The Elements of Euclid for the Use of Schools and Colleges/Notes

by Isaac Todhunter

NOTES ON EUCLID'S ELEMENTS.

 

The article Eucleides in Dr Smith's Dictionary of Greek and Roman Biography was written by Professor De Morgan; it contains an account of the works of Euclid, and of the various editions of them which have been published. To that article we refer the student who desires full information on these subjects. Perhaps the only work of importance relating to Euclid which has been published since the date of that article is a work on the Porisms of Euclid by Chasles; Paris, 1860.

Euclid appears to have lived in the time of the first Ptolemy B.C. 323—283, and to have been the founder of the Alexandrian mathematical school. The work on Geometry known as The Elements of Euclid consists of thirteen books; two other books have sometimes been added, of which it is supposed that Hypsicles was the author. Besides the Elements, Euclid was the author of other works, some of which have been preserved and some lost.

We will now mention the three editions which are the most valuable for those who wish to read the Elements of Euclid in the original Greek.

(1) The Oxford edition in folio, published in 1703 by David Gregory, under the title Εύκλείδον τά σωόμενα. "As an edition of the whole of Euclid's works, this stands alone, there being no other in Greek." De Morgan.

(2) Euclidis Elementorum Libri sex priores... edidit Joannes Gulielrmus Camerer. This edition was published at Berlin in two volumes octavo, the first volume in 1824 and the second in 1825. It contains the first six books of the Elements in Greek with a Latin Translation, and very good notes which form a mathematical commentary on the subject.

(3) Euclidis Elementa ex optimis libris in usum tironum Greece edita ab Ernesto Ferdinando August. This edition was published at Berlin in two volumes octavo, the first volume in 1826 and the second in 1829. It contains the thirteen books of the Elements in Greek, with a collection of various readings. A third volume, which was to have contained the remaining works of Euclid, never appeared. "To the scholar who wants one edition of the Elements we should decidedly recommend this, as bringing together all that has been done for the text of Euclid's greatest work." De Morgan.

An edition of the whole of Euclid's works in the original has long been promised by Teubner the well-known German publisher, as one of his series of compact editions of Greek and Latin authors; but we believe there is no hope of its early appearance.

Robert Simson's edition of the Elements of Euclid, which we have in substance adopted in the present work, differs considerably from the original. The English reader may ascertain the contents of the original by consulting the work entitled The Elements of Euclid with dissertations... by James Williamson. This work consists of two volumes quarto; the first volume was published at Oxford in 1781, and the second at London in 1788. Williamson gives a close translation of the thirteen books of the Elements into English, and he indicates by the use of Italics the words which are not in the original but which are required by our language.

Among the numerous works which contain notes on the Elements of Euclid we will mention four by which we have been aided in drawing up the selection given in this volume.

An Examination of the first six Books of Euclid's Elements by Willam Austin... Oxford, 1781.

Euclid's Elements of Plane Geometry with copious notes... by John Walker. London, 1827.

The first six books of the Elements of Euclid with a Commentary by Dionysius Lardner, fourth edition. London, 1834.

Short supplementary remarks on the first six Books of Euclid's Elements, by Professor De Morgan, in the Companion to the Almanac for 1849.

We may also notice the following works:

Geometry, Plane, Solid, and Spherical,... London 1830; this forms part of the Library of Useful Knowledge.

Théorèmes et Problèmes de Géométrié Eleméntaire par Eugène Catalan... Troisieme edition. Paris, 1858.

For, the History of Geometry the student is referred to Montucla's Histoire des Mathematiques, and to Chasles's Aperçu historique sur l'origine et le devèloppement des Méthodes en Gèométrie..

THE FIRST BOOK.

Definitions. The first seven definitions have given rise to considerable discussion, on which however we do not propose to enter. Such a discussion would consist mainly of two subjects, both of which are unsuitable to an elementary work, namely, an examination of the origin and nature of some of our elementary ideas, and a comparison of the original text of Euclid with the substitutions for it proposed by Simson and other editors. For the former subject the student may hereafter consult Whewell's History of Scientific Ideas and Mill's Logic, and for the latter the notes in Camerer's edition of the Elements of Euclid.

We will only observe that the ideas which correspond to the words point, line, and surface, do not admit of such definitions as will really supply the ideas to a person who is destitute of them. The so-called definitions may be regarded as cautions or restrictions. Thus a point is not to be supposed to have any size, but only position; a line is not to be supposed to have any breadth or thickness, but only length; a surface is not to be supposed to have any thickness, but only length and breadth.

The eighth definition seems intended to include the cases in which an angle is formed by the meeting of two curved lines, or of a straight line and a curved line; this definition however is of no importance, as the only angles ever considered are such as are formed by straight lines. The definition of a plane rectilineal angle is important; the beginner must carefully observe that no change is made in an angle by prolonging the lines which form it, away from the angular point.

Some writers object to such definitions as those of an equilateral triangle, or of a square, in which the existence of the object defined is assumed when it ought to be demonstrated. They would present them in such a form as the following: if there be a triangle having three equal sides, let it be called an equilateral triangle.

Moreover, some of the definitions are introduced prematurely. Thus, for example, take the definitions of a right-angled triangle and an obtuse-angled triangle; it is not shewn until I. 1 7, that a triangle cannot have both a right angle and an obtuse angle, and so cannot be at the same time right-angled and obtuse-angled. And before Axiom 11 has been given, it is conceivable that the same angle may be greater than one right angle, and less than another right angle, that is, obtuse and acute at the same time.

The definition of a square assumes more than is necessary. For if a four-sided figure have all its sides equal and one angle a right angle, it may be shewn that all its angles are right angles; or if a four-sided figure have all its angles equal, it may be shewn that they are all right angles.

Postulates. The postulates state what processes we assume that we can effect, namely, that we can draw a straight line between two given points, that we can produce a straight line to any length, and that we can describe a circle from a given centre with a given distance as radius. It is sometimes stated that the postulates amount to requiring the use of a ruler and compasses. It must however be observed that the ruler is not supposed to be a graduated ruler, so that we cannot use it to measure off assigned lengths. And we do not require the compasses for any other process than describing a, circle from a given point with a given distance as radius; in other words, the compasses may be supposed to close of themselves, as soon as one of their points is removed from the paper.

Axioms. The axioms are called in the original Common Notions. It is supposed by some writers that Euclid intended his postulates to include all demands which are peculiarly geometrical, and his common notions to include only such notions as are applicable to all kinds of magnitude as well as to space magnitudes. Accordingly, these writers remove the last three axioms from their place and put them among the postulates; and this transposition is supported by some manuscripts and some versions of the Elements.

The fourth axiom is sometimes referred to in editions of Euclid when in reality more is required than this axiom expresses. Euclid says, that if A and B be unequal, and C and D equal, the sum of A and C is unequal to the sum of B and D. What Euclid often requires is something more, namely, that if A be greater than B, and C and D be equal, the sum of A and C is greater than the sum of B and D. Such an axiom as this is required, for example, in I. 17. A similar remark applies to the fifth axiom.

In the eighth axiom the words "that is, which exactly fill the same space," have been introduced without the authority of the original Greek. They are objectionable, because lines and angles are magnitudes to which the axiom may be applied, but they cannot be said to fill space.

On the method of superposition we may refer to papers by Professor Kelland in the Transactions of the Royal Society of Edinburgh, Vols. xxi, and xxiii.

The eleventh axiom is not required before I. 14, and the twelfth axiom is not required before I. 29; we shall not consider these axioms until we arrive at the propositions in which they are respectively required for the first time.

The first book is chiefly devoted to the properties of triangles and parallelograms.

We may observe that Euclid himself does not distinguish between problems and theorems except by using at the end of the investigation phrases which correspond to q.e.f. and q.e.d respectively.

I. 2. This problem admits of eight cases in its figure. For it will be found that the given point may be joined with either end of the given straight line, then the equilateral triangle may be described on either side of the straight line which is drawn, and the sides of the equilateral triangle which are produced may be produced through either extremity. These various cases may be left for the exercise of the student, as they present no difficulty.

There will not however always be eight different straight fines obtained which solve the problem. For example, if the point A falls on BC produced, some of the solutions obtained coincide; this depends on the fact which follows from I. 32, that the angles of all equilateral triangles are equal.

I. 5. "Join FC" Custom seems to allow this singular expression as an abbreviation for "draw the straight line FC," or for "Join F to C by the straight line FC"

In comparing the triangles BFC, CGB, the words " and the base BC is common to the two triangles BFC, CGB" are usually inserted, with the authority of the original. As however these words are of no use, and tend to perplex a beginner, we have followed the example of some editors and omitted them.

A corollary to a proposition is an inference which may be deduced immediately from that proposition. Many of the corollaries in the Elements are not in the original text, but introduced by the editors. It has been suggested to demonstrate I. 5 by superposition. Conceive the isosceles triangle ABC to be taken up, and then replaced so that AB falls on the old position of AC, and AC falls on the old position of AB. Thus, in the manner of I. 4, we can shew that the angle ABC is equal to the angle ACB.

