376 GEOMETRY G none. ness. PART I.-PURE GEOMETRY. EOMETRY has been divided since the time of Euclid | still an open question. We shall consider only Euclid's into an elementary” and a “higher” part. The system. contents and limits of the former have been fixed by § 2. The axioms are obtained from inspection of space Euclid's Elements. The latter included at the time of and of solids in space,,hence from experience. The same the Greek mathematicians principally the properties of the source gives us the notions of the geometrical entities to conic sections and of a few other curves. The methods used which the axioms relate, viz., solids, surfaces, lines or in both were essentially the same. These began to be re- curves, and points. A solid is directly given by expe placed during the 17th century by more powerful methods, rience; we have only to abstract all material from it in invented by Roberval, Pascal, Desargues, and others. But order to gain the notion of a geometrical solid This bas the impetus which higher geometry received in their works shape, size, position, and may be moved. Its boundary or was soon arrested, in consequence of the discoveries of boundaries are called surfaces. They separate one part of Descartes,—the new calculus to which these gave rise space from another, and are said to have no thickness, absorbing the attention of mathematicians almost ex. Their boundaries are curves or lines, and these have clusively, until Monge, at the end of the 18th century, length only. Their boundaries, again, are points, which re-established "pure as distinguished from Descartes's have no magnitude but only position. We thus come in "coordinate” (or analytical) geometry. Since then the three steps from solids to points which have no magnitude; purely geometrical methods have been continuously ex- in each step we lose one extension. Henco we say a solid tended, especially by Poncelet, Steiner, Von Staudt, and has three dimensions, a surface two, a line one, and a point Cremona, and in England by Hirst and Henry Smith, to Space itself, of which a solid forms only a part , is mention only a few of the leading names. also said to be of three dimensions. The same thing is Whilst higher geometry thus made most rapid progress, intended to be expressed by saying that a solid bas length, the elementary part remained almost unaltered. It has breadth, and thickness, a surface length and breadth, a line been taught up to the present day on the basis of Euclid's length only, and a point no extension whatsoever. Elements, the latter being either used directly as a text- Euclid gives the essence of these statements as defini. book (in England), or being replaced (in most parts of the tions :Continent) by text-books which are essentially Euclid's Def. 1, l. A point is that which has no parts, or which has no mag. Elements rewritten, with a few additions about the men nitude. saration of the circle, cone, cylinder, and sphere. Def. 2, I. A line is length without breadth, Only Def. 6, 1. A superficies is that which has only length and breadih. within a very recent period have attempts been made to Def. 1, XI. A solid is that which has length, breadth, and thickchange the character of the elementary part by introducing some of the modern methods. If we allow motion in geometry,—and it seems impasWe shall give in this article—first, a survey of elemen- sible to avoid it,—we may generate these entities by tary geometry as contained in Euclid's Elements, and then, moving a point, a line, or a surface, thus :in form of an independent treatise, an introduction to The path of a moving point is a line. higher geometry, based on modern methods. In the The path of a moving line is, in general, a surface. former part we shall suppose that a copy of Euclid's The path of a moving surface is, in general, a solid. Elements is in the hands of the reader, so that we may And we may then assume that the lines, surfaces, and dispense, as a rule, with giving proofs or drawing figures. solids, as defined before, can all be generated in this manWê thus shall give only the contents of his propositions ner. From this generation of the entities it follows again grouped together in such a way as to show their connexion, that the boundaries—the first and last position of the morand often expressed in words which differ from the verbal ing element—of a line are points, and so on; and thus we translation in order to make their meaning clear. It will come back to the considerations with which we started. make little difference which of the many English editions Euclid points this out in his definitions, - Def. 3, I., of Euclid's Elements the reader takes. Of these we may Def . 6, 1, and Def. 2, XI. He does not, however, show mention Simson's, Potts's, and Todhunter's the connexion which these definitions have with those mentioned before. When points and lines have been SECTION 1.