straight lines diverge in such manner as to lie in one and the same straight line, or even to form a corner on the other side. If one hand of a watch stood still at twelve o'clock, while the other moved round, first to three, then to six, then to nine, and so on till it coincided with the position from which it set out, all the portions of plane thus passed over would be in geometry called angles, and the moving hand is the radius vectus. The principal use of introducing this last term, is to give the power of designating the larger and last-mentioned kind of angles ; which would be difficult without it. PROPOSITION I.-BOOK I. In this and other Propositions, the word “finite' has been omitted, as being unnecessary where magnitudes are stated to be either given' or equal to one another.' The observation that equality (meaning of magnitude, not of ratio) implies being finite, is as old as the Philebus of Plato. PROP. II.-BOOK I. This Proposition and the next have been sometimes objected to as unnecessary ; because, it is urged, a straight line might be transported from its place and applied to another straight line, and its two extremities marked off. There is no doubt that this might be done if it were a case of necessity. But Euclid desired to do without it, as being one demand the less. And the same reason appears valid for following his example. PROP. IV.-BOOK I. In this and several other Propositions, the term 'base' has been avoided as unnecessary; and its use confined to places where its meaning is distinctly laid down. PROP. V.-BOOK I. Professor Playfair (Elem. of Geom. Notes, p. 395) has noticed the ease with which this Proposition might be demonstrated, if it was allowed to suppose the isoskeles triangle transported to a new situation, and the new one applied to the old so as to apply different equal sides to one another. This might be mechanically exhibited, by cutting out the isoskeles triangle, as acb, with scissors, and applying it to the old situation ABC, so that the point a should coincide with A, and the side ac with AB; whereupon it is manifest that by reason of the equality of the angles K 1 3 B В a and A, and of the sides ab and AC, the point b will also coincide with C, and the third side cb with BC; whence the angles c and B, as coinciding with one another, will be equal. This species of superimposition might be practically objectionable, as difficult of comprehension for a beginner. It is at the same time worth noticing, how very little substantial difference there is in Euclid's method. For all he does is to append an additional triangle to ABC in two different quarters or directions; and with these appendages, he unscrupulously applies the triangle ABC to the triangle ACB as proposed above. The readiest way of seeing the truth of this, is to imagine BF (See the figure in the text of Prop. V.) to be diminished till it becomes evanescent. PROP. VI.-BOOK I. Cor. This Corollary may be used with advantage in the demonstration of the Proposition IV. 15. of Simson, which is to inscribe an equilateral and equiangular hexagon in a given circle. PROP. VII.-BOOK I. The sides terminated in the first-mentioned extremity of AB, have been made visibly equal in the figures ; which it was imagined would be advantageous to a beginner. PROP. IX.-BOOK I. The Proposition to bisect a given angle, has been moved to be Proposition XIV, for reasons stated in that place. Other alterations in numbers have been made in consequence; but to the smallest extent that could be contrived. PROP. XI.-BOOK I. The insertion of the matter of this Proposition as an Axiom, is an instance of the carelessness of the ancients on what may be called the philosophy of Geometry. It is true that Euclid does not seem to have troubled himself with the fact that two straight lines cannot have a common segment. But by taking EF equal to BC, and EH to BG, and applying C and G to F and H, the necessity of the two straight lines coinciding in order not to inclose a space, and of the point B coinciding with E in consequence of the equality of CB and FE, would have put the proof in the same train as here given. PROP. XIV.-BOOK I. This Proposition, as it stood as Prop. IX, was only a proposition to bisect an angle less than two right angles. It therefore became necessary to remove it to the other side of the Propositions numbered XII and XIII. PROP. XXI.-BOOK I. Referred to in the Proposition III. 8. of Simson, which shows that of straight lines from a point without the circle to the circumference, the least is that between the point without the circle and the diameter; &c. PROP. XXII.-BOOK I. Considerable changes have been made in the details of this Proposition, for the purpose of showing that the circles used in the construction will necessarily cut one another. And the Proposition which usually stood as Prop. XXIII has been inserted as Cor. 2, to make room for the insertion of another Proposition as XXIII, for reasons next hereafter stated. PROP. XXIII.