I. 6 is the converse of part of I. 5. One proposition is said to be the converse of another when the conclusion of each is the hypothesis of the other. Thus in I. 5 the hypothesis is the equality of the sides, and one conclusion is the equality of the angles; in I. 6 the hypothesis is the equality of the angles and the conclusion is the equality of the sides. When there is more than one hypothesis or more than one conclusion to a proposition, we can form more than one converse proposition. For example, as another converse of I. 5 we have the following: if the angles formed by the base of a triangle and the sides produced be equal, the sides of the triangle are equal; this proposition is true and will serve as an exercise for the student.

The converse of a true proposition is not necessarily true; the student however will see, as he proceeds, that Euclid shews that the converses of many geometrical propositions are true.

I. 6 is an example of the indirect mode of demonstration, in which a result is established by shewing that some absurdity follows from supposing the required result to be untrue. Hence this mode of demonstration is called the reductio ad absurdum. Indirect demonstrations are often less esteemed than direct demonstrations; they are said to shew that a theorem is true rather than to shew why it is true. Euclid uses the reductio ad absurdum chiefly when he is demonstrating the converse of some former theorem; see I. 14, 19, 25, 40.

Some remarks on indirect demonstration by Professor Sylvester, Professor De Morgan, and Dr Adamson will be found in the volumes of the Philosophical Magazine for 1852 and 1853.

I. 6 is not required by Euclid before he reaches II. 4; so that I. 6 might be removed from its present place and demonstrated hereafter in other ways if we please. For example, I. 6 might be placed after I. 18 and demonstrated thus. Let the angle ABC be equal to the angle ACB: then the side AB shall be equal to the side AC. For if not, one of them must be greater than the other; suppose AB greater than AC. Then the angle ACB is greater than the angle ABC, by I. 18. But this is impossible, because the angle ACB is equal to the angle ABC, by hypothesis. Or I. 6 might be placed after I. 16 and demonstrated thus. Bisect the angle BAC by a straight line meeting the base at D. Then the triangles ABD and ACD are equal in all respects, by I. 16.

I. 7 is only required in order to lead to I. 8. The two might be superseded by another demonstration of I. 8, which has been recommended by many writers.

Let ABC, DEF be two triangles, having the sides AB, AC equal to the sides DB, DF, each to each, and the base BC equal to the base EF: the angle BAC shall be equal to the angle EDF.

For, let the triangle DEF be applied to the triangle ABC, so that the bases may coincide, the equal sides be conterminous, and the vertices fall on opposite sides of the base. Let GBC represent the triangle DEF thus applied, so that G corresponds to D. Join AG. Since, by hypothesis, BA is equal to BG, the angle BAG is equal to the angle BGA, hy I. 5. In the same manner the angle CAG is equal to the angle CGA. Therefore the whole angle BAC is equal to the whole angle BGC, that is, to the angle EDF.

There are two other cases; for the straight line AG may pass through B or C, or it may fall outside BC: "these cases may be treated in the same manner as that which we have considered.

I. 8. It may be observed that the two triangles in I. 8 are equal in all respects; Euclid however does not assert more than the equality of the angles opposite to the bases, and when he requires more than this result he obtains it by using I. 4.

I. 9. Here the equilateral triangle DEF is to be described on the side remote from A, because if it were described on the same, side, its vertex, F, might coincide with A, and then the construction would fail. I. II. The corollary was added by Simson, It is liable to serious objection. For we do not know how the perpendicular BE is to be drawn. If we are to use I. 11 we must produce AB, and then we must assume that there is only one way of producing AB for otherwise we shall not know that there is only one perpendicular; and thus we assume what we have to demonstrate.

Simson's corollary might come after I. 13 and be demonstrated thus. If possible let the two straight lines ABC, ABD have the segment AB common to both. From the point B draw any straight line BE. Then the angles ABE and EBC are equal to two right angles, by I. 13, and the angles ABE and EBD are also equal to two right angles, by I. 13. Therefore the angles ABE and EBC are equal to the angles ABE and EBD. Therefore the angle EBC is equal to the angle EBD; which is absurd.

But if the question whether two straight lines can. have a common segment is to be considered at all in the Elements, it might occur at an earlier place than Simson has assigned to it. For example, in the figure to I. 5, if two straight lines could have a common segment AB, and then separate at B, we should obtain . two different angles formed on the other side of BC by these produced parts, and each of them would be equal to the angle BCG. The opinion has been maintained that even in I. 1, it is tacitly assumed that the straight lines AC and BC cannot have a common segment at C where they meet; see Camerer's Euclid, pages 30 and 36.

Simson never formally refers to his corollary until XI. I. The corollary should be omitted, and the tenth axiom should be extended so as to amount to the following; if two straight lines coincide in two points they must coincide both beyond and between those points.

I. 12. Here the straight line is said to be of unlimited length, in order that we may ensure that it shall meet the circle.

Euclid distinguishes between the terms at right angles and perpendicular. He uses the term at right angles when the straight line is drawn from a point in another, as in I. 11; and he uses the term perpendicular when the straight line is drawn from a point without another, as in I. 12. This distinction however is often disregarded by modern writers.

I. 14. Here Euclid first requires his eleventh axiom. For in the demonstration we have the angles ABC and ABE equal to two right angles, and also the angles ABC and ABD equal to two right angles; and then the former two right angles are equal to the latter two right angles by the aid of the eleventh axiom. Many modern editions of Euclid however refer only to the first axiom, as if that alone were sufficient; a similar remark applies to the demonstrations of I. 15, and I. 24. In these cases we have omitted the reference purposely, in order to avoid perplexing a beginner; but when his attention is thus drawn to the circumstance he will see that the first and eleventh axioms are both used.

We may observe that errors, in the references with respect to the eleventh axiom, occur in other places in many modern editions of Euclid, Thus for example in III. 1, at the step "therefore the angle FDB is equal to the angle GDBB," a reference is given to the first axiom instead of to the eleventh.

There seems no objection on Euclid's principles to the following demonstration of his eleventh axiom.

Let AB be at right angles to DAC at the point A, and EF at right angles to HEG at the point E: then shall the angles BAC and FEG be equal.

Take any length AC, and make AD, EH, HG all equal to AC. Now apply HEG to DAC, so that H may be on D, and HG on DC, and B and F on the same side of DC; then G will coincide with C, and E with A. Also EF shall coincide with AB; for if not, suppose, if possible, that it takes a different position as AK. Then the angle DAK is equal to the angle HEF, and the angle CAK to the angle GEF; but the angles HEF and GEF are equal, by hypothesis; therefore the angles DAK and CAK are equaL But the angles DAB and CAB are also equal, by hypothesis; and the angle CAB is greater than the angle CAK; therefore the angle DAB is greater than the angle CAK. Much more then is the angle DAK greater than the angle CAK. But the angle DAK was shewn to be equal to the angle CAK; which is absurd. Therefore EF must coincide with AB; and therefore the angle FEG coincides with the angle BAC, and is equal to it.

I. 18, I. 19. In order to assist the student in remembering which of these two propositions is demonstrated directly and which indirectly, it may be observed that the order is similar to that in I. 5 and I. 6.

I. 20. "Proclus, in his commentary, relates, that the Epicureans derided Prop, ao, as being manifest even to asses, and needing no demonstration; and his answer is, that though the truth of it be manifest to our senses, yet it is science which must give the reason why two sides of a triangle are greater than the third: but the right answer to this objection against this and the 21st, and some other plain propositions, is, that the number of axioms ought not to be increased without necessity, as it must be if these propositions be not demonstrated." Simson.

I. 21. Here it must be carefully observed that the two straight lines are to be drawn from the ends of the side of the triangle. If this condition be omitted the two straight lines will not necessarily be less than two sides of the triangle.

I. '22. " Some authors blame Euclid because he does not demonstrate that the two circles made use of in the construction of this problem must cut one another: but this is very plain from the determination he has given, namely, that any two of the straight lines DF, FG, GH, must be greater than the third. For who is so dull, though only beginning to learn the Elements, as not to perceive that the circle described from the centre F, at the distance FD, must meet FH betwixt F and H, because FD is less than FH; and that for the like reason, the circle described from the centre G, at the distance GH...must meet DG betwixt D and G; and that these circles must meet one another, because FD and GH are together greater than FG?" Simson.

The condition that B and C are greater than A, ensures that the circle described from the centre G shall not fall entirely within the circle described from the centre F; the condition that A and B are greater than C, ensures that the circle described from the centre F shall not fall entirely within the circle described from the centre; the condition that A and C are greater than B, ensures that one of these circles shall not fall entirely without the other. Hence the circles must meet. It is easy to see this as Simson says, but there is something arbitrary in Euclid's selection of what is to be demonstrated and what is to be seen, and Simson's language suggests that he was really conscious of this.

I. 24. In the construction, the condition that DE is to be the side which is not greater than the other, was added by Simson; unless this condition be added there will be three cases to consider, for F may fall on EG, or ahove EG, or below EG. It may be objected that even if Simson's condition be added, it ought to be shewn that F will fall below EG. Simson accordingly says "...it is very easy to perceive, that DG being equal to DF, the point G is in the circumference of a circle described from the centre D at the distance DF, and must be in that part of it which is above the straight line EF, because DG falls above DF, the angle EDG being greater than the angle EDF." Or we may shew it in the following manner. Let H denote the point of intersection of DF and EG. Then, the angle DHG is greater than the angle DEG, by I. 16; the angle DEG is not less than the angle DGE, by I. 19; therefore the angle DHG is greater than the angle DGH. Therefore DH is less than DG, by I. 20. Therefore DH is less than DF.

