Elementary or Common Geometry, is that which is 27. A right angled triangle, is that which has a employed in the consideration of right lines and right angle. plane surfaces, with the solid3 generated from 28. An obtuse angled triangle, is that which has them. And the an obtuse angle. Higher or Sublime Geometry, is that which is em- 29. An acute angled triangle, is that which has ployed in the consideration of curve lines, conic three acute angles. sections, and the bodies formed of them. This 30. Of four sided figures, a square is that which part has been chiefly cultivated by the moderns, has all its sides equal, and all its angles right by help of the improved state of algebra, and the angles. modern analysis or fuxions. 31. An oblong, is that which has all its angles The standard author on the elements of geome. right angles, tu: has not all its sides equal. try is Euclid. See Euclid and ELEMENTS. The 32. A rhombus, is that which has all its sides definitions and propositions of his first six, and equal, but its angles are not right angles. eleventh and twelfth books, are as below. 33. A rhomboid, is that which has its opposite Book I. Def, 1.-A point is that which hath no sides equal to one another, but all its sides are not magnitude, equal, nor its angles right angles. 2. A line is length without breadth. 34. All otiser four sided figures besides these, 3. The extremities of a line are points. are called trapeziums. 4. A straight line is that which lies evenly be- 35. Parallel straight lines, are such as are in the tween its extreme points. same plane, and which, being produced ever so far 5. A superficies is that which hath only length both ways, do not meet. and breadth. Postulates.-1. Let it be granted that a straight 6. The extremities of a superficies are lines. lime ray be drawn froir. any one point to any 7. A plane superficies is that in whicn any two other point. points being taken, the straight line between them 2. That a terminated straight line may be prolies wholly in that saperficies. duced to any length in a straight line. 8. “A plane angle is the inclination of two lines 3. And that a circle may be described from any to one another in a plane, which meet together, centre, at any distan, e from that centre. but are not in the same directior.." Arioms.-1. Things which are equal to the same 9. A plane rectilineal angle is the inclination of are equal to one another, two straight lines to one another, which meet to- 2. If equals be added to equals, the wbəles are get her, but are not in the same straight line. equal. 10. When a straight line standing on another 3. If equais be taken from equals, the remainstraight line makes the adjacent angles equal to ders are equal. one another, each of the angles is called a right 4. If equals be added to unequals, the wholes angle; and the straight line which stands on the are unequal. other is called a perpendicular to it. 5. If equals be taken from unequals, the re11. An obtuse angle is that which is greater mainders are unequal. than a right angle. 6. Things which are double of the same, are 12. An acute angle is that which is less than a equal to one another. right angle. 7. Things which are hal; es of the same, are 13. “A term or boundary is the extremity of equal to one another, any thing." 8. Magnitudes which coincide with one another, 14. A figure is that which is inclosed by one or that is, which exactly fill the saine space, are more boundaries. equal to one another. 15. A circle is a plane figure contained by one 9. The whole is greater than its part. line, which is called the circumference, and is 10. Two straight lines cannot inclose a space. such that all straight lines drawn from a certain 11. All right angles are equal to one another. point within the figure to the circumference, are 12. “If a straight line meets twu straigot lines, equal to one another. so as to make the two interior angles on the same 16. And this point is called the centre of the side of it taken together less than two right ancircle. gles, these straight lines being continually pro17. A diameter of a circle is a straight line duced, shall at length meet upon that side ou drawn through the centre, and terminated both which are the angles which are less than two right ways by the circumference. angles. See the notes on Prop. XXIX. of Book L.” 18. A semicircle is the figure contained by a Prop. I. Prob. To describe un equilateral diameter and the part of the circumference cut off triangle upon a giren finite straight line. by the diameter. Prop. II. Prob. Froni a given point to draw a 19. “A segment of a circle is the figure con- straight line equal to a given straight line. tained by a straight line, and the circumference it Prop. III. Prob. From the greater of two giren cuts off," straight lines to cut off a part equal to the less. 20. Rectilineal figures are those which are con- Prop. IV. Theor. If two triangles have two tained by straight lines. sides of the one equal to two sides of the other, 21. Trilateral figures, or triangles, by three each to each; and have likewise the angles constraight lines. tained by those sides equal to one another; they 22. Quadrilateral, by four straight lines. shall likewise have their bases, or third sides, 