-ELEMENTARY OR EUCLIDIAN GEOMETRY. defined, a statement like Def. 3, I, “The extremities of a line are points,” is a proposition which either has to be The Axioms. proved, and then it is a theorem, or which has to be taken § 1. The object of geometry is to investigate the proper- for granted, in which case it is an axiom. And so with ties of space. The first step must consist in establishing Def. 6, I., and Def. 2, XI. those fundamental properties from which all others follow § 3. Euclid's definitions mentioned above are attempts to by processes of deductive reasoning. They are laid down describe, in a few words, notions which we have obtained in the Axioms, and these ought to form such a system that by inspection of and abstraction from solids. A few more nothing need be added to them in order fully to charac- notions have to be added to these, principally those of the terize space, and that nothing may be left out without mak simplest line-the straight line, and of the simplest surface ing the system incomplete. They must, in fact, completely —the flat surface or plane. These notions we possess, but “ define” space. Several such systems are conceivable. to define them accurately is difficulte Euclid's Definition Euclid has given one, others have been put forward in 4, 1., “A straight line is that which lies evenly between recent times by Riemann (Abhandl. der königl . Gesellsch. its extreme points," must be meaningless to any one who zu Göttingen, vol. xiii.), by Helmholz (Göttinger Nach- has not the notion of straightness in his mind. Neither richten, June 1868), and by Grassmann (Ausdehnungslehre does it state a property of the straight line which can be von 1841). How many axioms the system ought to con- used in any further investigation. Such a property is tain, and which system is the simplest, may be said to be 1 given in Axiom 10, 1." It is really this axion, together That every with Postulates 2 and 3, which characterizes the straight thing descriped. Many of them overdetermine a figura line. (Compare notes to definitions in Simson's or Todhunter's Whilst for the straight line the verbal definition and edition.) axiom are kept apart, Euclid mixes them up in the case of § 5. Euclid's Elements are contained in thirteen books. the plane. Here the Definition 7, I., includes an axiom. Of these the first four and the sixth are devoted to "plane It defines a plane as a surface which has the property that geometry,” as the investigation of figures in a plane is every straight line which joins any two points in it lies generally called. The 5th book contains the theory of altogether in the surface. But if we take a straight line proportion which is used in Book VI. The 7th, 8th, and a point in such a surface, and draw all straight lines and 9th books are purely arithmetical, whilst the 10th which join the latter to all points in the first line, the contains a most ingenious treatment of geometrical irrational surface will be fully determined. This construction is quantities. These four books will be excluded from our therefore sufficient as a definition. other survey. The remaining three books relate to figures in straight line which joins any two points in this surface space, or, as it is generally called, to “solid geometry.” The lies altogether in it is a further property, and to assume it 7th, 8th, 9th, 10th, 13th, and part of the 11th and 12th gives another axiom. books are now generally omitted from the school editions Thus a number of Euclid's axioms are bidden among of the Elements. In the first four and in the 6th book his first definitions. A still greater confusion exists in it is to be understood that all figures are drawn in a the present editions of Euclid between the postulates and plane. axioms so-called, but this is due to later editors and not Door I. Or Euclid's “ ELEMENTS." to Euclid himself. The latter had the last three axioms put together with the postulates (airuara), so that these $ 6. According to the third postulate it is possible to draw in were meant to include all assumptions relating to space. any plane a circle which has its centre at any given point, and its radius equal to the distance of this point from any other point The remaining assumptions which relate to magnitudes in general , viz, the first eight“ axioms” in modern editions, siven in the plane. This makes it possible (Prop. 1) to construct on a given line AB an equilateral triangle, by drawing first a were called "common notions” (xouvai čvvocal). Of the latter circle with A as centre and AB 28 radius, and then a circle with B a few may be said to be detinitions. Thus the eighth as centre and BA as radius. The point where these circles intermight be taken as a definition of “equal,” and the seventh sect—that they intersect Euclid quietly assumes—is the vertex of of halves. If we wish to collect the axioms used in circle may be drawn which has its radius equal to the distance be the required triangle. Euclid does not suppose, however, that a Euclid's Elements, we have therefore to take the three tween any two points unless one of the points be the centre. This postulates, the last three axioms as generally given, a few implies also that we are not supposed to be able to make any axioms hidden in the definitions, and an axiom used by straight line equal to any other straight line, or to carry a distance about in space. Euclid therefore next solves the problem: It Euclid in the proof of Prop. 4 and on a few other occa is required along a given straight line from a point in it to set sions, viz., that figures may be moved in space without off a distance equal to the length of another straight line given change of shape or size. anywhere in the plane.. This is done in two steps. It is shown in We shall not enter into the investigation how far ute Prop. 2 how a straight line may be drawn from a given point equal assumptions which would be included in such a list are in longth to another given straight line not drawn