If Simson's condition be omitted, we shall have two other cases to consider besides that in Euclid. If F falls on EG, it is obvious that EF is less than EG. If F falls above EG, the sum of DF and EF is less than the sum of DG and EG, by I. 21; and therefore EF is less than EG.

I. 26. It will appear after I. 32 that two triangles which have two angles of the one equal to two angles of the other, each to each, have also their third angles equal. Hence we are able to include the two cases of I. 26 in one enunciation thus; if two triangles have all the angles of the one respectively equal to all the angles of the other, each to each, and have also a side of the one, opposite to any angle, equal to the side opposite to the equal angle in the other, the triangles shall he equal in all respects.

The first twenty-six propositions constitute a distinct section of the first Book of the Elements. The principal results are those contained in Propositions 4, 8, and 26; in each of these Propositions it is shewn that two triangles which agree in three respects agree entirely. There are two other cases which wdll naturally occur to a student to consider besides those in Euclid; namely, (1) when two triangles have the three angles of the one respectively equal to the three angles of the other, (2) when two triangles have two sides of the one equal to two sides of the other, each to each, and an angle opposite to one side of one triangle equal to the angle opposite to the equal side of the other triangle. In the first of these two cases the student will easily see, after reading I. 29, that the two triangles are not necessarily equal. In the second case also the triangles are not necessarily equal, as may be shewn by an example; in the figure of I. 11, suppose the straight line FB drawn; then in the two triangles FBE, FBD, the side FB and the angle FBC are common, and the side FE is equal to the side FD, but the triangles are not equal in all respects. In certain cases, however, the triangles will be equal in all respects, as will be seen from a proposition which we shall now demonstrate.

If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles opposite to a pair of equal sides equal; then if the angles opposite to the other pair of equal sides be both acute, or both obtuse, or if one of them be a right angle, the two triangles are equal in all respects.

Let ABC and DEF be two triangles; let AB be equal to DE, and BC equal to EF, and the angle A equal to the angle D.

First, suppose the angles C and F acute angles.

If the angle B be equal to the angle E, the triangles ABC, DEF are equal in all respects, by I. 4. If the angle B be not equal to the angle E, one of them must be greater than the other; suppose the angle B greater than the angle E, and make the angle ABG equal to the angle E. Then the triangles ABG, DEF are equal in all respects, by I. 26; therefore BG is equal to EF, and the angle BGA is equal to the angle EFD. But the angle EFD is acute, by hypothesis; therefore the angle BGA is acute. Therefore the angle BGC is obtuse, by I. 13. But it has been shewn that BG is equal to EF; and EF is equal to BC, by hypothesis; therefore BG is equal to BC. Therefore the angle BGC is equal to the angle BCG, by I. 5; and the angle BCG is acute, by hypothesis; therefore the angle BGC is acute. But BGC was shewn to be obtuse which is absurd. Therefore the angles ABC, DEF are not unequal; that is, they are equal. Therefore the triangles ABC, DEF are equal in all respects, by I. 4.

Next, suppose the angles at C and F obtuse angles. The demonstration is similar to the above.

Lastly, suppose one of the angles a right angle, namely, the angle C, If the angle B be not equal to the angle E, make the angle ABG equal to the angle E. Then it may be shewn, as before, that BG is equal to BC, and therefore the angle BGC is equal to the angle BCG, that is, equal to a right angle. Therefore two angles of the triangle BGC are equal to two right angles; which is impossible, by I. 17. Therefore the angles ABC and DEF are not unequal; that is, they are equal. Therefore the triangles ABC, DEF are equal in all respects, by I. 4.

If the angles A and D are both right angles, or both obtuse, the angles C and F must be both acute, by I. 17. If AB is less than BC, and DE less than EF, the angles at C and F must be both acute, by I. 18 and I. 17,

The propositions from I. 27 to I. 34 inclusive may be said to constitute the second section of the first Book of the Elements. They relate to the theory of parallel straight lines. In I. 29 Euclid uses for the first time his twelfth axiom. The theory of parallel straight lines has always been considered the great difficulty of elementary geometry, and many attempts have been made to overcome this difficulty in a better way than Euclid has done. We shall not give an account of these attempts. The student who wishes to examine them may consult Camerer's Euclid, Gergonne's Annales de Mathématiques, Volumes sx and xvi, the work by Colonel Perronet Thompson entitled Geometry without Axioms, the article Parallels in the English Cyclopædia, a memoir by Professor Baden Powell in the second volume of the Memoirs of the Ashmolean Society, an article by M. Bouniakofsky in the Bulletin de l' Académie Impériale, Volume v, St Petersbourg, 1863, articles in the volumes of the Philosophical Magazine for 1856 and 1857, and a dissertation entitled Sur un point de Vhisioire de la Geometrie chez les Grecs par A. J. H. Vincent. Paris, 1857.

Speaking generally it may be said that the methods which differ substantially from Euclid's involve, in the first place an axiom as difficult as his, and then an intricate series of propositions; while in Euclid's method after the axiom is once admitted the remaining process is simple and clear.

One modification of Euclid's axiom has been proposed, which appears to diminish the difficulty of the subject. This consists in assuming instead of Euclid's axiom the following; two intersecting straight lines cannot he both parallel to a third straight line.

The propositions in the Elements are then demonstrated as in Euclid up to I. 28, inclusive. Then, in I. 29, we proceed with Euclid up to the words, "therefore the angles BGH, GHD are less than two right angles." We then infer that BGH and GHD must meet: because if a straight line be drawn through G so as to make the interior angles together equal to two right angles this straight line will be parallel to CB, by I. 28; and, by our axiom, there cannot be two parallels to CB, both passing through G.

This form of making the necessary assumption has been recommended by various eminent mathematicians, among whom may be mentioned Playfair and De Morgan. By postponing the consideration of the axiom until it is wanted, that is, until after I. 28, and then presenting it in the form here given, the theory of parallel straight lines appears to be treated in the easiest manner that has hitherto been proposed.

I. 30. Here we may in the same way shew that if AB and EF are each of them parallel to CD, they are parallel to each other. It has been said that the case considered in the text is so obvious as to need no demonstration; for if AB and CD can never meet EF which lies between them, they cannot meet one another.

I. 32. The corollaries to I. 32 were added by Simson. In the second corollary it ought to be stated what is meant by an exterior angle of a rectilineal figure. At each angular point let one of the sides meeting at that point be produced; then the exterior angle at that point is the angle contained between this produced part and the side which is not produced. Either of the sides may be produced, for the two angles which can thus be obtained are equal, by I. 15.

The rectilineal figures to which Euclid confines himself are those in which the angles all face inwards; we may here however notice another class of figures. In the accompanying diagram the angle AFC faces, outwards, and it is an angle less than two right angles; this angle however is not one of the interior angles of the figure AEDCF. We may consider the corresponding interior angle to be the excess of four right angles above the angle AFC; such an angle, greater than two right angles, is called a re-entrant angle.

The first of the corollaries to I. 32 is true for a figure which has a re-entrant angle or re-entrant angles; but the second is not.

I. 32. If two triangles have two angles of the one equal to two angles of the other each to each they shall also have their third angles equal. This is a very important result, which is often required in the Elements. The student should notice how and 2 one pair of right angles is equal to any other pair of right angles. Then, by I. 32, the three angles of one triangle aro together equal to the three angles of any other triangle. Then, by Axiom 2, the sum of the two angles of one triangle is equal to the sum of the two equal angles of the other; and then, by Axiom 3, the third angles are equal.

After I. 32 we can draw a straight line at right angles to a given straight line from its extremity, without producing the given straight line.

Let AB be the given straight line. It is required to draw from A a straight line at right angles to AB. On AB describe the equilateral triangle ABC. Produce BC to D, so that CD may be equal to CB. Join AD. Then AD shall be at right angles to AB. For, the angle CAD is equal to the angle CDA, and the angle CAB is equal to the angle CBA, by I. 5. Therefore the angle BAD is equal to the two angles ABD, BDA, by Axiom 2. Therefore the angle BAD is a right angle, by I. 32.

The propositions from I. 35 to I. 48 inclusive may be said to constitute the third section of the first Book of the Elements. They relate to equality of area in figures which are not necessarily identical in form.

I. 35. Here Simson has altered the demonstration given by Euclid, because, as he says, there would be three cases to consider in following Euclid's method. Simson however uses the third Axiom in a peculiar manner, when he first takes a triangle from a trapezium, and then another triangle from the same trapezium, and infers that the remainders are equal. If the demonstration is to be conducted strictly after Euclid's manner, three cases must be made, by dividing the latter part of the demonstration into two. In the left-hand figure we may suppose the point of intersection of BE and DC to be denoted by G. Then, the triangle ABE is equal to the triangle DCF; take away the triangle DGE from each; then the figure ABGD is equal to the figure EGCF; add the triangle GBC to each; then the parallelogram ABGD is equal to the parallelogram EBCF. In the right-hand figure we have the triangle AEB equal to the triangle DEC; add the figure BEDC to each; then the parallelogram ABCD is equal to the parallelogram EBCF.