23. Multilateral figures, or polygons, by more equal; and the two triangles shall be equal ; and than four straight lines. their other angles shall be equal, each to each, 24. Of three sided figures, an equilateral trian- viz. those to which the equal sides are opposite. gle is that which has three equal sides. Prop. V. Theor. The angles at the base of an -25. An isosceles triangle, is that which has only isosceles triangle are equal to one another; and, two sides equal. if the equal sides be produced, the angles upon 26. A scalene triangle, is that which has three the other side of the base shall be equal. unequal sides. Prop. VI. Theor. If two angles of a triangle-be equal to one another, the sides also which subtend, be greater than the angle contained by the sides ör are opposite to, the equal angles, shall be equal equal to them, of the other. to one another. Prop. XXVI. Theor. If two triangles have two Prop. VII. Theor. Upon the same base, and angles of one equal to two angles of the other, each on the same side of it, there cannot be two trian- to each; and one side equal to one side, viz. either gles that have their sides which are terminated in the sides adjacent to the equal angles, or the sides one extremity of the base equal to one another, opposite to equal angles in each ; then shall the and likewise those which are terminated in the other sides be equal, each to each; and also the other extremity. third avgle of the one to the third angle of the Prop. VIII. Theor. If two triangles have two other. sides of the one equal to two sides of the other, Prop. XXVII. Theor. If a straight line falling each to each, and have likewise their bases equal; upon two other straight lines makes the alternate the angle which is contained by the two sides of angles equal to one another, these two straight the one shall be equal to the angle contained by lines shall be parallel. the two sides equal to them, of the other. Prop. XXVIII. Theor. If a straight line falling Prop. IX. Prob. To bisect a given rectilineal upon iwo others traight lines makes the exterior angle, that is, to divide it into two equal angles. angle equal to the interior and opposite upon the Prop. X. Prob. To bisect a given finite straight saine side of the line; or makes the interior anline, that is, to divide it into two equal parts. gles upon the same side together equal to two right Prop. XI. Prob. To draw a straight line at angles; the two straight lines shall be parallel to right angles to a given straight line, from a given one another. point in the same. Prop. XXIX. Theor. If a straight line fall Prop. XII. Prob. To draw a straight line per- upon two parallel straight lines, it makes the alpendicular to a given straight line of an unlimit. ternate angles equal to one another; and the exed length, from a given point without it. terior angle equal to the interior and opposite Prop. XIll. Theor. The angles which one upon the same side; and likewise the two interior straight line makes with another upon the one side angles upon the same side together equal to two of it, are either two right angles, or are together right angles. equal to two right angles. Prop. XXX. Theor. Straight lines which are Prop. XIV. Theor. If, at a point in a straight parallel to the same straight line are parallel to Jipe, two other straight lines, upon the opposite one another. sides of it, make the adjacent angles togetber Prop. XXXI. Prob. To draw a straight line equal to two right angles, these two straight lines through a given point parallel to a given straight shall be in one and the same straight line. line. Prop. XV. Theor. If two straight lines cut one Prop. XXXII. Theor. If a side of any triangle another, the vertical, or opposite, angles shall be be produced, the exterior angle is equal to the two equal. interior and opposite angles: and the three inteProp. XVI. Theor. If one side of a triangle rior angles of every triangle are equal to two right be produced, the exterior angle is greater than angles. either of the interior opposite angles. Prop. XXXIII. Theor. The straight lines Prop. XVII. Theor. Any two angles of a trian- which join the extremities of two equal and pagle are together less than two right angles. rallel straight lines, towards the same parts, are Prop. XVIII. Theor. The greater side of every also themselves equal and parallel. triangle is opposite to the greater angle. Prop. XXXIV. Theor. The opposite sides and Prop. XIX. Theor. The greater angle of every angles of parallelograms are equal to one another, triangle is subtended by the greater side, or has and the diameter bisects them, that is, divides the greater side opposite to it. them in two equal parts. Prop. XX. Theor. Any two sides of a triangle Prop. XXXV. Theor. Parallelograms upon are together greater than the third side. the saine base and between the same parallels, are Prop. XXI. Theor. If, from the ends of the equal to one another. side of a triangle, there be drawn two straight Prop. XXXVI. Theor. Parallelograms upon lines to a point within the triangle, these shall be equal bases, and between the same parallels, are less than the other two sides of the triangle, but equal to one another. shall contain a greater angle. Prop. XXXVII. Theor. Triangles