-BOOK I. The Proposition here numbered XXIII was usually given as Prop. XII, and is removed to avoid the necessity of proving that a straight line which cuts the circumference of a circle but without passing through the centre, cuts it only in a point ; a thing necessary to the establishment of the old demonstration, but which, to be proved, would require to be presented as a Corollary to some of the Propositions in the Third Book. COR. 1. Is inserted as being of frequent practical application; and as capable of being referred to on the subject of straight lines equally distant from the centre,' in the Third Book. PROP. XXIV.BOOK I. The construction and proof of this Proposition are altered, for the purpose of showing that the point Gwill necessarily fall on the other side of BC from the point A, or AG be greater than AH; which is essential to the demonstration, and is not so clearly established by Simson in his Note to this Proposition. PROP. XXVI.-BOOK I. The truth of the First Case is apparent almost upon inspection, if the two triangles are supposed to be applied to one another, beginning with the sides that are equal in each. But to establish this application with accuracy, would involve the reintroduction of the subject of the coincidence of planes, in the same manner as in Prop. IV; and therefore it may be a question on the whole, whether the shortest way is not to follow Euclid in employing the agency of that Proposition. This Proposition is the last in which Euclid establishes the equality of triangles and their sides and angles respectively, from the equality of certain of their parts. He never notices the case of two triangles having two sides of the one equal to two sides of the other respectively, and an angle opposite to equal sides equal in both. This is probably, first, because he never found any necessity for making use of the proposition ; and secondly, because it is burthened with an additional condition, which is, that the other angle opposite to equal sides shall not be acute in one of the triangles and obluse in the other. Without this proviso, the equality does not necessarily exist, for there may often be two different triangles answering the conditions; but with it, the demonstration is easy by the help of the Propositions in the Third Book. PROP. XXVII.-BOOK I. The demonstration of this Proposition is altered, and the figure changed ; in order to avoid the representation of two straight lines with an angle in the middle of each, which was peculiarly abhorrent to beginners, and may reasonably be so to all other persons. Euclid also has omitted to say anything of the lines being in the same plane ; which is necessary to their being parallel. PROP. XXVIII.-BOOK I. This Proposition has been removed, in order to bring it on the hither side of that part of the question of parallel lines which has given so much trouble to geometers. Which clearly ought to be done where practicable. To preserve the numbers of the Propositions as nearly as possible, the Proposition usually numbered XXVIII has been added as a Corollary to Prop. XXVII, which it really is; and the two theorems which have usually been joined under the title of Prop. XXXII, have been separated. PROP. XXX-BOOK I. This Proposition has hitherto always been demonstrated in the case which needed no demonstration. For if two straight lines are each parallel to a straight line that is between them ; because they can never meet the intermediate straight line, they can never meet one another. This Proposition is referred to, in the Proposition numbered XLV of the First Book by Simson, but omitted in this work for the purpose of removing to the Sixth Book as hereafter noted. PROP. XXXII.-BOOK I. COR. 2. Referred to in the Appendix, on the question of Parallel Lines. If any of the angles of the figure are re-entering or greater than two right angles, the assertion in the Corollary becomes inapplicable altogether. (From Lardner's Euclid.) PROP. XXXIII.-BOOK I. This Proposition and the next following it have been transposed, for a supposed improvement in the order. COR. 1. From Lardner's Euclid. The division of a given straight line into any assigned number of equal parts, has generally been effected by means of the principle of proportionals in the Sixth Book (VI. 9. of Simson). But as it can be performed upon the earlier principle, it ought to be. PROP. XXXIV bis.-BOOK I. This Proposition, in addition to its use in throwing light on the properties of quadrilateral figures, is noticeable as containing the principle of the parallel ruler. PROP. XXXV.-BOOK I. COR. Inserted as being the foundation of the mensuration of the area of parallelograms in general. PROP. XXXIX.-BOOK I. What has usually been given as the demonstration, does not provide for the case where the straight line drawn through the point A parallel to the base, should not fall within the straight line that joins the vertices of the triangles. PROP. XL.-BOOK I. What has usually been given as the demonstration, has the same defect as is pointed out in the preceding Proposition; and further, it was not shown that AG being prolonged will of necessity meet DE, and not fall upon EB at some point between E and B. This Proposition does not appear to be ever afterwards referred |