The equality of the parallelograms in I. 35 is an equality of area, and not an identity of figure. Legendre proposed to use the word equivalent to express the equality of area, and to restrict the word equal to the case in which magnitudes admit of superposition and coincidence. This distinction, however, has not been generally adopted, probably because there are few cases in which any ambiguity can arise; in such cases we may say especially, equal in area, to prevent misconception.

Cresswell, in his Treatise of Geometry, has given a demonstration of I. 35 which shews that the parallelograms may be divided into pairs of pieces admitting of superposition and coincidence; see;also his Preface, page x.

I. 38. An important case of I. 38 is that in which the triangles are on equal bases and have a common vertex.

I. 40. We may demonstrate I. 40 without adopting the indirect method. Join BD, CD. The triangles DEC and DEF are equal, by I. 38; the triangles ABC and DEF are equal, by hypothesis; therefore the triangles DBC and ABC are equal, by the first Axiom. Therefore AD is parallel to BC, by I. 39. Philosophical Magazine, October 1850.

I. 44. In I. 44, Euclid does not shew that AH and FG will meet. "I cannot help being of opinion that the construction would have been more in Euclid's manner if he had made GH equal to BA and then joining HA had proved that HA was parallel to GB by the thirty-third proposition." Williamson.

I. 47. Tradition ascribed the discovery of I. 47 to Pythagoras. Many demonstrations have been given of this celebrated proposition; the following is one of the most interesting.

Let ABCD, AEFG be any two squares, placed so that their bases may join and form one straight line. Take GH and EK each equal to AB, and join HC, CK, KF, FH.

Then it may be shewn that the triangle HBC is equal in all respects to the triangle FEK, and the triangle KDG to the triangle FGH. Therefore the two squares are together equivalent to the figure CKFH. It may then be shewn, with the aid of I. 32, that the figure CKFH is a square. And the side CH is the hypotenuse of a right-angled triangle of which the sides CB, BH are equal to the sides of the two given squares. This demonstration requires no proposition of Euclid after I. 32, and it shews how two given squares may be cut into pieces which will fit together so as to form a third square. Quarterly Journal of Mathematics, Vol. i.

A large number of demonstrations of this proposition are collected in a dissertation by Joh. Jos. Ign. Hoffinann, entitled Der Pythagorische Lehrsatz...Zweyte...Ausgabe. Mainz. 1821.

THE SECOND BOOK.

The second book is devoted to the investigation of relations between the rectangles contained by straight lines divided into segments in various ways.

When a straight line is divided into two parts, each part is called a segment by Euclid. It is found convenient to extend the meaning of the word segment, and to lay down the following definition. When a point is taken in a straight line, or in the straight line produced, the distances of the poin.t from the ends of the straight line are called segments of the straight line. When it is necessary to distinguish them, such segments are called internal or external, according as the point is in the straight line, or in the straight line produced.

The student cannot fail to notice that there is an analogy between the first ten propositions of this book and some elementary facts in Arithmetic and Algebra.

Let ABCD represent a rectangle which is 4 inches long and 3 inches broad. Then, by drawing straight lines parallel to the sides, the figure may be divided into 12 squares, each square being described on a side which represents an inch in length. A square described on a side measuring an inch is called, for shortness, a square inch. Thus if a rectangle is 4 inches long and 3 inches

broad it may be divided into 12 square inches; this is expressed by saying, that its area is equal to 12 square inches, or, more briefly, that it contains 12 square inches. And a similar result is easily seen to hold in all similar cases. Suppose, for example, that a rectangle is 12 feet long and 7 feet broad; then its area is equal to 12 times 7 square feet, that is to 84 square feet; this may be expressed briefly in common language thus; if a rectangle measures 12 feet by 7 it contains 84 square feet. It must be carefully observed that the sides of the rectangle are supposed to be measured by the same unit of length. Thus if a rectangle is a yard in length, and a foot and a half in breadth, we must express each of these dimensions in terms of the same unit; we may say that the rectangle measures 36 inches by 18 inches, and contains 36 times 18 square inches, that is, 648 square inches.

Thus universally, if one side of a rectangle contain a unit of length an exact number of times, and if an adjacent side of the rectangle also contain the same unit of length an exact number of times, the product of these numbers will be the number of square units contained in the rectangle.

Next suppose we have a square, and let its side be 5 inches in length. Then, by our rule, the area of the square is 5 times 5 square inches, that is 25 square inches. Now the number 25 is called in Arithmetic the square of the number 5. And universally, if a straight line contain a unit of length an exact number of times, the area of the square described on the straight line is denoted by the square of the number which denotes the length of the straight line.

Thus we see that there is in general a connexion between the product of two numbers and the rectangle contained by two straight lines, and in particular a connexion between the square of a number and the square on a straight line; and in consequence of this connexion the first ten propositions in Euclid's Second Book correspond to propositions in Arithmetic and Algebra.

The student will perceive that we speak of the square described on a straight line, when we refer to the geometrical figure, and of the square of a number when we refer to Arithmetic. The editors of Euclid generally use the words "square described upon" in 1. 47 and I. 48, and afterwards speak of the square of a straight line. Euclid himself retains throughout the same form of expression, and we have imitated him.

Some editors of Euclid have added Arithmetical or Algebraical demonstrations of the propositions in the second book, founded on the connexion we have explained. We have thought it unnecessary to do this, because the student who is acquainted with the elements of Arithmetic and Algebra will find no difficulty in supplying such demonstrations himself, so far as they are usually given. We say so far as they are usually given, because these demonstrations usually imply that the sides of rectangles can always be expressed exactly in terms of some unit of length; whereas the student will find hereafter that this is not the case, owing to the existence of what are technically called incommensurable magnitudes. We do not enter on this subject, as it would lead us too far from Euclid's Elements of Geometry with which we are here occupied.

The first ten propositions in the second book of Euclid may be arranged and enunciated in various ways; we will briefly indicate this, but we do not consider it of any importance to distract the attention of a beginner with these diversities.

II. 1 and II. 3 are particular cases of II. i.

II. 4 is very important; the following particular case of it should be noticed; the square described on a straight line made up of two equal straight lines is equal to four times the square described on one of the two equal straight lines.

II. 5 and II. 6 may be included in one enunciation thus; the rectangle under the sum and difference of two straight lines is equal to the difference of the squares described on those straight lines; or thus, the rectangle contained by two straight lines together with the square described on half their difference, is equal to the square described on half their sum.

II. 7 may be enunciated thus; the square described on a straight line which is the difference of two other straight lines is less than the sum of the squares described on those straight lines by twice the rectangle contained by those straight lines. Then from this and II. 4, and the second Axiom, we infer that the square described on the sum of two straight Lines, and the square described on their difference, are together double of the sum of the squares described on the straight lines; and this enunciation includes both II. 9 and II.10, so that the demonstrations given of these propositions by Euclid might be superseded.

II. 8 coincides with the second form of enunciation which we have given to II. 5 and II. 6, bearing in mind the particular case of II. 4 which we have noticed.

II. II. When the student is acquainted with the elements of Algebra he should notice that II. 11 gives a geometrical construction for the solution of a particular quadratic equation.

II. 12, II. 13. These are interesting in connexion with I. 47; and, as the student may see hereafter, they are of great importance in Trigonometry; they are however not required in any of the parts of Euclid's Elements which are usually read. The converse of I. 47 is proved in I. 48; and we can easily shew that converses of II. 12 and II. 13 are true.

Take the following, which is the converse of II. 12; if the square described on one side of a triangle be greater than the sum of the squares described on the other two sides, the angle opposite to the first side is obtuse.

For the angle cannot be a right angle, since the square described on the first side would then be equal to the sum of the squares described on the other two sides, by I. 47; and the angle cannot be acute, since the square described on the first side would then be less than the sum of the squares described on the other two sides, by II. 13; therefore the angle must be obtuse.

Similarly we may demonstrate the following, which is the converse of II. 13; if the square described on one side of a triangle be less than the sum of the squares described on the other two sides, the angle opposite to the first side is acute.

II. 13. Euclid enunciates II. 13 thus; in acute-angled triangles, &c.; and he gives only the first case in the demonstration. But, as Simson observes, the proposition holds for any triangle; and accordingly Simson supplies the second and third cases. It has, however, been often noticed that the same demonstration is applicable to the first and second cases; and it would be a great improvement as to brevity and clearness to take these two cases together. Then the whole demonstration will be as follows.

Let ABC be any triangle, and the angle at B one of its acute angles; and, if AC be not perpendicular to BC, let fall on BC, produced if necessary, the perpendicular AD from the opposite angle: the square on AC opposite to the angle B, shall be less than the squares on CB, BA, by twice the rectangle CB, BD.

First, suppose AC not perpendicular to BC.
The squares on CB, BD are equal to twice the rectangle CB, BD, together with the square on CD. [II. 7.
To each of these equals add the square on DA.
Therefore the squares on CB, BD, DA are equal to twice the rectangle CB, BD, together with the squares on CD, DA.
But the square on AB is equal to the squares on BD, DA,
and the square on AC is equal to the squares on CD, DA,
because the angle BDA is a right angle. [I. 47.
Therefore the squares on CB, BA are equal to the square on AC,
together with twice the rectangle CB, BD;
that is, the square on AG alone is less than the squares on CB, BA, by twice the rectangle CB, BD.