from that point. And then the problein itself is solved in Prop. 3, by drawing first sufficient, and how far they are necessary. It may be through the given point some straight line of the required length, sufficient here to stato that from the beginning of a and then about the same point as centre a circle having this length geometrical science to the present century attempts without as radius. This circle will cut off from the given straight line a end have been made to prove the last of Euclid's axioms, through this long process, we take a pair of compasses and set off length equal to the required one. Now-a-days, instead of going that only at the beginning of the present century the the given length by its aid. This assumes that we may move a futility of this attempt was shown, and that only within the length about without changing it. But Euclid has not assumed it, last twenty years the true nature of the connexion between and this proceeding would be fully justified by his desire not to the axioms has become known through the researches of take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though Riemann and Helmholz, although Grassmann bad pub- without stating it. lished already, in 1844, his classical but long.neglected § 7. We now come (in Prop. 4) to the first theorem. It is the fundamental theorem of Euclid's whole system, there being only a § 4. The assumptions actually made by Euclid may be very few propositions (like Props. 13, 14, 15, 1.), except those in the 5th book and the first half of the 11th, which do not depend upon It is stated very accurately, though sumewhat clumsily, as 1. Straight lines exist which have the property that any one of follows: thein may be produced both ways without limit, that through any If tro triangles have two sides of the one equal to two sides of the two points in space such a line may be drawn, and that any tyd of other, cach to each, and have also the angles contained by those sides thern coincide throughout their indefinite extensions as soon as two equal to one another, they shall also have their bascs or third sides points in the one coincide with two points in the other. (This cqual; and the two triangles shall be equal ; ar l their other angles gives the contents of Def. 4, part of par. 85, the first two Postulates, shall be equal, each to eacả, namely, those to which the equal sides are opposite. 2. Plane snrfaces or planes exist having the property laid down That is to say, the triangles are identically" equal, and one in Def. 7, that every straight line joining any two points in such a may be considered as a copy of the other. The proof is very simple. The first triangle is taken up and placed on the second, so that the 3. Right angles, as defined in Def. 10, are possible, and all right parts of the triangles which are known to be equal fall upon each angles are equal; that is to say, wherever in space we take a other. It is then easily seen that also the remaining parts of one plane, and wherever in that plane we construct a right angle, all coincide with those of the other, and that they are therefore equal. angles thus constructed will be equal, so that any one of them may This process of applying one figure to another Euclid scarcely uses be made to coincide with any other. (Axiom 11.) again, though many proofs would be simplified by doing so. The 4. The 12th Axiom of Euclid. This we shall not state now, but process introduces motion into geometry, and includes, as already only introduce it when we cannot proceed any further without it. stated, the axiom that figures may be moved without change of 6. Figures may be freely moved in space without change of shape or size. shape of size. This is assumed by Euclid, but not stated as an If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop: 5), i two sides 6. In any plane a circle may be described, having any point in of a triangle are equal, then the angles opposite these sides are that plane as centre, and its distance from any other point in that equal . plane as radius. (Postulate 3.) Euclid's proof is somewhat complicated, and a stumbling-Wock The definitions which have not been mentioned are all Aulehnungslehre. stated as follows : it. and surfare lies altogether in it. axiom. to many schoolboys. The proof becomes much simpler if we con sider the isosceles triangle ABC (AB-AC) twice over, once as a “ nominal definitions," that is to say, they fix a name for å triangle BAC, and once as a triangle CAB; and now remember that 10--15* AB, AC in the first are equal respectively to AC, AB in the second, vestigation, which starts from Prop. 16, will become clearer if a few and the angles included by these sides are equal. Hence the tri- names be explained which are not all. used by Euclid. If two angles are equal, and the angles in the one are equal to those in the straight lines be cut by a third, the latter is now generally called a other, viz., those which are opposite equal sides, i.e., angle ABC in "transversal ” of the figure. It forms at the two points where it the first equals angle ACB in the second, as they are opposite the cuts the given lines four angles with each. Those of the angles equal sides AC, and AB in the two triangles. which lie between the given lines are called interior angles, and of There follows the converse theorem (Prop. 6). If two angles in these, again, any two which lie on opposite sides of the transversal a triangle are