apon the Prop. XXII. Prob. To make a triangle of same base, and between the same parallels, are which the sides shall be equal to three given equal to one another. straight lines, but any two whatever of these must Prop. XXXVIII. Theor. Triangles upon equal be greater than the third. bases, and between the same parallels, are equal Prop. XXIII. Prob. At a given point in a given to one another. straight line, to make a rectilineal angle equal to a Prop. XXXIX. Theor. Equal triangles upon given rectilineal angle. Lhe same base, and upon the same side of it, are Prop. XXIV. Theor. If two triangles have two ' between the same parallels. sides of the one equal to two sides of the other, Prop. XL. Theor. Equal triangles upon equal each to each, but the angle contained by the two bases, in the same straight line, and towards the sides of one of them greater than the angle con- same parts, are between the same parallels. tained by the two sides equal to them, of the Prop. XLI. Theor. If a parallelogram and otber; the base of that which has the greater triangle be upon the same base, and between the angle shall be greater than the base of the other. same parallels; the parallelogram shall be double Prop. XXV. Theor. If two triangles have two of the triangle. sides of the one equal to two sides of the other, Prop. XLII. Prob. To describe a parallelogram each to each, but the base of the one greater than that shall be equal to a given triang'e, and have the base of the other ; the angle also contained by one of its angles equal to a given rectilineal the sides of that which has the greater base, sball angle, VOL. V. S Prop. XLIII. Theor. The complements of the double of the square of half the line, and of the parallelograms which are about the diameter of square of the line between the points of section. any parallelogram, are equal to one another. Prop. X. Theor. If a straight line be bisected, Prop. XLIV. Prob. To a given straight line to and produced to any point, the square of the apply a parallelogram, which shall be equal to a whole line thus produced, and the square of the given triangle, and have one of its angles equal to part of it produced, are together double of the a given rectilineal angle. square of balf the line bisected, and of the square Prop. XLV. Prob. To describe a parallelo- of the line made up of the half and the part program equal to a given rectilineal figure, and hav- duced. ing an angle equal to a given rectilineal angle. Prop. XI. Prob. To divide a given straight line Prop. XLVI. Prob. To describe a square upon into two parts, so that the rectangle contained by a given straight line. the whole, and one of the parts, shall be equal to Prop. XLVII. Theor, In any right angled the square of the other part. triangle, the square which is described upon the Prop. XII. Theor. In oltuse angled triangles, if side subtending the right angle, is equal to the a perpendicular be drawn from any of the acute squares described upon the sides which contain angles to the opposite side prodaced, the square the right angle, of the side subtending the obtuse angle is greater Prop. XLVIII. Theor. If the square described than the squares of the sides containing the ob upon one of the sides of a triangle, be equal to the tuse angle, by twice the rectangle contained by squares described upon the other two sides of it; the side upon which, when produced, the perpenthe angle contained by these two sides is a right dicular falls, and the straight line intercepted angle. without the triangle between the perpendicular Book II. Def. 1.-Every right angled parallelo. and the obtuse angle. gram is said to be contained by any two of the Prop. XIII. Theor. In every triangle, the square straight lines which contain one of the right angles. of the side subtending any of the acute angles, iš 2. In every parallelogram, any of the parallelo- less than the squares of the sides containing that grams about a diameter, together with the two angle, by twice the rectangle contained by either complements, is called a gnomon. of these sides, and the straight line intercepted Prop. I. Theor. If there be two straight lines, between the perpendicular let fall upon it from one of which is divided into any number of parts; the opposite angle, and the acute angle. the rectangle contained by the two straight lines, Prop. XIV. Prob. To describe a square that shall is equal to the rectangles contained by the undi- be equal to a given rectilineal figure. vided line, and the several parts of the divided Book III. Def. 1.-Equal circles are those of line. which the diameters are equal, or from the centres Prop. II. Theor. If a straight line be divided of which the straight lines to the circumferences 'into any two parts, the rectangles contained by are equal. the whole and each of the parts, are together This is not a definition but a theorem, the equal to the square of the whole line. truth of wbich is evident; for, if the circles be Prop. III. Theor. If a straight line be divided applied to one another, so that their centres coininto any two parts, the rectangle contained by the cide, the circles inust likewise coincide, since the whole and one of the parts, is equal to the rectan- straight lines from the centres are equal. gle contained by the two parts, together with the 2. A straight line is said to touch a circle, when square of the foresaid part. it meets the circle, and being produced does not Prop. IV. Theor. If a straight line be divided cut it. into any two parts, the square of the whole line 3. Circles are said to touch one another, which is equal to the squares of the two parts, together meet, but do not cut one another. with twice the rectangle contained by the parts. 