ABC, ABE, are together equal to two right angles; but the angles LBC, ABD, are likewise together equal to two right angles; therefore the angles CBA, ABE, are equal to the angles CBA, ABD; take away the common angle ABC, the remaining angle, ABE is equal to the remaining angle ABD, the less to the greater, which is impossible; therefore BE is not in the same straight line with BC. And, in like manner, it may be demonstrated that no other can be in the same straight line with it but BD, which therefore is m the same straight line with CB.

XV.-If two straight lines cut one another, the vertical, or opposite, angles shall be equal.

Let the two straight lines AB, CD, cut one another in the point E; the angle AEC shall be equal to the angle DEB, and CEB to AED.

Because the straight line AE makes with CD the angles CEA, AED, these angles are together equal to two right angles. Again, because

the straight line DE makes with AB

the angles AED, DEB, these also are A E
together equal to two right angles;
and CEA, AED, have been shown to

B

be equal to two right angles; wherefore the angles CEA, AED, are equal to AED, DEB. Take away the common angle AED, and the remaining angle CEA is equal to DEB. In the same manner it can be demonstrated that the angles CEB, AED, are equal.

COROLLARY 1.-From this it is manifest that, if two straight lines cut one another, the angles they make at the point where they cut each other are together equal to four right angles.

COROLLARY 2.-And consequently that all the angles made by any number of lines, meeting in one point, are together equal to four right angles.

W. STEWART & Co., Educational Publishers, Holborn Viaduct Steps, London, E.C.

Constable's Educational Series.

THE EDUCATIVE READING BOOKS.

By SIMON S. LAURIE, M.A., Professor of Education in the University of Edinburgh.

SCHOOL BOARD EDITION.

Adopted by the London and the leading Provincial School Boards.

A

CHEAPER EDITION of these highly-approved READING BOOKS, excluding such Religious Lessons as are objected to by School Boards, and carefully adapted to the requirements of the New Code, has just been issued.

"The best of all the many Series of Reading Books when the Educating of the Pupil is kept in view, as contra-distinguished from the mere learning to Read."

EDINBURGH AND GLASGOW: J. MENZIES & Co.

By W. STEWART ROSS.

History of England to the Accession of Henry VII. for Standards IV. and V., New Code. Cloth, price 6d.

History of England from the Accession of Henry VII. for Standard VI., New Code. Cloth, price 6d.

Multum in Parvo History of Scotland, by W. Stewart Ross, being specially adapted to the requirements of examinations, New Code (Scotland), and containing an "Outline of the History of Great Britain from the Union of the Crowns," as required by Standard V. Extra subject. Crown 8vo, strongly bound in cloth, 4d.

Undoubtedly the most compendious School History yet published."

Outlines of British History, from the Union to the Death of George III., for New Code (Scotland), Standards V. and VI. Cloth, price 6d.

Comprehensive History of England, from the earliest period

to the present day, with marginal dates, sketches of Constitutional and Colonial History, &c. By W. Stewart Ross. Cloth, price ls. "The best Multum in Parvo of English History we have yet seen."-School Board Chronicle.

Stewart's Biographical History of the British Empire, chronologically arranged in the successive reigns, and containing a succinct notice of each person of note from the Saxon Invasion to the present day. By W. Stewart Ross (University of Glasgow), Author of "History of Scotland," &c. Price 9d.

"A sequel to all Histories, and most invaluable to those preparing for Examinations." Stewart's First History of England, with Questions. Price 4d. Stewart's Multum in Parvo History of England, based upon

the Papers set for Pupil Teachers', Middle Class Local, College of Preceptors', and Civil Service Examinations. By W. Stewart Ross, Author of "Last Century of British History," &c. Price 4d. "A perfect marvel of judicious condensation."

Stewart's History of English Literature, containing Biographical and Critical Sketches of Authors from the Saxon Period to the Present Time. By W. Stewart Ross. Price 18. 6d.

Specimens free by post for published price in stamps.

LONDON: W. STEWART & Co.. THE HOLBORN VIADUCT STEPS, E. C.

EUCLID.

SECOND STAGE.

EUCLID.-Book I.

LONDON:

W. STEWART AND CO., HOLBORN VIADUCT STEPS, E.C EDINBURGH AND GLASGOW: J. MENZIES AND CO.