equal, then the sides opposite them are equal, -i..., the but one at each of the two points are called "alternate angles", trianglo is isosceles. The proof given consists in what is called a We may now state Prop. 16 thus :-If two straight lines which reductio ad absurdum, a kind of proof often used by Euclid, enect are cut by a transversal, their alternate angles are unequal. and principally in proving the converse of a previous theorem. For the lines will form a triangle, and one of the alternate angles It assumes that the theorem to be proved is wrong, and then will be an exterior angle to the triangle, the other interior and shows that this assumption leads to an absurdity, i.e., to & con. opposite to it. clusion which is in contradiction to a proposition proved before- From this follows at once the theorem contained in Prop. 27. that therefore the assumption made cannot be true, and hence that If two straight lines which are cut by a transversal maks allenate the theorem is true. It is often stated that Euclid invented this angles cqual, the lines cannot meet, however far they be produced, kind of proof, but the method is most likely much older. honce they are parallel. This proves the existence of parallel lines. 8 8. It is next proved that two triangles which have the three sides Prop. 28 states the same fact in different forms. If a straight of the one equal respectively to those of the other are identically equal, line, falling on two other straight lines, make the exterior angle squa! hence that the angles of the one are oqual respectively to those of the to the interior and opposite angle on the same side of te lire, or other, those being equal which are opposite equal sidcs. This is make the interior angles on the same side together equal to two right Prop. 8, Prop. 7 containing only a first step towards its proof. angles, the two straight linos shall be parallel to one another. These theorems allow now of the solution of a number of pro- Hence we know that, "if two straight lines which are cut by a blems, viz. : transversal meet, their alternate angles are not equal"; and bence To bisect a given angle (Prop. 9). that, “if alternate angles are equal, then the lines are parallel." To bisect a given finite straight line (Prop. 10). The question now arises, Are the propositions converse to these To draw a straight line perpendicularly to a given straight line truo or not? That is to say, “If alternate angles are unequal, do titrough a given point in it (Prop. 11), and also through a given the lines mcet ?" And if the lines are parallel, are alternate point not in it (Prop. 12). angles nocessarily equal ?" The solutions all depend upon properties of isosceles triangles. l'he answer to either of these two questions inplies the answer 8 9. The next three theorems relate to angles only, and might have to the other. But it has been found impossible to prove that the been proved before Prop. 4, or even at the very beginning. The negation or the affirmation of either is true. first (Prop. 18) says, The angles which one straight lins makes with The difficulty which thus arises is overcome by Euclid assuming another straight line on one side of it either are two right angles or that the first qnestion has to be answered in the affirmative. This are together equal to tuo right angles. This theorem would have gives his last axiom (12), which we quote in his own words. been unnecessary if Euclid had admitted the notion of an angle Axiom 12.-If a straight line meet two straight lines, so as to make such that its two lin.its are in the same straight line, and had the two interior angles on the same side of it taken together less thas besides defined the sum of two angles. two right angles, ihest straight lines, being continually produced Its converse (Prop. 14) is of great use, inasmuch as it enables us shall at length moct on that side on which are the angles which art in many cases to prove that two straight lines drawn from the same loss than two right angles. point are one the continuation of the other. So also is The answer to the second of the above questions follows from this, Prop. 15. If two straight lines cut one another, the vertical or and gives the theorem Prop. 29. If a straight line fall on txo opposite angles shall be equal. parallel straight lines, it makes the alternate angles equal to on § 10. Euclid returns now to properties of uriangles. Of great another, and the exterior angle equal to the interior and opposite importance for the next steps (though afterwards superseded by a angle on the samo side, and also the two interior angles on the sme more complete theorem) is sido together equal to two right angles. Prop. 18. If one side of a triangle be proauced, the exterior angle § 14. With this a new part of elementary geometry begins. shall be greater than either of the interior opposite angles. The earlier propositions are independent of this axiom, and would Prop. 17, Any two angles of a triangls are together less than two be true even if a wrong assumption had been made in it right angles, is an immediate consequence of it. By the aid of They all relate to figures in a plane. But a plane is only one these two, the following fundamental properties of triangles are among an infinite number of conceivable surfaces. We may drav easily proved : figures on any one of them and study their properties. We may, Prop. 18. The greater sido oj' every triangle has the greater