4. Straight lines are said to be equally distant Prop. V. Theor. If a straight line be divided from the centre of a circle, when the perpendicuinto two equal parts, and also into two unequal lars drawn to them from the centre are equal. parts, the rectangle contained by the unequal 5. And the straight line on which the greater parts, together with the square of the line be- perpendicular falls, is said to be farther from the iween the points of section, is equal to the square centre. of half the line. 6. A segment of a circle is the figure contained Prop. VI. Theor. If a straight line be bisected, by a straight line and the circunference it cuts and produced to any point; the rectangle con- off. tained by the whole line thus produced, and the 7. The angle of a segment is that which is part of it produced, together with the square of contained by the straight line and the circumferhalf the line bisected, is equal to the square of the ence. straight line which is made up of the half and the 8. An angle in a segment is the angle contained part produced. 'by two straight lines drawn from any point in the Prop. VII. Theor. If a straight line be divided circumference of the segment, to the extremities into any two parts, the squares of the whole line, of the straight line which is the base of the seg. and of one of the parts, are equal to twice the ment, rectangle contained by the whole and that part, 9. And an angle is said to insist or stand upon together with the square of the other part. the circumference intercepted between the straight Prop. VIII. Theor. If a straight line be divided lines that contain the angle. into any two parts, four times the rectangle con- 10. The sector of a circle is the figure contaio. tained by the whole line, and one of the parts, to ed by two straight lines drawn from the centre, gether with the square of the other part, is equal and the circumference between them. to the square of the straight line which is made up 11. Similar segments of a circle, are those in of the whole and that part. which the angles are equal, or which contain Prop. IX. Theor, If a straight line be divided equal angles. into two equal, and also into two unequal parts; Prop. I. Prob. To find the centre of a given cir. the square of the two unequal parts are together cle. Prop. II. Theor. If any two points be taken in angle with the diameter at its extremity, or so the circumference of a circle, the straight line small an angle with the straight line which is at which joins them shall fall within the circle. right angles to it, as not to cut the circle. Prop. III. Theor. If a straight line drawn through Prop. XVII. Prob. To draw a straight line from the centre of a circle bisect a straight line in it a given point, either without or in the circumferwbich does not pass through the ceotre, it shall ence, which shall touch a giren circle. cut it at right angles; and, if it cuts it at right Prop. XVIII. Theor. If a straight line touches a angles, it shall bisect it. circle, the straight line drawn from the centre to Prop. IV. Theor. If in a circle two straight lines the point of contact, shall be perpendicular to the cut one another which do not both pass through line touchiog the circle. the centre, they do not bisect each other. Prop. XIX. Theor. If a straight line touches a Prop. V. Theor, If two circles cut one another, circle, and from the point of contact a straight they shall not have the same centre. line be drawn at right angles to the touching line, Prop. VI. Theor. If two circles touch one an- the centre of the circle shall be in that line. other internally, they shall not have the same Prop. XX. Theor. The angle at the centre of a centre, circle is double of the angle at the circumference, Prop. VII. Theor. If any point be taken in the upon the same base, that is, upon the same part diameter of a circle, which is not the centre, of all of the circumference. the straight lines which can be drawn from it to Prop. XXI. Theor. The angles in the same segthe circumference, the greatest is that in which ment of a circle are equal to one another. the centre is, and the other part of that diameter Prop. XXII. Theor. The opposite angles of any is the least; and, of any others, that which is quadrilateral figure described in a circle, are tonearer to the line which passes through the centre gether equal to two right angles. is always greater than one more remote: and from Prop. XXIII. Theor. Upon the same straight the same point there can be drawn only two line, and upon the same side of it, there cannot straight lines that are equal to one another, one be two similar segments of circles, not coinciding upon each side of the shortest line. with one another. Prop. VIII. Theor. If any point be taken with. Prop. XXIV. Theor. Similar