AND COLLEGIATE STUDENTS, THE NATIONAL TEACHERS, EDITED BY P. M. EGAN, Author of the famous "TASK BOOK SERIES," for Schools; "AGRICULTURAL HOME for Collective Teaching, etc., etc. KILKENNY-P. M. EGAN, High Street. DUBLIN A. E. CHAMNEY, 4, Lower Ormond Quay. 1883. PREFACE. THE difficulties attending Examinations in Euclid are admitted to be every day increasing. We have in various Programmes certain portions of the subject specified, but the Examiners, to the great perplexity of Students, never limit themselves to the prescribed pages of the text, rather making it an apparent hobby to puzzle the Scholar upon the broad and seeming unlimited field for exercises on the subject. It is to supply the want for a book which will help the Student to cope with this serious obstacle to promotion that the present work is published. With that view the number of Exercises here given is by far the largest which has hitherto been published in any country upon the Third, Fourth, Fifth, and Sixth Books of Euclid, so as to give the Student a complete mastery over the extensive range which Examiners are wont to travel. Every Exercise worthy of note proposed for solution in any important work upon the subject, or which has been given at Competitive Examinations, will be found in these pages, the solution of some of which we presume to be the clearest and most scientific that has yet appeared, many being worked in two or more ways so as to give the Student a selection, and, at the same time, aid his powers of comparison and invention. It will be observed that originality is a feature of the work for which credit may be taken, as some of the most difficult. Exercises are for the first time divested of the cumbrous machinery of construction in which they are usually clouded, and are here demonstrated in a style which, we trust, will at once recommend itself for simplicity and originality. In the selection, those Exercises which may be considered as the essence of Euclid are all accorded a place; while the arrangement has been to place them, as far as consistent, in the order of their comparative difficulty, at the same time grouping propositions in families so as to afford greater facilities for acquiring the analysis of the subject, and to help the memory. The separation of the Cuts from the Text, and placing them together in convenient plates, has been approved by eminent educationists, it being obvious that more scope will be thereby given for attempting the questions as Test Papers without the help which would necessarily follow if the Cuts were in the same page with the enunciations. P. M. E. If two circles cut each other, the line joining the points of intersection is bisected at right angles by the line joining their centres. (Fig. 1, Plate I.)-Let ABR, ABL be two circles intersecting in the points A and B; from the centres M and N draw MK, NK to the point of bisection of the common chord AB. In the circle ABR, because the line MK is drawn from the centre to the middle point of the chord AB, it therefore cuts it at right angles (III. 3); for a similar reason NK bisects AB at right angles, and .. MK, NK are in one and the same straight line (I. 14). EXERCISE 2. AB is the diameter and C the centre of a semicircle: show that K the centre of any circle inscribed in the semicircle is equidistant from C and from the tangent to the semicircle parallel to AB. (Fig. 2, Plate 1.)-Produce DK to G, and join CF. Because the circles touch, the straight line joining their centres C and K passes through their point of contact E (III. 11); and since CF-CE, and CF= DG, .. CE- DG; but DK = EK, and, taking these equals from the equals CE and DG, the remainder CK = the remainder GK. EXERCISE 3. Through a given point within a circle, to draw a chord which shall be bisected in that point, and prove it to be the least. (Fig. 3, Plate I.)-Let N be the centre, and ANB the |