OF EUCL I D: BOOKS IV., V., VI. BY THE REV. JOSEPH A. GALBRAITH, M. A., FELLOW OF TRINITY COLLEGE, AND ERASMUS SMITH'S PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY AND THE REV. SAMUEL HAUGHTON, F. R. S., FELLOW OF TRINITY COLLEGE, AND PROFESSOR OF GEOLOGY IN THE UNIVERSITY OF DUBLIN. LONGMAN, BROWN, GREEN, LONGMANS, & ROBERTS. 1859. 183. C. 11. ELEMENTS OF EUCLID. PREFACE. In the following Edition of the Fourth, Fifth, and Sixth Books of Euclid, we have restored the pure text of Euclid, particularly in the Fifth Book, which had become so overladen by the rubbish heaped upon it by commentators as to be unrecognisable as the work of Euclid. We have added to it an Algebraical Commentary. To the Fourth and Sixth Books we have subjoined Appendices, which we hope will be found of use by the learner; and to the whole we have appended a Geometrical Gymnasium, in which the young geometer may practise himself in solving the most useful and elegant, of the geometrical theorems in use in the Universities of Dublin and Cambridge. The reader may rely on the matter in large type being the genuine production of Euclid; what is printed in small type is either our own, or borrowed from the recognised commentators on Euclid. TRINITY COLLEge, Dublin, November, 1858. Ὅρος τ-Σχήμα δὲ εὐθύγραμμον περὶ κύκλον περιγράφεσθαι λέγεται, ὅταν ἑκάστη πλευρά του περιγραφομένου ἐφάπτηται τῆς τοῦ κύκλου περιφερείας. DEFINITION 4-A rectilinear figure is said to be circumscribed about a circle when all its sides touch the periphery of the circle. In the figure already given the larger square is said to be circumscribed about the circle. Ορος ε'.--Κύκλος δὲ εἰς σχῆμα ὁμοίως λέγεται ἐγγράφεσθαι, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης πλευρᾶς τοῦ εἰς ὃ ἐγγράφεται ἄπτηται. DEFINITION 5.—In like mann inner a circle is said to be inscribed in a rectilinear figure when the periphery of the circle touches all the sides of the figure. In the figure already given the circle is said to be inscribed in the greater square. "Ορος σ'.—Κύκλος δὲ περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης γωνίας τοῦ περὶ ὃ περιγρά φεται ἅπτηται. DEFINITION 6.-A circle is said to be circumscribed about a figure when the periphery of the circle passes through all the angles of the figure. Thus the circle said, in the preceding figure, to be circumscribed about the lesser square. "Ορος ζ'.—Εὐθεῖα εἰς κύκλον ἐναρμόζεσθαι λέγεται, ὅταν τὰ πέρατα αὐτῆς ἐπὶ τῆς περιφέρειας ἢ τοῦ κύκλου. DEFINITION 7.-A right line AB is said to be applied in a circle when its extremities rest on the periphery of the circle. B PROPOSITION I.-PROBLEM. Προτασις ά.—Εἰς τὸν δοθέντα κύκλον τῇ δοθέισῃ εὐθεία, μὴ μείζονι οὔση τῆς τοῦ κύκλου διαμέτρου, ἴσην εὐθεῖαν ἐναρμόσαι. In a given circle BCA to apply a given right line D, which is not greater than the diameter of the circle. Draw a diameter AB of the circle; and if this be equal to the given line D, the problem is solved. Construction.-If not, take in it the segment AE equal to D (I. 3); from the centre A with the radius AE describe a B D circle ECF, and draw to either intersection of it with the given circle the line AC. Proof. This line is equal to AE by construction, and therefore to the given line D. PROPOSITION II.-PROBLEM. Προτασις β'.—Εἰς τὸν δοθέντα κύκλον τῷ δοθέντι τριγώνῳ ἰσογώνιον τρίγωνον ἐγγράψαι. In a given circle BCA to inscribe a triangle equiangular to a given triangle EDF. H Construction.-Draw the line GH a tangent to the given circle in any point A; at the point A with the line AH make the angle HAC equal to the angle E, and at the same point with the line AG make the angle GAB equal to the B angle F; and draw BC. Proof. Because the angle D E F C E is equal to HAC by construction, and HAC is equal to the angle B in the alternate segment (III. 32), the |