angle for instance, take a sphere instead of the plane, and obtain opposite to it ; spherical" in the place of “plane" geometry. If on one of Its converse, Prof. 19. The grealer angle of every triangle is these surfaces lines and figures could be drawn, answering to all subtended by the greater side, or has the greuter side opposite to it, the definitions of our plane figures, and if the axioms with the er: Prop. 20. Any two sides of a triangle are together greater than ception of the last all hold, then all propositions up to the 28th will the third side ; be true for these figures. This is the case in spherical geometry if And also Prop. 21. If from the ends of the side of a triangle there we substitute "shortest line "great circle” for “ straight be drawn two straight lines to a point within the triangle, these shall line," "small circle” for “circle," and if, besides, we limit all be less than the other two sides of the triangle, but shall contain a figures to a part of the sphere which is less than a hemisphere, so greater angle. that two points on it cannot be opposite ends of a diameter, and $ 11, Having solved two problems (Props. 22, 23), he returns to two therefore determine always one and only one great circle. triangles which have two sides of the one equal respectively to two For spherical triangles, therefore, all the important propositions sides of the other. It is known (Prop. 4) that if the included | 4, 8, 26 ; 5 and 6 ; and 18, 19, and 20 will hold good. angles are equal then the third sides are equal; and conversely This remark will be sufficient to show the impossibility of proving (Prop. 8), if the third sides are equal, then the angles included by Euclid's last axiom, which would mean proving that this axiom iš the first sides are equal. From this it follows that if the included a consequence of the others, and henco that the theory of parallels angles are not equal, the third sides are not equal, and conversely, would hold on a spherical surface, where the other axioms do hold, that if the third sides are not equal, the included angles are not whilst parallels do not even exist. equal. Euclid now completes this knowledge by proving, that “if It follows that the axiom in question states an inherent difference the included angles are not equal, thon the third side in that triangle between the plane and other surfaces, and that the plane is only is the greater which contains the greater angle;” and conversely, that fully characterized when this axiom is added to the other assum[“if the third sides are unequal, that triangle contains the greater tions. angle which contains the greater side. These are Prop. 24 and 8 15. The introduction of the new axiom and of parallel lines Prop. 25. leads to a new class of propositions. $ 12. The next theorem (Prop. 26) says that if tivo triangles have After proving (Prop. 30) that “two lines which are each parallel one side and two angles of the one equal respectively to one side and to a third are parallel to each other," we obtain the new properties two angles of the other, viz., in both triangles either the angles ad. of triangles contained in Prop. 32. · Of these the second part is the jacent to the equal side, or one angle adjacent and one angle opposite most important, viz., the theorem, The three interior angles of it, then the tvio trianglos are identically equal. every triangle are together equal to troo right anglas. This theorem belongs to a group with Prop. 4 and Prop. 8. Its As easy deductions not given by Euclid but added by Simson first case might have been given immediately after Prop. 4, but the follow the propositions about the angles in polygons, they are second case requires Prop. 16 for its proof. given in English editions as corollaries to Prop. 32. § 13. We come now to the investigation of parallel straight lines, These theorems do not hold for spherical figures. The sum of the i.c., of straight lines which lie in the same plane, and cannot be interior angles of a spherical triangle is always greater than two made to meet however far they be produced either way. The in. right angles, and increases with the ares, or & a $ 16. The theory of parallels as such may be said to be finished | equal to the product of the numbers which measure tie sides, & with Props. 33 and 34, which state properties of the parallelogram, unit square being the square on the unit line. If, however, no i.e., of a quadrilateral formed by two pairs of parallels. They are- such unit.can be found, this process requires that connexion between Prop. 33. The straight lines which join the extremities of two equal lines and numbers which is only established by aid of ratios of lines, and parallel straighi lines towards the same parts are themselves and which is therefore at this stage altogether inadmissible. But equal and parallel ; and · there exists another way of connecting these propositions with Prop. 31. The opposite sides and angles of a parallelogram are algebra, based on modern notions which seem destined greatly to equal to one another, and the diameter (diagonal) bisccts the parallelo- change and to simplify mathematics. We shall introduce here as gram, that is, divides it into two equal parts. much of it as is required for our present purpose. 