Next, suppose AC perpendicular to BC. Then BC is the straight line intercepted between the perpendicular and the acute angle at B.
And the square on AB is equal to the squares on AC, CB. [I. 47.
Therefore the square on AC is less than the squares on AB, BC, by twice the square on BC.

II. 14. This is not required in any of the parts of Euclid's Elements which are usually read; it is included in VI. 22.

 

THE THIRD BOOK.

The third book of the Elements is devoted to properties of circles.

Different opinions have been held as to what is, or should be, included in the third definition of the third book. One opinion is that the definition only means that the circles do not cut in the neighbourhood of the point of contact, and that it must be shewn that they do not cut elsewhere. Another opinion is that the definition means that the circles do not cut at all; and this seems the correct opinion. The definition may therefore be presented more distinctly thus. Two circles are said to touch internally when their circumferences have one or more common points, and when every point in one circle is within the other circle, except the common point or points. Two circles are said to touch externally when their circumferences have one or more common points, and when every point in each circle is without the other circle, except the common point or points. It is then shewn in the third Book that the circumferences of two circles "which touch can have only one common point.

A straight line which touches 'a circle is often called a tangent to the circle, or briefly, a tangent.

It is very convenient to have a word to denote a portion of the boundary of a circle, and accordingly we use the word arc. Euclid himself uses circumference both for the whole boundary and for a portion of it.

Ill, I. In the construction, DC is said to be produced to E; this assumes that D is within the circle, which Euclid demonstrates in III. 1.

III. 3. This consists of two parts, each of which is the converse of the other; and the whole proposition is the converse of the corollary in III. 1.

III. 5 and III. 6 should have been taken together. They amount to this, if the circumferences of two circles meet at a point they cannot have the same centre, so that circles which have the same centre and one point in their circumferences common, must coincide altogether. It would seem as if Euclid had made three cases, one in which the circles cut, one in which they touch internally, and one in which they touch externally, and had then omitted the last case as evident.

III. 7, III. 8. It is observed by Professor De Morgan that in III. 7 it is assumed that the angle FEB is greater than the angle FEC, the hypothesis being only that the angle DFB is greater than the angle DFC; and that in III. 8 it is assumed that K falls within the triangle DLM, and E without the triangle DMF. He intimates that these assumptions may be established by means of the following two propositions which may be given in order after I.

The perpendicular is the shortest straight line which can he drawn from a given point to a given straight line; and of others that which is nearer to the perpendicular is less than the more remote, and the converse; and not more than two equal straight lines can he drawn from the given point to the given straight line, one on each side of the perpendicular.

Every straight line draWn from the vertex of a triangle to the base is less than the greater of the two sides, or than either of them if they he equal.

The following proposition is analogous to III. 7 and III. 8.

If any point he taken on the circumference of a circle, of all the straight lines touch can he drawn from it to the circumference, the greatest is that in which the centre is; and of any others, that which is nearer to the straight line which passes through the centre is always greater than one more remote; and from the same points there can, be drawn to the circumference two straight lines, and only two, which are equal to one another, one on each side of the greatest line.

The first two parts of this proposition are contained in III. 15; all three parts might be demonstrated in the manner of III. 7, and they should be demonstrated, for the third part is really required, as we shall see in the note on III. 10.

III. 9. The point E might be supposed to fall within the angle ADC. It cannot then be shewn that DC is greater than DBF and DB greater than DA, but only that either DC or DA is less than DB; this however is sufficient for establisiting the proposition,

Euclid has given two demonstrations of III. 9, of which Simson has chosen the second. Euclid's other demonstration is as follows. Join D with the middle point of the straight line AB; then it may be shewn that this straight line is at right angles to AB; and therefore the centre of the circle must lie in this straight line, by III. 1, Corollary. In the same manner it may be shewn that the centre of the circle must lie in the straight line which joins D with the middle point of the straight line BC. The centre of the circle must therefore be at D, because two straight lines cannot have more than one common point.

III. 10. Euclid has given two demonstrations of III. 10, of which Simson has chosen the second. Euclid's first demonstration resembles his first demonstration of III. 9. He shews that the centre of each circle is on the straight line which joins K with the middle point of the straight line BG, and also on the straight line which joins K with the middle point of the straight line BH; therefore K must be the centre of each circle.

The demonstration which Simson has chosen requires some additions to make it complete. For the point K might be supposed to fall without the circle DEF, or on its circumference, or within it; and of these three suppositions Euclid only considers the last. If the point K be supposed to fall without the circle DEF we obtain a contradiction of III. 8; which is absurd. If the point K be supposed to fall on the circumference of the circle DEF we obtain a contradiction of the proposition which we have enunciated at the end of the note on III. 7 and III. 8; which is absurd.

What is demonstrated in III. 10 is that the circumferences of two circles cannot have more than two common points; there is nothing in the demonstration which assumes that the circles cut one another, but the enunciation refers to this case only because it is shewn in III. 13 that if two circles touch one another, their circumferences cannot have more than one common point.

III. II, III. 12. The enunciations as given by Simson and others speak of the point of contact; it is however not shewn until III. 13 that there is only one point of contact. It should be observed that the demonstration in III. 1 1 will hold even if D and H be supposed to coincide, and that the demonstration in III. 12 will hold even if C and D be supposed to coincide. We may combine III. 11 and III. 12 in one enunciation thus.

If two circles touch one another their circumferences cannot have a common point out of the direction of the straight line which joins the centres.

III. II may be deduced from III. 7. For GH is the least line that can be drawn from G to the circumference of the circle whose centre is F, by III. 7. Therefore GH is less than GD, that is, less than GD; which is absurd. Similarly III. 12 may be deduced from III. 8.

III. 13. Simson observes, "As it is much easier to imagine that two circles may touch one another within in more points than one, upon the same side, than upon opposite sides, the figure of that case ought not to have been omitted; but the construction in the Greek text would not have suited with this figure so well, because the centres of the circles must have been placed near to the circumferences; on which account another construction and demonstration is given, which is the same with the second part of that which Campanus has translated from the Arabic, where, without any reason, the demonstration is divided into two parts."

It would not be obvious from this note which figure Simson himself supplied, because it is uncertain what he means by the "same side" and "opposite sides." It is the left-hand figure in the first part of the demonstration. Euclid, however, seems to be quite correct in omitting this figure, because he has shewn in III. 1 1 that if two circles touch internally there cannot be a point of contact out of the direction of the straight line which joins the centres. Thus, in order to shew that there is only one point of contact, it is sufficient to put the second supposed point of contact on the direction of the straight line which joins the centres. Accordingly in his own demonstration Euclid confines himself to the right-hand figure; and he shews that this case cannot exist, because the straight line BD would be a diameter of both circles, and would therefore be bisected at two different points; which is absurd.

Euclid might have used a similar method for the second part of the proposition; for as there cannot be a point of contact out of the straight line joining the centres, it is obviously impossible that there can be a second point of contact when the circles touch externally. It is easy to see this; but Euclid preferred a method in which there is more formal reasoning.

We may observe that Euclid's mode of dealing with the contact of circles has often been censured by commentators, but apparently not always with good reason. For example, Walker gives another demonstration of III. 13; and says that Euclid's is worth nothing, and that Simson fails; for it is not proved that two circles which touch cannot have any arc common to both circumferences. But it is shewn in III. 10 that this is impossible; Walker appears to have supposed that III. 10 is limited to the case of circles which cut. See the note on III. 10.

III. 17. It is obvious from the construction in III. 17 that two straight lines can be drawn from a given external point to touch a given circle; and these two straight lines are equal in length and equally inclined to the straight line which joins the given external point with the centre of the given circle.

After reading III. 31 the student will see that the problem in III. 17 may be solved in another way, as follows: describe a circle on AE as diameter; then the points of intersection of this circle with the given circle will be the points of contact of the two straight lines which can be drawn from A to touch the given circle.

III. 18. It does not appear that III. 18 adds anything to what we have already obtained in III. 16. For in III. 16 it is shewn, that there is only one straight line which touches a given circle at a given point, and that the angle between this straight line and the radius drawn to the point of contact is a right angle.

III. 20. There are two assumptions in the demonstration of III. 20. Suppose that A is double of B and C double of D; then in the first part it is assumed that the sum of A and C is double of the sum of B and D, and in the second part it is assumed that the difference of A and C is double of the difference of B and D. The former assumption is a particular case of V. 1, and the latter is a particular case of V. 5.

An important extension may be given to III. 20 by introducing angles greater than two right angles. For, in the first figure, suppose we draw the straight lines BF and CF. Then, the angle BEA is double of the angle BFA, and the angle CEA is double of the angle CFA; therefore the sum of the angles BEA and CEA is double of the angle BFC. The sum of the angles BEA and CEA is greater than two right angles; we will call the sum, the re-entrant angle BEC. Thus the re-entrant angle BEC is double of the angle BFC. (See note on I. 3-2). If this extension be used some of the demonstrations in the third book maybe abbreviated. Thus III. 21 maybe demonstrated without making two cases; III. 22 will follow immediately from the fact that the sum of the angles at the centre is equal to four right angles; and III. 31 will follow immediately from III. 20.

