
 which arrange at a right angle from the base, as shown; the other sides 
 may then be introduced to make good the square. 
 
 PLATE L. 
 
 Is the lines A and B shown formed into squares ; the line A forming one, 
 and the line B two, which the diagrams serve sufficiently to elucidate. 
 
 PLATE LI. 
 
 - 
 
 The square compounded with angles. The full square A having as 
 much abstracted from it as makes the angles CDEF ; consequently the 
 solidity is not exceeded by the alteration: but the covering surface is in- 
 finitely more. The triangle, in which the letter A is placed in this diagram, 
 being a solid, the next on either side are vacant, as well as their opposites, 
 the inner angles of CDBE are also vacant, and by rendering A a cubical 
 figure, the appearance and disposition of the abstracted angles would 
 form an interesting object, and one worthy of attention. 
 
 PLATE LII. 
 
 In this figure we have something similar to the foregoing, showing an 
 octagon ; and the eight triangles being reversed, serve to introduce a solid 
 exterior, leaving the whole of the original space void: consequently the 
 
agate pecs [he sb: eee 
 
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 London; Jom Bennstt,4,Three Tin Fassage, ivy Lane. 
 
, 1 > 
 
 THE deals COMPOUNDED WITH ANGLE 
 | | 
 
 The till square A, having as much abstracted Tom Ww, as makes ee 
 
 FIG.51. 
 
 Davented and drawn by John Bernat. 
 
 “London, Jchn Bennett. 4. Taree Ton Passage Ivy -su¢ 
 
 angles ©. D.E.F; anseyuently, the solidity ts not exceeded by the alteration. 
 but, the corering surtiue ts tifinitely more. . 
 
 4) 
 
 : 
 ; 
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 Sa oe 
 
* | - 
 
 OCTAGON. 
 
 . . f “ é 
 The cants, oP angular divisions, of the octagonal figure ; 
 reversed ,and thrown outwards. as this diagram SHOWS . 
 
 “F1G.32. 
 
 Duented and drawn by John Lernete. 
 
 J.ondon; John Bennet,4,Three Tum Pssage. Ivy Lane 
 
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 wh | , - 7 as ,.: 
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 Invented and drawn by John Bennett. 
 
 * 
 
 London; John Bennett, 4, Three Tun Passage, vy Lane, 1837. 
 
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 London ; John Bennett, 4, Three Tun Passage, Ivy Lane, 18377. 
 
 Jb 
 
geno 
 
DESCRIPTION OF PLATES. , 21 
 
 outer figure, although it appears considerably more extensive, contains not 
 the least particle of additional area to the inner, By introducing lines 
 from the extremes of the points parallel to the internal diagonal lines, it 
 will then be found to produce half a solid contained under forty-eight 
 triangular sides, and this figure can be made to show half, or twenty-four. 
 
 PLATE LIII. 
 
 The surfaces:of two cubes only, transformed. The cube has six sides, 
 and the whole of which are uniform : therefore, the surfaces of two cubes 
 will make altogether twelve sides ; and the twelve sides are in this figure 
 so disposed as to form the annexed plate. Numbers one, two, upper ; 
 three, centre; four and five, lower, occupy the interior. Six, seven, 
 eight, and nine, make the angles one, two, four, five. Ten and eleven 
 enclose the centre, and twelve make the four exterior angles. The eight 
 protruding points are composed of half a side, as is shown by the dotted 
 figure below. The student, in performing this figure, will be particular 
 in observing the precise areas ; and, in order to ascertain the correct ad- 
 measurement, he may readily consolidate the cubes and also the figure ; 
 he will then find their areas to correspond. 
 
 PLATE LIV. 
 
 The circle compounded, or divided into four parts; viz. one, two, 
 three, and four; the same are disposed below in another figure, and 
 are represented in different positions upon a similar arrangement, as 
 Figure 53. 
 
 G 
 
22 DESCRIPTION OF PLATES. 
 
 PLATE LV. 
 
 — 
 
 The spiral, or volute. This figure is formed in the simplest and 
 plainest manner; and by this method any figure of like nature may 
 thus be fabricated: it is plainly shown at the centre, or commencement 
 of the circle, three quarters of an inch in diameter, the origin of the spiral; 
 and the line proceeds from four to one, which sets off one division from 
 the centre circle ; another takes place at figure two, two divisions from 
 the centre circle, three divisions at three, four divisions at four, and 
 so upwards, until the end of the line, at twenty-three. The quarter cir- 
 cles, forming the spiral, may be struck with the dividers from a centre, 
 obtained so as to take in the extreme points regularly. 
 
TS 
 
 27 
 
 * Designed and drawn by John Bennett . 
 
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