segments of circles out a circle, and straight lines be drawn from it to upon eqnal straight lines, are equal to one anthe circumference, whereof one passes through the other. centre; of those wbich fall upon the concave cir- Prop. XXV. Prob. A segment of a circle being camference, the greatest is that which passes given, to describe the circle of which it is the through the centre; and of the rest, that which is segment. nearer to that through the centre is always greater Prop. XXVI. Theor. In eqnal circles, equal than the more remote: but of those which fall angles stand upon equal circumferences, whether upon the convex circumference, the least is that they be at the centres or circumferences. between the point without the circle, and the dia- Prob. XXVII. Theor. In equal circles, the anmeter; and of the rest, that which is nearer to gles which stand upon equal circumferences are the least is always less than the more remote: equal to one another, whether they be at the and only two equal straight lines can be drawn centres or circumferences. from the point unto the circunference, one upon Prop. XXVIII. Theor. In equal circles, equal each side of the least. straight lines cut off equal circumferences, the Prop. IX. Prob. If a point be taken within a greater equal to the greater, and the less to the circle, from which there fall more than two equal less. straight lines to the circumference, that point is Prop. XXIX. Theor. In equal circles equal cirthe centre of the circle. cumferences are subtended by equal straight Prop. X. Theor. One circumference of a circle lines. cannot cut another in more than two points. Prop. XXX. Prob. To bisect a given circumfer Prop. XI. Theor. If two circles touch cach other ence, that is, to divide it into two equal parts. internally, the straight line which joins their cen- Prop. XXXI. Theor. In a circle, the angle in a tres being produced shall pass through the point semicircle is a right angle; but the angle in a segof contact. ment greater than a semicircle is less than a right Prop. XII. Theor. If two circles touch each angle; and the angle in a segment less than a other externally, the straight line which joins their semicircle is greater than a right angle. centres shall pass through the point of contact. Prop. XXXII. Theor. If a straight line touches Prop. XIII. Theor. One circle cannot touch a circle, and from the point of contact a straight another in more points than one, whether it line be drawn cutting the circle, the angles made touches it on the inside or outside. by this line with the line touching the circle, shall Prop. XIV. Theor. Equal straight lines in a be equal to angles which are in the alternate segcircle are equally distant from the centre; and ments of the circe. those which are equally distant from the centre, Prop. XXXIII. Prob. Upon a given straight line are equal to one another. to describe a segment of a circle, containing an Prop. XV. Theor. The diameter is the greatest angle equal to a given rectilineal angle. straight line in a circle; and, of all others, that Prop. XXXIV. Prob. To cut off a segment from which is nearer to the centre is always greater a given circle which shall contain an angle equal than one more remote; and the greater is nearer to a given rectilineal angle. to the centre than the less. Prop. XXXV. Theor. If two straight lines within Prop. XVI. Theor. The straight line drawn at a circle cut one another, the rectangle contained > right angles to the diameter of a circle, from the by the segments of one of them is equal to the extremity of it, falls without the circle; and no rectangle contained by the segments of the other. straight line can be drawn between that straight Prop. XXXVI. Theor. If from any point withline and the circumference from the extremity, so out a circle two straight lines be drawn, one of as not to cut the circle; or, which is the same which cuts the circle, and the other touches it; thing, no straight line can make so great an acute the rectangle contained by the whole line which cute the circle, and the part of it without the cir- 2. A greater magnitude is said to be a multiple cle, shall be equal to the square of the line which of a less, when the greater is measured by the less, touches it. that is, when the greater contains the less a cerPop. XXXVII. Theor. If from a point without tain number of times exactly. a circle there be drawn into straight lines, one of 3. Ratio is a mutual relation of two magniwhich cuts the circle, and the other inéets it; if tudes of the same kind to one another, in respect the rectangle contained by the whole line which of quantity: cuts the c rcle, and the part of it without the cir- 4. Magnitudes are said to have a ratio to one cle be equal to the square of the line which meets another, when the less can be multiplied so as to it, the line which meets shall touch the circle. exceed the other. Book [V. Def. 1.