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NOTES AND ILLUSTRATIONS.
ELEMENTS, BOOK I.
To what is stated in the foot notes, it may here be added, that several changes have been made in the arrangement of the definitions. The definition of parallel straight lines is placed after the definitions of an angle and its species, the inclination of lines and their parallelism equally depending on their mutual positions. The definitions of rectilineal figures are made to precede the definition of the circle ; and the definition of the parallelogram is inserted, and is placed before those of its species, the rectangle, square, rhombus, and rhomboid.
Euclid has defined a straight line to be that which lies evenly, or equally, between its extreme points. This definition affords no assistance in arriving at the properties of straight lines; and therefore Playfair's definition has been adopted in this edition. From him also the definition of a point has been taken. In Dr. Simson's edition, a point is defined to be " that which has no parts, or no magnitude.” This is objectionable, as being wholly negative.
The following illustrations, taken in substance from Dr. Simson, are well deserving of attention.
It is necessary to consider a solid, that is, a magnitude which has length, breadth, and thickness, in order to understand aright, the definitions of a surface, line, and point, for these all arise from a solid, and exist in it. The boundary or boundaries which contain a solid, are surfaces ; or the boundary which is common to two contiguous solids, or which divides one solid into two contiguous parts, is a surface. Thus, if CD be a boundary of the solid CM, or the common boundary of the two solids CM, FD, it is
called a surface ; and it is in the one solid as well as in the other, and has no thickness. For if it have any, this thickness must be either a part of the thickness of the solid CM, or FD, or of the thickness of both. It cannot be a part of the thickness of CM; for if this solid be removed from FD, the surface CD, the boundary of DF, remains as it was : and it would appear in a similar manner, that its thickness can be no part of the solid FD. The surface CD, therefore, has only length and breadth, without any thickness.
The boundary of a surface is a line; or a line is the common boundary of two contiguous surfaces, or it is that which divides one surface into two contiguous parts. Thus, if CD be a boundary of the surface ACD, or of the contiguous surfaces ACD, BCD, it is called a line, and it has no breadth. For if it have any, this must be part of the breadth, either of the surface ACD, or BCD, or of both. It is not a part of the breadth of ACD; for if this surface Á be removed from BCD, the line CD, which is a boundary of BCD, remains the same as it was. In like manner, it would be shown, that the supposed breadth is not a part of BCD. Therefore the line CD has no breadth ; neither has it any thickness, because the surface in which it is, has no thickness. A line, therefore, has only length, without breadth or thickness.
Lastly, the boundary of a line is called a point; or a point is the common boundary or extremity of two contiguous lines. Thus, in the last figure, D, one extremity of the line AD, or the common extremity of the two lines AD, BD, is a point; and it bas no thickness or breadth, because it is in a line, and a line has neither thickness nor breadth, Neither has the point D any length. For if it had any, this would be a part of the length, either of AD, or BD, or of both. It is not a part of the length of AD; for if AD were removed, D would still remain as before, the boundary of DB ; and it would be shown in a similar manner, that its length is not a part of BD. Hence, a point has a certain position, but it has not length, breadth or thickness.
GEOMETRY has been defined to be that science which treats of the properties of extension-of solids, surfaces, and lines. It forms the subject of this volume, with the exception of Euclid's fifth book and the Supplement to it, which are not confined to the consideration of geometrical magnitudes, but are equally applicable in arithmetic.
Plane geometry investigates the properties of lines and figures lying in the same plane; while solid geometry treats of solids, and of lines lying in different planes. The first, second, third, fourth,
and sixth books of Euclid belong exclusively to the former : the eleventh and twelfth chiefly to the latter.
Elementary geometry treats of the straight line and circle, of solid figures bounded by planes, and of the “ three round bodies,” the cylinder, the cone, and the sphere. The higher geometry treats of the conic sections, and other curves.
PROP. VIII. AND XXVI.
In addition to what is demonstrated by Euclid in these two propositions, it is proved, in each, that the areas of the triangles are equal. This shortens the demonstrations of several subsequent propositions.
No subject in elementary geometry has so much perplexed mathematicians, both ancient and modern, as the theory of parallel lines. There is no difficulty in showing, that if straight lines in the same plane make certain angles equal, the lines will never meet, and are therefore parallel, as is done in the 27th and 28th propositions of this book; but to establish the converse, by showing, that if straight lines be parallel, they make certain angles equal, which is the object of the 29th proposition, has never been effected in a manner perfectly unobjectionable. The truth of the conclusions is universally admitted ; but to demonstrate these conclusions with that rigour, of which geometrical reasoning is, in other cases, susceptible, has been found to present difficulties of no ordinary kind. Euclid has been unable to effect this by means of his definition of parallels, and has been obliged to assume, in his 12th axiom, a principle, which, though readily admitted after the 28th proposition is understood, is by no means self-evident.