8 17. The rest of the first book relates to areas of figures. At the beginning of the second book we find a definition accord The theory is made to depend upon the theoreins ing to which " a rectangle is said to be contained by the two sides Prop. 35. Parallelograms on the same base and between the same which contain one of its right angles"; in the text this phraseparallels are equal to one another; and ology is extended by speaking of rectangles contained by any two Prop. 36. Parallelograms on equal bases, and between the same straight lines, meaning the rectangle which has two adjacent sides parallels, are equal to one another. equal to the two straight lines. As each parallelogram is bisected by a diagonal, the last theorems We shall denote a finite straight line by a single small letter, hold also if the word parallelogram be replaced by “triangle," as is a, b, c, . . . . x, and the area of the rectangle contained by two 'done in Props. 37 and 38. lines a and 6 by ab, and this we shall call the product of the two It is to be remarked that Euclid proves these propositions only lines a and b. It will be understood that this definition has nothing in the case when the parallelograms or triangles have their bases in to do with the definition of a product of numbers. the same straight line. We define as follows :The theorems converse to the last form the contents of the next The sum of two straight lines a and b means a straight line o three propositions, viz. : which may be divided in two parts equal respectively to a and b. THEOREM (Props. 40 and 41).—Equal triangles, on the same or on This sum is denoted by a +b. equal bases, in the same straight line, and on the same side of it, are The difference of two lines a and 6 (in symbols, a – 6) means a between the same parallels. line c which when added to b gives a ; that is, That the two cases here stated are given by Euclid in two sepa a-b=c if b+c=a. rate propositions proved separately is characteristic of his method. g 18. To compare areas of other figures, Euclid shows first, in The product of two lines a and 6 (in symbols, ab) means the area Prop. 42, how to draw a parallelogram which is equal in area to a of the rectangle contained by the lines a and 6.' For aa, which given triangle, an has one of its angles equal to a given angle. If means the square on the line a, we write a'. the given angle is right, then the problem is solved to draw a rect- 8 21. The first ten of the fourteen propositions of the second angle" equal in area to a given triangle. book may then be written in the form of formulæ as follows: Next this parallelogram transformed into another parallelogram, which has one of its sides equal to a given straight line, whilst Prop. 1. a(6+c+d+ ...)-ab+ac+ad+ ... its angles remain unaltered. This may be done by aid of the 2. ab + ac-aif b+c-a. theorem in 3. a(a+b)-a2 + ab. Prop. 43. The complements of the parallelograms which are about 4. (a+b): - a2 + 2ab +63. the diameter of any parallelogram are equal to one another. 5. (a+b) (a - b) +62 - a?. Thus the problem (Prop. 44) is solved to construct a parallelogram 6. (a+b) (a - b) + b3 -- a?. on a given line, which is equal in area to a given triangle, and which 7. a? +(a - b)2 = 2a(a - b)+69. has onc angle equal to a given angle (generally a right angle). 8. 4(a + b)a + 6*- (2a +b). As every polygon can be divided into a number of triangles, 9. (a + b)2 + (a - b)2 - 2a2 + 20%. we can now construct a parallelogram having a given angle, say a 10. (a + b)2 + (a - b)- 2a + 25%. right angle, and being equal in area to a given polygon. For each It will be seen that 5 and 6, and also 9 and 10, are identical.' of the triangles into which the polygon has been divided, a parallelo. In Euclid's statement they do not look the same, the figures being gram may be constructed, having one side equal to a given straight arranged differently. line, and one angle equal to a given angle. If these parallelograms If the letters a, b, c, . . . denoted numbers, it follows from be placed side by side, they may be added together to form a single algebra that each of these formulæ is true. But this does not parallelogram, having still one side of the given length. This is prove them in our case, where the letters denote lines, and their done in Prop. 45. products areas without any reference to numbers. To prove them Herewith a means is found to compare areas of different polygons. we have to discover the laws which rule the operations introduced, We need only construct two rectangles equal in area to the given viz., addition and multiplication of segments. This we shall do polygons, and having each one side of given length. By comparing now; and we shall find that these laws are the same with those the unequal sides we are enabled to judge whether the areas are which hold in algebraical addition and multiplication. equal, or which is the greater. Euclid does not state this con $ 22. In a sum of numbers we may change the order in which sequence, but the problem is taken up again at tho end of the the numbers are added, and we may also add the numbers together second book, where it is shown how to construct a square equal in groups, and then add these groups. But this also holds for the in area to a given polygon. 2 19. The first book concludes with one of the most important tion shows. That the sum of rectangles has always a meaning sum of segments and for the sum of rectangles, as a little consideratheorems in the whole of geometry, and one which has been cele follows from the Props. 