-A rectilineal figure is said to 5. The first of four magnitudes is said to hare bc inscribed in another rectilineal figure, when all the same ratio to the second, which the third has the angles of the inscribed figure are upon the side to the fourth, when any equimultiples whatsoever of the figure in which it is inscribed, each upon each. of the first and third being taken, and any equie 2. Iu like manner, a figure is said to be describ- multiples whatsoever of the second and fourth *; if ed about another figure, w' en all the sides of the the multiple of the first be less than that of the circumscribed figure pass through the angular second, the multiple of the third is also less than points of the figure about which it is described, that of the fourth; or, if the multiple of the first each through each. be equal to that of the second, the multiple of the 3. A rcctilineal figure is said to be inscribed in third is also equal to that of the fourth; or, if the a circle, when all the angles of the inscribed fi. multiple of the first be greater than that of the gure are upon the ciicumference of the circle. second, the multiple of the third is also greater 4. A rectilineal figure is said to be described than that of the fourth. about a circle, when each side of the circumscrib- 6. Maguitudes which have the same ratio are ed figure touches the circumference of the circle. called proportionals. N. B. When four magoi 5. In like manner, a circle is said to be inscribed tudes are proportionals, it is usually expressed by in a rectilineal figure, when the circumserence of saying, the first is to the second, as the third to the circle touches each side of the figure. the fourth. 6. A circle is said to be described about a rec- 7. When of the equimultiples of four magnitudes tilineal figure, when the circumference of the cire (taken as in the fifth definition) the multiple of cle passes through all the angular points of the the first is greater than that of the second, but ligure about which it is described. the multiple of the third is not greater than the 7. A straight line is said to be placed in a circle, multiple of the fourth; then the first is said to when the extremities of it are in the circumference have to the second a greater ratio than the third of the circle. magnitude has to the fourth; and, on the contraProp. I. Piob. In a given circle to place a straight ry, the third is said to have to the fourth a less line, equal to a given straight line not greater ratio than the first has to the second. than the diameter of the circle. 8. Analogy, or proportion, is the similitude Prop. II. Prob. In a given circle to inscribe a of ratios. triangle equiangular to a given triangle. 9. Proportion consists in three terms at least. Prop. III. Prob. About a given circle to describe 10. When three magnitudes are proportionals, a triangle equiangular to a given triangle. the first is said to have to the third the duplicate Prop. iv. Prob. To inscribe a circle in a giren ratio of that which it has to the second. triangle. 11. When four niagnitudes are continual proProp. V. Prob. To describe a circle about a portionals, the first is said to have to the fourth given triangle. the triplicate ratio of that which it has to the Prop. VI. Prob. To inscribe a square in a given second, and so 041, quadruplicate, &c. increasing circle. the denomination stilt by unity, in any number of Prop. VII. Prob. To describe a square about a proportionals. given circle. Definition 4. to wit, of compound ratio. Prop. VIII. Prob. To inscribe a circle in a given When there are any number of magnitudes, of square, the same kind, the first is said to have to the last Prop. IX. Prob. To describe a circle about a of them the ratio compounded of the ratio which given square the first has to the second, and of the ratio which Prop. x. Prob. To describe an isosceles trian. the second has to the third, and of the ratio which gle, having each of the angles at the base double the third has to the fourth, and so on unto the of the third angle. last magnitude. Prop. XI. Prob. To inscribe an equilateral and For example, if A, B, C, D be four magnitudes equiangular pentagon in a given circle. of the same kind, the first is said to have to the Prop. XII. Prob. To describe an equilateral and last Đ the ratio compounded of the ratio of A to equiangular pentagon about a given circle. D, and of the ratio of B to C, and of the ratio of Prop. XII. Prob. To inscribe a circle in a given C to D; or, the ratio of A to D is said to be comcquilateral and equiangular pentagon. pounded of the ratios of A to B, B to C, and C to Prop. XIV. Prob. To describe a circle about a D: and if A has to B the same ratio wbich E has given equilateral and equiangular pentagon. to F; and B to C, the same ratio that G has to H; Prop. XV. Prob. To inscribe an equilateral and ' and C to D, the same that K has to L; then, by equiangular bexagon in a given circie. this definition, A is said to have to D the ratio Prop. XVI. Prob. To inscribe an equilateral and compounded of ratios which are the same with egaiangular quindecagon in a given circle. the ratios of E to F, G to H, and K to L: and the Book y. Def. 1.-A less magnitude is said to be same thing is to be understood when it is more a part of a greater magnitude, when the less mea- brielly expressed, by saying A has to D the ratio sures the greater, that is, when the less is con- compounded of the ratios of E to F, G to H, and tained a certain number of times exactly in the. K to L. Treater. In like manner, the same things being supposed, |