III. 21. In III. 21 Euclid himself has given only the first case; the second case has been added by Simson and others. In either of the figures of III. 2 1 if a point be taken on the same side of BD as A, the angle contained by the straight lines which join this point to the extremities of BD is greater or less than the angle BAD, according as the point is within or without the angle BAD; this follows from I. 21.

We shall have occasion to refer to IV. 5 in some of the remaining notes to the third Book; and the student is accordingly recommended to read that proposition at the present stage.

The following proposition is very important. If any number of triangles be constructed on the same base and on the same side of it, with equal vertical angles, the vertices will all lie on the circumference of a segment of a circle.

For take any one of these triangles, and describe a circle round it, by IV. 5; then the vertex of any other of the triangles must be on the circumference of the segment containing the assumed vertex, since, by the former part of this note, the vertex cannot be without the circle or within the circle.

III. 22. The converse of III. 22 is true and very important; namely, if two opposite angles of a quadrilateral be together equal to two right angles, a circle may be circumscribed about the quadrilaeral. For, let ABCD denote the quadrilateral. Describe a circle round the triangle ABC, by IV. 5. Take any point E, on the circumference of the segment cut off by AC, and on the same side of AC as D is. Then, the angles at B and F are together equal to two right angles, by III. 72; and the angles at B and D are together equal to two right angles, by hypothesis. Therefore the angle at E is equal to the angle at D. Therefore, by the preceding note D is on the circumference of the same segment as E.

III. 32. The converse of III. 32 is true and important; namely, if a straight line meet a circle, and from the point of meeting a straight line be drawn cutting the circle, and the angle between the two straight lines be equal to the angle in the alternate segment of the circle, the straight line which meets the circle shall touch the circle.

This may be demonstrated indirectly. For, if possible, suppose that the straight line which meets the circle does not touch it. Draw through the point of meeting a straight line to touch the circle. Then, by III. 32 and the hypothesis, it will followthat two different straight lines pass through the same point, and make the same angle, on the same side, with a third straight line which also passes through that point; but this is impossible.

III. 35, III. 36. The following proposition constitutes a large part of the demonstrations of III. 35 and III. 36. If any point be taken in the base, or the base produced, of an isosceles triangle, the rectangle contained by the segments of the base is equal to the difference of the square on the straight line joining this point to the vertex and the square on the side of the triangle.

This proposition is in fact demonstrated by Euclid, without using any property of the circle; if it were enunciated and demonstrated before III. 35 and III. 36 the demonstrations of these two propositions might be shortened and simplified.

The following converse of III. 35 and the Corollaray of III. 36 may be noticed. If two straight lines AB, CD intersect at 0, and the rectangle AO, OB be equal to the rectangle CO, OD, the circumference of a circle will pass through the four points A, B, C, D.

For a circle may be described round the triangle ABC, by IV. 5; and then it may be shewn indirectly, by the aid of III. 35 or the Corollary of III. 36 that the circumference of this circle will also pass through D.

THE FOURTH BOOK.

The fourth Book of the Elements consists entirely of problems. The first five propositions relate to triangles of any kind; the remaining propositions relate to polygons which have all their sides equal and all their angles equal. A polygon which has all its sides equal and all its angles equal is called a regular polygon.

IV. 4, By a process similar to that in IV. 4 we can describe a circle which shall touch one side of a triangle and the other two sides produced. Suppose, for example, that we wish to describe a circle which shall touch the side BC, and the sides AB and AC produced: bisect the angle between AB produced and BC, and bisect the angle between AC produced and BC; then the point at which the bisecting straight lines meet will be the centre of the required circle. The demonstration will be similar to that in IV. 4.

A circle which touches one side of a triangle and the other two sides produced, is called an escribed circle of the triangle.

We can also describe a triangle equiangular to a given triangle, and such that one of its sides and the other two sides suppose AK produced to meet the circle again; and at the point of intersection draw a straight line touching the circle; this straight line with parts of NB and NC, will form a triangle, which will be equiangular to the triangle MLN, and therefore equiangular to the triangle EDF and one of the sides of this triangle, and the other two sides produced, will touch the given circle. the part which shews that DF and EF will meet. It has also been proposed to shew this in the following way: join DE; then the angles EDF and DEF are together less than the angles ADF and AEF, that is, they are together less than two right angles; and therefore DF and EF will meet, by Axiom 12. This assumes that ADE and AED are acute angles; it may however be easily shewn that BE is parallel to BC, so that the triangle ADE is equiangular to the triangle ABC and we must therefore select the two sides AB and AC such that ABC and ACB may be acute angles.

IV. 10. The vertical angle of the triangle in IV. 10 is easily seen to be the fifth part of two right angles; and as it may be bisected, we can thus divide a right angle geometrically into five equal parts.

It follows from what is given in the fourth Book of the Elements that the circumference of a circle can be divided into 3, 6, 12, 24, .... equal parts; and also into 4, 8, 16, 32, .... equal parts; and also into 5, 10, 20, 40, .... equal parts; and

also into 15, 30, 60, 120,........ equal parts. Hence also regular polygons having as many sides as any of these numbers may be inscribed in a circle, or described about a circle. This however does not enable us to describe a regular polygon of any assigned number of sides; for example, we do not know how to describe geometrically a regular polygon of 7 sides.

It was first demonstrated by Gauss in 1801, in his Disquisitiones Arithmeticae, that it is possible to describe geometrically a regular polygon of $2^{n}+1$ sides, provided $2^{n}+1$ be a prime number; the demonstration is not of an elementary character. As an example, it follows that a regular polygon of 17 sides can be described geometrically; this example is discussed in Catalan's Théorèmes et Problèmes de Géometrié Elémentaire.

For an approximate construction of a regular heptagon see the Philosophical Magazine for February and for April, 1864.

THE FIFTH BOOK.

The fifth Book of the Elements is on Proportion. Much has been written respecting Euclid's treatment of this subject; besides the Commentaries on the Elements to which we have already referred, the student may consult the articles Ratio and Proportion in the English Cyclopædia, and the tract on the Connexion of Number and Magnitude by Professor De Morgan.

The fifth Book relates not merely to length and space, but to any kind of magnitude of which we can form multiples.

V. Def. 1. The word part is used in two senses in Geometry. Sometimes the word denotes any magnitude which is less than another of the same kind, as in the axiom, the whole is greater than its part. In this sense the word has been used up to the present point, but in the fifth Book Euclid confines the word to a more restricted sense. This restricted sense agrees with that which is given in Arithmetic and Algebra to the term aliquot part, or to the term submultiple.

V. Def. 3. Simson considers that the definitions 3 and 8 are "not Euclid's, but added by some unskilful editor." Other commentators also have rejected these definitions as useless. The last word of the third definition should be quantuplicity, not quantity; so that the definition indicates that ratio refers to the number of times which one magnitude contains another. See De Morgan's Differential and Integral Calculus, page 18.

V. Def. 4. This definition amounts to saying that the quantities must be of the same kind.

V. Def. 5 The fifth definition is the foundation of Euclid's doctrine of proportion. The student will find in works on Algebra a comparison of Euclid's definition of proportion with the simpler definitions which are employed in Arithmetic and Algebra. Euclid's definition is applicable to incommensurable quantities, as -well as to commensurable quantities.

We should recommend the student to read the first proposition of the sixth Book immediately after the fifth definition of the fifth Book; he will there see how Euclid applies his definition, and will thus obtain a better notion of its meaning and importance.

Compound Ratio. The definition of compound ratio was supplied by Simson. The Greek text does not give any definition of compound ratio here, but gives one as the fifth definition of the sixth Book, which Simson rejects as absurd and useless.

V. Defs. 18, 19, 10. The definitions 18, 19, 20 are not presented by Simson precisely as they stand in the original. The last sentence in definition 18 was supplied by Simson. Euclid does not connect definitions 19 and 10 with definition 18. In 19 he defines ordinate proportion, and in 20 he defines perturbate proportion. Nothing would be lost if Euclid's definition 18 were entirely omitted, and the term ex æquali never employed. Euclid employs such a term in the enunciations of V. 20, 21, 11, 23; but it seems quite useless, and is accordingly neglected by Simson and others in their translations.

The axioms given after the definitions of the fifth Book are not in Euclid; they were supplied by Simson.

The propositions of the fifth Book might be divided into four sections. Propositions 1 to 6 relate to the properties of equimultiples. Propositions 7 to 10 and 13 and 14 connect the notion of the ratio of magnitudes with the ordinary notions of greater, equal, and less. Propositions 11, 12, 15 and 16 may be considered as introduced to shew that, if four quantities of the same kind be proportionals they will also he proportionals when taken alternately. The remaining propositions shew that magnitudes are proportional by composition, by division, and ex aequo.

In this division of the fifth Book propositions 13 and 14 are supposed to be placed immediately after proposition 10; and they might be taken in this order without any change in Euclid's demonstrations.

The propositions headed A, B, C, D, E were supplied by Simson.

