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OF

E UCL ID:

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

IN THE UNIVERSITY OF DUBLIN ;

AND

THE REV. SAMUEL HAUGHTON, F. R. S.,

FELLOW OF TRINITY COLLEGE,
AND PROFRSSOR OF GEOLOGY IN THE UNIVERSITY OF DUBLIN.

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LONDON:
LONGMAN, BROWN, GREEN, LONGMANS, & ROBERTS.

1859.

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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.

Όρος 7.-Σχήμα δε ευθύγραμμον περί κύκλον περιγράφεσθαι λίγεται, όταν εκάστη πλευρά του περιγραφομένου εφάπτηται της του 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 εί-Κύκλος δε εις σχήμα ομοίως λέγεται εγγράφεσθαι, όταν ή του κύκλου περιφέρεια έκάστης πλευράς του εις και εγγράφεται άπτηται.

DEFINITION 5.-In like manner 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.

"Όρος σ'.-Κύκλος δε περί σχήμα περιγράφεσθαι λέγεται, όταν ή του κύκλου περιφέρεια εκάστης γωνίας του περί και περιγράφεται άπτηται.

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 is said, in the preceding figure, to be circumscribed about the lesser square.

"Ορος ζ'.-Ευθεία εις κύκλον έναρμόζεσθαι λέγεται, όταν τα πέρατα αυτής επί της περιφέρειας και του 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.

В.

Προτασις ά.-Εις τον δοθέντα κύκλον τη δοθειση ευθεία, μη μείζονι ούση της του κύκλου διαμέτρου, ίσην ευθείαν έναρμόσαι.

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

A in it the segment AE equal to D (I. 3); from the centre A with the radius AE describe a

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.

H

Προτασις β'.-Εις τον δοθέντα κύκλον τη δοθέντι τριγώνω ισογώνιον τρίγωνον έγγράψαι. .

In a given circle BCA to inscribe a triangle equiangular to a given triangle EDF.

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 E is equal to HAC by construction, and HAC is equal to the angle B in the alternate segment (III. 32), the

E

F

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