43–45 in the first book. These laws about brated since the earliest times. It is stated, but on doubtful addition are reducible to the twoauthority, that Pythagoras discovered it, and it has been called by his name. If we call that side in a right-angled triangle which is a+b=b+ (1), opposite the right angle the hypotenuse, we may state it as a +(b+c) - a to : : (2); follows: or, when expressed for rectangles, THEOREM OF PYTHAGORAS (Prop. 47).— In every right-angled tri ab + ed-ed + ab angle the square on the hypotenuse is equal to the sum of the squares of (3), the other sides. ab +(cd + ef )-ab+o+ef (4). And conversely The brackets mean that the terms in the bracket have been added Prop. 48. If the square described on one of the sides of a triangle be together before they are added to another term. The more general equal to the squares described on the other sides, then the angle contained cases for more terms may be deduced from the above. by these two sides is a right angle. For the product of two numbers we have the law that it remains On this theorem (Prop. 47) almost all geometrical measurement unaltered if the factors be interchanged. This also holds for our depends, which cannot be directly obtained. geometrical product. For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area Book II. of the rectangle which has 6 as base and a as altitude. But in a rectangle we may take either of the two lines which contain it as $ 20. The propositions in the second book are yery different in base, and then the other will be the altitude. This gives character from those in the first; they all relate to areas of rectangles and squares. Their true significance is best seen by ab-ba. (5). stating them in an algebraic form. This is often done by expressing In order further to multiply a sum by a number, we have in algebra the lengths of lines by aid of numbers, which tell how many times the rule:-Multiply each term of the sum, and add the products a chosen unit is contained in the lines. If there is a unit to be thus obtained. That this holds for our geometrical products is found which is contained an exact number of times in each side of a shown by Euclid in his first proposition of the second book, where rectangle, it is easily seen, and generally shown in the teaching of he proves that the area of a rectangle whose base is the sum of a arithmetic, that the rectangle contains a number of unit squares I number of segments is equal to the sum of rectangles which have a . } . + these segments separately as bases. In symbols this gives, in the defined in Book I., Def. 15. We restate it here in slightly diffesiraplest case, rent words :a(b+c)- ab + ac) Definition. The circunference of a circle is a plane curre such (6). and (b + c)a=ba + ca that all points in it have the same distance from a fixed point in To these laws which have been investigated by Sir William the plane. This point is called the "centre" of the circle. Hamilton and by Hermann Grassmann, the former has given of the third book, a few only require special mention. The first, of the new definitions, of which eleven are given at the beginning special names. He calls the laws expressed in which says that circles with equal radii are equal, is in rart a (1) and (3) the commutative law for addition; theorem, but easily proved by applying the one circle to the other. (5) multiplication ; Or it may be considered proved by aid of Prop. 24, equal circles not (2) and (4) the associative laws for addition; being used till after this theorem. (6) the distributive law. In the second definition is explained what is meant by a line § 23. Having proved that these six laws hold, we can at once which “ touches" a circle. Such a line is now generally called a prove every one of the above propositions in their algebraical tangent to the circle. The introduction of this name allows us to form. state many of Euclid's propositions in a much shorter form. The first is proved geometrically, it being one of the funda- For the sume reason we shall call a straight line joining two mental laws. The next two propositions are only special cases of points on the circumference of a circle a “chord.” the first of the others we shall prove one, viz., the fourth : Definitions 4 and 5 may be replaced with a slight generalization by the following:(a+b)-(a+b) (a + b) = (a + b)a + (a + b)6 by (6) Definition.-By the distance of a point from a line is meant the But (a +.b)a=aa + ba by (6), length of the perpendicular drawn from the point to the line. - aa + ab by (5); $ 27. From the definition of a circle it follows that every circle and (a + b)b - ab + bb by (6). has a centre. Prop. 1 requires to find it when the circle is given, Therefore (a+b)*=aa + ab +(ab +66) i.e., when its circumference is drawn. - aa +(ab + ab) + bb by (4). To solve this problem a chord is drawn (that is, any two points in - aa + 2ab + bb the circunference are joined), and through the point where this is This gives the theorem in question. bisected a perpendicular to it is erected. Euclii then proves, first, In the same manner every one of the first ten propositions is that no point of this perpendicular can be the centre, hence that the proved. centre must lie in this line ; and, secondly, that of the points on the It will be seen that the operations performed are exactly the perpendicular one only can be the centre, viz., the one which game as if the letters denoted numbers. bisects the part of the perpendicular bounded by the circle. In Props. 