To obviate these objections, some writers have endeavoured to prove the 12th axiom and the 29th proposition by means of the properties of lines established by Euclid, in what precedes that proposition ; others have assumed some other axiom, supposed to be more obvious than Euclid's; and some have given a new definition of parallels. A short statement and examination of a few of the principal attempts of this kind, will make the student aware of the difficulty of the subject, and will show the necessity of great care and caution in all such inquiries.
1. The earliest attempt with which we are acquainted, to establish the theory of parallels, without a new definition or axiom, is that of Ptolemy, the celebrated author of the astronomical system
which is called after his name. His proof, as given by Proclus in the fourth book of his Commentaries, is in substance as follows. Let the straight line EGHF fall on the two parallels AB, CD; the angles BGH, DHG are together equal to two right angles. For, if not, let them be greater. Then, because the lines GB, HD are not more parallel tban GA, HC, the angles AGH, CHG are also greater than two right angles ; and therefore the four interior angles at G and H, are together greater than four right angles, which (I. 13.) is absurd. Therefore BGH, DHG are not greater than two right angles ; and it may be shown, in a similar manner, that they are not less: they are therefore equal to two right angles.
Now this cannot be regarded as a proof arising from the definition of parallel lines. By that definition, AB, CD never meet; but we cannot hence infer immediately, that the interior angles on one side of GH are greater or less than two right angles, according as those on the other side are: and though we readily admit that this is likely to be true, it is from the loose idea which we have concerning parallel lines, as lying in the same direction, and not from their definition. It is therefore to be regarded as a disguised axiom, independent of the definition; or, at best, as an illustration, showing the reasonableness of the last part of the 29th proposition.
2. In 1787, an attempt, to prove the 12th axiom without employing any new principle, was published by Franceschini, professor of mathematics in Bologna. This geometer demonstrates, that if B be a right angle, and A an acute one, and if CD, EF be perpendiculars to AD from the points C, E, AF must be greater than AD. This he does indirectly, by showing, that a straight line joining ED is not perpendicular to AB; and secondly, that one drawn from E to any point between A and D, is likewise oblique to AB. He then infers, that as AE increases, AF must also increase, and that, since AE may be increased without limit, so AF may become greater than any given line, and may therefore exceed AB; wherefore, since the perpendiculars to AB, from points beyond B meet AE produced, the perpendicular BG must also meet it: and thus he conceives that he has proved the 12th axiom.
This reasoning, however, is fallacious, as magnitudes may continually increase, and yet never become equal to a given finite magnitude. Of this the 41st proposition of the fourth book of the Appendix affords an instance, as it is there shown, that the sum of all the lines CG, FL, GM, LN, &c., how many soever be taken, will never become so great as the radius CF. Other instances are also furnished by the 40th and 42d propositions, by numberless infinite series', and by curves that have asymptotes, that is, lines which continually approach them, and yet never meet
of them may
them. From the mere fact that a magnitude continually increases, we are not entitled to conclude, without considering the comparative magnitudes of the additions, that it will ever exceed a given amount: and therefore, in the present instance, we cannot legitimately infer, that, however AE is increased, the point F will fall beyond B.
3. In the Notes to Legendre's Eléments de Géométrie, a new method of investigating the theory of parallels is given, of which the following is an abstract. It is proved by superposition (or as in Euc. I. 26.) that two triangles are in every respect equal, when a side and the two adjacent angles of the one are respectively equal to a side and the two adjacent angles of the other. If, therefore, p be a side of a triangle, and A and B the adjacent angles, the third angle C is determined if A, B, and p be given ; as otherwise there might be different triangles which would have each a side equal to p, and the adjacent angles equal to A and B. Therefore the angle C must be a determinate function of A, B and p; or, as it may be expressed C=° (A, B, p), where is used to denote a function or combination of A, B and p. Now
р must be rejected from this equation ; for, if not, we might have, conversely, the value of p expressed in terms of A, B and C, which is absurd, p and the angles A, B, C being heterogeneous, and such, that no combination whatever of the angles could give the value of p. Hence, therefore, we have simply C= Q(A, B); that is, C is determined by A and B, independently of the sides.
From this it follows, that if two angles of one triangle be equal to two angles of another, their third angles are also equal.
Let now ABC be a triangle having the angle BAC a right angle, and draw the perpendicular AD. Then, in the triangles ABC,DBA, the right angles BAC, BDA are equal, and the angle B is common; therefore BAD is equal to C. In like mannerit would be shown that CAD is equal to B: whence (I. ax. 2.) the angle BAC is equal to the two angles B and C; but BAC is a right B angle, and therefore the three angles of the right-angled triangle are together equal to two right angles.
The same property is easily shown to belong to any other triangle, as it may be divided by a perpendicular from the greatest angle to the opposite side, into two right-angled triangles, the angles of which will be equal to the angles of the assumed triangle, together with the two right angles made by the perpendicular-and this proves the 32d of the first book of Euclid. *
The author next proceeds to prove, on similar principles, that, in equiangular triangles, the sides which are similarly situated, are
* It is evident, that the remaining part of the theory might be established without proportion, as in page 362.