V. 1, 2, 3, 5, 6. These are simple propositions of Arithmetic, though they are here expressed in terms which make them appear less familiar than they really are. For example, V. i "states no more than that ten acres and ten roods make ten times as much as one acre and one rood." De Morgan.

In V. 5 Simson has substituted another construction for that given by Euclid, because Euclid's construction assumes that we can divide a given straight line into any assigned number of equal parts, and this problem is not solved until VI. 9.

V. 18. This demonstration is Simson's. We will give here Euclid's demonstration.

Let AE be to EB as CF is to FD: AB shall be to BE as CD is to DF.
For, if not, AB will be to BE as CD is to some magnitude less than DF, or greater than DF.

First, suppose that AB is to BE as CD is to DG, which is less than DF.

Then, because AB is to BE as CD is to DG, therefore AE is to EB as CG is to GD. [V. 17.
But AE is to EB as CF is to FD, [Hypothesis.
therefore CG is to GD as CF is to FD. [V. 11.
But CG is greater than CF; [Hypothesis.
therefore GD is greater than FD. [V. 11.
But GD is less than FD; which is impossible.

In the same manner it may be shewn that AB is not to BE as CD is to a magnitude greater than DF.
Therefore AB is to BE as CD is to DF.

The objection urged by Simson against Euclid's demonstration is that "it depends upon this hypothesis, that to any three magnitudes, two of which, at least, are of the same kind, there may be a fourth proportional:...... Euclid does not demonstrate it, nor does he shew how to find the fourth proportional, before the 12th Proposition of the 6th Book.... "

The following demonstration is given by Austin in his Examination of the first six books of Euclid's Elements.

Let AE be to EB as CF is to FD: AB shall be to BE as CD is to DF.

For, because AE is to EB as CF is to FD, therefore, alternately, AE is to CF as EB is to FD. [V. 16.

And as one of the antecedents is to its consequent so is the sum of the antecedents to the sum of the consequents; [V. 12.

therefore as EB is to FD so are AE and EB together to CF and FD together, that is, AB is to CD as EB is to FD.

Therefore, alternately, AB is to EB as CD is to FD. [V. 16.

V. 25. The first step in the demonstration of this proposition is "take AG equal to E and CH equal to F"; and here a reference is sometimes given to I. 3. But the magnitudes in the proposition are not necessarily straight lines, so that this reference to I. 3 should not be given; it must however be assumed that we can perform on the magnitudes considered, an operation similar to that which is performed on straight lines in I. 3. Since the fifth Book of the Elements treats of magnitudes generally, and not merely of lengths, areas, and angles, there is no reference made in it to any proposition of the first four Books.

Simson adds four propositions relating to compound ratio, which he distinguishes by the letters F, G, H, K; it seems however unnecessary to reproduce them as they are now rarely read and never required.

 

THE SIXTH BOOK.

The sixth Book of the Elements consists of the application of the theory of proportion to establish properties of geometrical figures,

VI. Def. I. Eor an important remark bearing on the first definition, see the note on VI. 5.

VI. Def. 1. The second definition is useless, for Euclid makes no mention of reciprocal figures. VI. Def. 4. The fourth definition is strictly only applicable to a triangle, because no other figure has a point which can be exclusively called its vertex. The altitude of a parallelogram is the perpendicular drawn to the base from any point in the opposite side.

VI. 2. The enunciation of this important proposition is open to objection, for the manner in which the sides may be cut is not sufficiently limited. Suppose, for example, that AD is double of DB, and CE double of EA; the sides are then cut proportionally, for each side is divided into two parts, one of which is double of the other; but DE is not parallel to BC. It should therefore be stated in the enunciation that the segments terminated at the vertex of the triangle are to he homologous terms in the ratios, that is, are to he the antecedents or the consequents of the ratios.

It will be observed that there are three figures corresponding to three cases which may exist; for the straight line drawn parallel to one side may cut the other sides, or may cut the other sides when they are produced through the extremities of the base, or may cut the other sides when they are produced through the vertex. In all these cases the triangles which are shewn to be equal have their vertices at the extremities of the base of the given triangle, and have for their common base the straight line which is, either by hypothesis or by demonstration, parallel to the base of the triangle. The triangle with which these two triangles are compared has the same base as they have, and has its vertex coinciding with the vertex of the given triangle.

VI. A. This proposition was supplied by Simson.

VI. 4. We have preferred to adopt the term " triangles which are equiangular to one another," instead of "equiangular triangles," when the words are used in the sense they bear in this proposition, Euclid himself does not use the term equiangular triangle in the sense in which the modern editors use it in the Corollary to I. 5, so that he is not prevented from using the term in the sense it bears in the enunciation of VI. 4. and elsewhere; but modern editors, having already employed the term in one sense ought to keep to that sense. In the demonstrations, where Euclid uses such language as "the triangle ABC is equiangular to the triangle DEF," the modern editors sometimes adopt it, and sometimes change it to "the triangles ABC and DEF are equiangular."

In VI. 4 the manner in which the two triangles are to be placed is very imperfectly described; their bases are to be in the same straight line and contiguous, their vertices are to be on the same side of the base, and each of the two angles which have a common vertex is to be equal to the remote angle of the other triangle.

By superposition we might deduce VI. 4 immediately from VI. 2.

VI. 5. The hypothesis in VI. 5 involves more than is directly asserted; the enunciation should be, "if the sides of two triangles, taken in order, about each of their angles;" that is, some restriction equivalent to the words taken in order should be introduced. It is quite possible that there should be two triangles ABC, DEF, such that AB is to BC as DE is to EF, and BC to CA as DF is to ED, and therefore, by V. 23, AB to AC as DF is to EF; in this case the sides of the triangles about each of their angles are proportionals, but not in the same order, and the triangles are not necessarily equiangular to one another. For a numerical illustration we may suppose the sides of one triangle to be 3, 4 and 5 feet respectively, and those of another to be 12, 15 and 20 feet respectively. Walker.

Each of the two propositions VI. 4 and VI. 5 is the converse of the other. They shew that if two triangles have either of the two properties involved in the definition of similar figures they will have the other also. This is a special property of triangles. In other figures either of the properties may exist alone. For example, any rectangle and a square have their angles equal, but not their sides proportional; while a square and any rhombus have their sides proportional, but not their angles equal.

VI. 7. In VI. 7 the enunciation is imperfect; it should be, "if two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals, so that the sides subtending the equal angles are homologous; then if each ....." The imperfection is of the same nature as that which is pointed out in the note on VI. 5. Walker.

The proposition might be conveniently broken up and the essential part of it presented thus: if two triangles have two sides of the one proportional to two sides of the other, and the angles opposite to one pair of homologous sides equal, the angles which are opposite to the other pair of homologous sides shall either be equal, or be together equal to two right angles.

For, the angles included by the proportional sides must be either equal or unequal. If they are equal, then since the triangles have two angles of the one equal to two angles of the other, each to each, they are equiangular to one another. We have therefore only to consider the case in which the angles included by the proportional sides are unequal.

Let the triangles ABC, DEF have the angle at A equal to the angle at D, and AB to BC as DE is to EF, but the angle ABC not equal to the angle DEF: the angles ACB and DFE shall be together equal to two right angles.

For, one of the angles ABC, DEF must be greater than the other; suppose ABC the greater; and make the angle ABG equal to the angle DEF. Then it may be shewn, as in VI. 7, that BG is equal to BC, and the angle BGA equal to the angle EFD. Therefore the angles ACB and DFE are together equal to the angles BGC and AGB, that is, to two right angles.

Then the results enunciated in VI. 7 will readily follow. For if the angles ACB and DFE are both greater than a right angle, or both less than a right angle, or if one of them be a right angle, they must be equal.

VI. 8. In the demonstration of VI. 8, as given by Simson, it is inferred that two triangles which are similar to a third triangle are similar to each other; this is a particular case of VI. 2 1, which the student should consult, in order to see the validity of the inference.

VI. 9. The word part is here used in the restricted sense of the first definition of the fifth Book. VI. 9 is a particular case of VI. 10.

VI. 10. The most important case of this proposition is that in which a straight line is to be divided either internally or externally into two parts which shall be in a given, ratio.

The case in which the straight line is to be divided internally is given in the text; suppose, for example, that the given ratio is that of AE to EC; then AB is divided at G in the given ratio.

Suppose, however, that AB is to be divided externally in a given ratio; that is, suppose that AB is to be produced so that the whole straight line made up of AB and the part produced may be to the part produced in a given ratio. Let the given ratio be that of AC to CE. Join EB; through C draw a straight line parallel to EB; then this straight line will meet AB, produced through B, at the required point.

VI. II. This is a particular case of VI. 12.

VI. 14. The following is a full exhibition of the steps which lead to the result that FB and BG are in one straight line.

The angle DBF is equal to the angle GBE; [Hypothesis.
add to each the angle FBE;
therefore the angles DBF, FBE are together equal to the angles GBE, FBE. [Axiom 2.
But the angles DBF, FBE are together equal to two right angles; [I. 13.
therefore the angles GBE, FBE are together equal to two right angles; [Axiom 1.
therefore FB and BG are in one straight line. [I. 14.