5 and 6 may also be written thus the second part Euclid silently assumes that the perpendicular there used does cut the circunference in two, and only in two points (a+b)(a - b)-a The proof therefore is incomplete. The proof of the first part, Prop. 7, which is an easy consequence of Prop. 4, may be trans- however, is exact. By drawing two non-parallel chords, and the formed. If we denote by c the line a +b, so that perpendiculars which bisect them, tho centre will be found as the C=a+b,-c-6, point where these perpendiculars intersect. we get § 28. In Prop. 2 it is proved that a chord of a circle lies altogether G* +(0-6)* - 2cc-6) +88 within the circle. - 20 - 26c+. What we have called the first part of Euclid's solution of Prop. 1 Subtracting c from both sides, and writing a forc, we get may be stated as a theorem : THEOREM.— Every straight line which bisccis a chord, and is at (a-6)- a2 - 2ab +6%. right angles to it, passes through the centre of the circle. In Euclid's Elements this form of the theorem does not appear, The converse to this gives Prop. 3, which may be stated thus:all propositions being so stated that the notion of subtraction does If a straight line through the centre of a circle biscct a chord, then not enter into them. it is perpendicular to the chord, and if it be perpendicular to the 3 24. The remaining two theorems (Props. 12 and 13) connect chord it bisects it. the square on one side of a triangle with the sum of the squares on An easy consequence of this is the following theorem, which is the other sides, in case that the angle between the latter is acute or essentially the same as Prop. 4 :obtuse. They are important theorems in trigonometry, where it is THEOREM (Prop. 4). — Two chords of a circle, of which neither possible to include them in a single theorem. passes through the centre, cannot biscct each other. 8 25. There are in the second book two problems, Props. 11 These last three theorems are fundamental for the theory of the and 14. circle. It is to be remarked that Euclid never proves that a If written in the above symbolic language, the former requires to straight line cannot have more than two points in common with a find a line z such that ala- x)=x". Prop. 11 contains, therefore, circumference. the solution of a quadratic equation, which we may write z* + at - a. $ 29. The next two propositions (5 and 6) might be replaced by The solution is required later on in the construction of a regular a single and a simpler theorem, viz. :decagon. THEOREM.-Two circles which have a common centre, and whose More important is the problem in the last proposition (Prop. 14). circumferences have one point in common, coincide. It requires the construction of a square equal in area to a given Or, more in agreement with Euclid's form: rectangle, hence a solution of the equation Theorem. — I'wo different circles, whose circumferences have a 2 - ab. point in common, cannot have the same centre. That Euclid treats of two cases is characteristic of Greek mathe. In Book I., 42-45, it has been shown how a rectangle may be matics. constructed equal in area to a given figure bounded by straight lines. By aid of the new proposition we may therefore now deter. They may be combined thus : The next two propositions (7 and 8) again belong together. mine a line such that the square on that line is equal in area to any THEOREM. - If from a point in a plane of a circle, which is not given rectilinear figure, or we can square any such figure. the contre, straight lines be drawn to the different points of the cir. As of two squares that is the greater which has the greater side, cumference, then of all these lines one is the shortest, and one the it follows that now the comparison of two areas has been reduced longest, and these lie both in that straight line which joins the given to the comparison of two lines. point to the centre. Of all the remaining lines cach is equal to one The problem of reducing other areas to squares is frequently met and only one other, and these cqual lines lie on opposite sides of the with among Greek mathematicians. We need only mention tho shortest or longest, and make equal angles with them. problem of squaring the circle. Euclid distinguishes the two cases where the given point lies In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common within or without the circle, omitting the case where it lies in the circumference. base. Their altitudes give then a measure of their areas. The construction of a rectangle having the base u, and being than two equal straight lines can be drawn to the circumference, From the last proposition it follows that if from a point more equal in area to a given rectangle, depends upon Prop. 43, I. This therefore gives a solution of the equation this point must be the centre. This is Prop. 9. As a consequence of this we get THEOREM. - If the circumferences of the two circles have three where z denotes the unkuown altitude. points in common they coincide. For in this case the two circles have a common centre, because Book III. from the centre of the one three equal lines can be drawn to points on the circumference of the other. But two circles which have a 8 26. The third book of the Elements relates exclusively to pro- common centre, and whose circunferences have a point in common, perties of the circle. A circle and its circumference have been l coincido, (Compare above statement of Props. 6 and 6.) ab=ux, |