VI. 15. This may be inferred from VI. 14, since a triangle is half of a parallelogram, with the same base and altitude.

It is not difficult to establish a third proposition conversely connected with the two involved in VI. 14, and a third proposition similarly conversely connected, with the two involved in VI. 15. These propositions are the following.

Equal parallelograms wJiich have their sides reciprocally proportional, have their angles equal, each to each. Equal triangles which have the sides about a pair of angles reciprocally proportional, have those angles equal or together equal to two right angles.

We will take the latter proposition.

Let ABC, ADE be equal triangles; and let CA be to AD as AE is to AB: either the angle BAC shall be equal to the angle DAE, or the angles BAC and DAE shall be together equal to two right angles.

[The student can construct the figure for himself. ]

Place the triangles so that CA and AD may be in one straight line; then if EA and AB are in one straight line the angle BAC is equal to the angle DAE. [I. 15.
If EA and AB are not in one straight line, produce BA through A to F, so that AF may be equal to AE; join DF and EF.
Then because CA is to AD as AE is to AB, [Hypothesis.
and AF is equal to AE, [Construction.
therefore CA is to AD as AF is to AB. [V. 9, V.11
Therefore the triangle DAF is equal to the triangle BAC. [VI. 15.
But the triangle DAE is equal to the triangle BAC. [Hypothesis.
Therefore the triangle DAE is equal to the triangle DAF. [Ax. 1.
Therefore EF is parallel to AD. [I. 39.

Suppose now that the angle DAE is greater than the angle DAF.
Then the angle CAE is equal to the angle AFE, [I. 29.
and therefore the angle CAE is equal to the angle AFE, [I. 5.
and therefore the angle CAE is equal to the angle BAC. [I. 29.
Therefore the angles BAC and DAE are together equal to two right angles.

Similarly the proposition may he demonstrated if the angle DAE is less than the angle DAF.

VI. 16. This is a particular case of VI. 14.

VI. 17. This is a particular case of VI, 16.

VI. 22. There is a step in the second part of VI. 12 which requires examination. After it has been shewn that the figure SR is equal to the similar and similarly situated figure NH, it is added "therefore PR is equal to GH." In the Greek text reference is here made to a lemma which follows the proposition. The word lemma is occasionally used in mathematics to denote an auxiliary proposition. From the unusual circumstance of a; reference to something following, Simson probably concluded that the lemma could not be Euclid's, and accordingly he takes no notice of it.

The following is the substance of the lemma.

If PR be not equal to GH, one of them must be greater than the other; suppose PR greater than GH.

Then, because SR and NH are similar figures, PR is to PS as GH is to GN. [VI. Definition 1.
But PR is greater than GH, [Hypothesis.
therefore PS is greater than GN. [V. 14.
Therefore the triangle RPS is greater than the triangle HGN. [I. 4, Axiom 9.
But, because SR and NH are similar figures, the triangle RPS is equal to the triangle HGN; [VI. 20.
which is impossible.
Therefore PR is equal to, GH.

VI. 23. In the figure of VI. 23 suppose BD and GE drawn. Then the triangle BCD is to the triangle GCE as the parallelogram ACis to the parallelogram CF. Hence the result may be extended to triangles, and we have the following theorem, triangles which have one angle of the one equal to one angle of the other, have to one another the ratio which is compounded of the ratios of their sides.

Then VI. 19 is an immediate consequence of this theorem. For lett ABC and DEF be simmilar triangles, so that AB is to DE as BC is to EF; and therefore, alternately, AB is to DE as BC is to EF. Then, by the theorem, the triangle ABC has to the triangle DEF the ratio which is compounded of the ratios of AB to BE and of BC to EF, that is, the ratio which is compounded of the ratios of BC to EF and of BC to EF. And, from the definitions of duplicate ratio and of compound ratio, it follows that the ratio compounded of the ratios of BC to EF and of BC to EF is the duplicate ratio of BC to EF.

VI. 25. It will be easy for the student to exhibit in detail the process of shewing that BC and CF are in one straight line, and also LE and EM; the process is exactly the same as that in I. 45, by which it is shewn that KH and HM are in one straight line, and also FG and GL.

It seems that VI. 25 is out of place, since it separates propositions so closely connected as VI. 24 and VI. 26. We may enunciate VI. 25 in familiar language thus:

to make a figure which shall have the form of one figure and the size of another.

VI. 26. This proposition is the converse of VI. 24; it might be extended to the case of two similar and similarly situated parallelograms which have a pair of angles vertically opposite.

We have omitted in the sixth Book Propositions 27, 28, 29, and the first solution which Euclid gives of Proposition 30, as they appear now to be never required, and have been condemned as useless by various modern commentators; see Austin, Walker, and Lardner, Some idea of the nature of these propositions may be obtained from the following statement of the problem proposed by Euclid in VI, 29. AB is a given straight line; it has to be produced through B to a point O, and a parallelogram described on AO subject to the following conditions; the parallelogram is to be equal to a given rectilineal figure, and the parallelogram on the base BO which can be cut of by a straight line through B is to be similar to a given parallelogram.

VI. 32. This proposition seems of no use. Moreover the enunciation is imperfect. For suppose ED to be produced through D to a point F, such that DF is equal to DE; and join CF. Then the triangle CDF will satisfy all the conditions in Euclid's enunciation, as well as the triangle CDE; but CF and CB are not in one straight line. It should be stated that the bases must lie on corresponding sides of both the parallels; the bases CF and BC do not lie on corresponding sides of the parallels AB and BC, and so the triangle CDF would not fulfil all the conditions, and would therefore be excluded.

VI. 33. In VI. 33 Euclid implicitly gives up the restriction, which he seems to have adopted hitherto, that no angle is to be considered greater than two right angles. For in the demonstration the angle BGL may be any multiple whatever of the angle BGC, and so may be greater than any number of right angles.

VI. B, C, D. These propositions were introduced by Simson. The important proposition VI. B occurs in the Μεγάλη Σύνταξις of Ptolemy.

THE ELEVENTH BOOK.

In addition to the first six Books of the Elements it is usual to read part of the eleventh Book. For an account of the contents of the other Books of the Elements the student is referred to the article Eucleides in Dr Smith's Dictionary of Greek and Roman Biography, and to the article Irrational Quantities in the English Cyclopcaedia. We may state briefly that Books VII, VIII, IX treat on Arithmetic, Book X on Irrational Quantities, and Books XI, XII on Solid Geometry.

XI. Def. 10. This definition is omitted by Simson, and justly, because, as he shews, it is not true that solid figures contained by the same number of similar and equal plane figures are equal to one another. For, conceive two pyramids, which have their bases similar and equal, but have different altitudes. Suppose one of these bases applied exactly on the other; then if the vertices be put on opposite sides of the base a certain solid is formed, and if the vertices be put on the same side of the base another solid is formed. The two solids thus formed are contained by the same number of similar and equal plane figures, but they are not equal.

It will be observed that in this example one of the solids has a re-entrant solid angle; see page 264. It is however true that two convex solid figures are equal if they are contained by equal plane figures similarly arranged; see Catalan's Théorèmes et Problèmes de Géométrie Elémentaire. This result was first demonstrated by Cauchy, who turned his attention to the point at the request of Legendre and Malus; see the Journal de I' École Polytechnique, Cahier 16.

XI. Def. 26. The word tetrahedron is now often used to denote a solid bounded by any four triangular faces, that is, a pyramid on a triangular base; and when the tetrahedron is to be such as Euclid defines, it is called a regular tetrahedron.

Two other definitions may conveniently be added.

A straight line is said to be parallel to a plane when they do not meet if produced.

The angle made by two straight lines which do not meet is the angle contained by two straight lines parallel to them, drawn through any point.

XI. 21. In XI. 21 the first case only is given in the original. In the second case a certain condition must be introduced, or the proposition will not be true; the polygon BCDEF must have no re-entrant angle. See note on I. 32.

The propositions in Euclid on Solid Geometry which are now not read, contain some very important results respecting the volumes of soHds. We will state these results, as they are often of use; the demonstrations of them are now usually given as examples of the Integral Calculus.

We have already explained in the notes to the second Book how the area of a figure is measured by the number of square inches or square feet which it contains. In a similar manner the volume of a solid is measured by the number of cubic inches or cubic feet which it contains; a cubic inch is a cube in which each of the faces is a square inch, and a cubic foot is similarly defined.

The volume of a prism is found by multiplying the number of square inches in its base by the number of inches in its altitude; the volume is thus expressed in cubic inches. Or we may multiply the number of square feet in the base by the number of feet in the altitude; the volume is thus expressed in cubic feet. By the base of a prism is meant either of the two equal, similar, and parallel figures of XI. Definition 13; and the altitude of the prism is the perpendicular distance between these two planes. The rule for the volume of a prism involves the fact that prisms, on equal bases and between the same parallels are equal in volume.

A parallelepiped is a particular case of a prism. The volume of a pyramid is one third of the volume of a prism on the same base and having the same altitude.

For an account of what are called the five regular solids the student is referred to the chapter on Polyhedrons in the Treatise on Spherical Trigonometry.

THE TWELFTH BOOK.

Two propositions are given from the twelfth Book, as they are very important, and are required in the University Examinations. The Lemma is the first proposition of the tenth Book, and is required in the demonstration of the second proposition of the twelfth Book,