And these are adjacent angles; but when one straight line falling on another straight line makes the adjacent angles equal, each of them is a right angle, and the straight line which stands on the other is called a perpendicular to it (Def. 10).

Wherefore CH is perpendicular to A B, and it is drawn from the point C.

Q.E.F.

CONCLUSION.

The writer has now completed the task which he undertook, viz., to give a familiar explanation of the first twelve propositions of Euclid. His object all through has been to teach beginners how to learn Euclid. The aim of his explanations and observations has been throughout to prevent Euclid being, to the reader, merely a book of words, and to make it to him a book of sound and solid meaning.

He hopes that the student will go through the propositions that follow successfully and with benefit to himself, by following the method of learning them here pointed out. Namely:

First. By fixing in his mind what is given, and what has to be done or to be proved.

Secondly. By observing the construction by which Euclid professes to effect what he wants to effect; building up his figure according to Euclid's directions.

Thirdly. By following out the demonstration in order to discover if Euclid therein shows that the construction has effected what he says it has.

He would even recommend the learner to do more than this. He would advise him, in his further progress through the First Book, to carry on the method of reading Euclid suggested by the writer's treatment of the last four propositions.

When the student has mastered the enunciation and construction, let him close his book and try whether he can make out for himself the proof that the construction has effected what was required.

If he cannot succeed without help, then let him note all the references, and observe the order in which they succeed each other in Euclid's demonstration. These references

will afford suggestive hints analogous to those given in the preceding pages, in order to encourage and help the learner to make out for himself the demonstrations of the Ninth, Tenth, Eleventh, and Twelfth Propositions, before reading over the demonstrations as given in Euclid.

When Euclid is thus read it becomes much more interesting. It has all the entertainment of solving riddles. Riddles, indeed, of a very high order are Euclid's problems and theorems.

This method has also the effect of fixing the propositions in the learner's mind far more permanently than if he merely reads through Euclid's demonstration, so as to follow his meaning, but without exerting independent thought.

Still, the learner must be cautioned while reading Euclid in this way not to hurry on too much, but to compare his demonstration with Euclid's, and to write it out in the best form with attention and care.

It is related in Sir David Brewster's 'Life of Sir Isaac Newton,' that, having purchased a copy of Euclid, soon after entering at Trinity College, he examined the problems, and found the truths which they enunciated so self-evident, that he expressed his astonishment that any person should have taken the trouble of writing a demonstration of them. He therefore threw aside Euclid as a 'trifling book,' and set himself to the study of Descartes' Geometry. The neglect which he had shown of the elementary truths of geometry he afterwards regarded as a mistake in his mathematical studies, and he expressed to Dr. Pemberton his regret that he had applied himself to the works of Descartes and other algebraical writers before he had considered the elements of Euclid with that attention that so excellent a writer deserved.'-Sir Isaac Newton's words as given in Pemberton's view of his philosophy.

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EXPLANATION OF THE FRONTISPIECE.

Obs. The pegs from which the chains hang may be considered to represent the axioms on which the propositions depend.

1. The First Proposition is the first link of the chain.

2. The Second Proposition, depending on the First, is the second link.

3. The Third Proposition, depending directly on the Second, and through the Second on the First, is the next link.

4. These three propositions are represented by smaller links, they being propositions subsidiary to those of the main chain of propositions, which starts from the Fourth Proposition as from a fresh beginning.

5. Upon the Fourth hangs the Fifth Proposition; therefore the link (5) in the main chain is suspended from (4).

6. Upon the Fourth hangs also the Sixth Proposition; but no proposition among the first Twelve depends upon the Sixth, and so no link hangs from that numbered (6).

Obs. In the construction of the Fifth and Sixth Propositions it is required from the greater of two straight lines to cut off a part equal to the less (Prop. III.). Therefore a second and special link connects those numbered (5) and (6) with the one numbered (3).

7. The Seventh Proposition depends on the Fifth; therefore link (7) is hung on (5),

8. On the Seventh Proposition depends the Eighth; therefore link (8) is hung on (7).

9. The Eighth Proposition enables us to solve the Ninth; therefore link (9) hangs from (8).

10. For the construction of the Tenth Proposition the Ninth is required; therefore link (10) is hung from (9); but as it is demonstrated by the Fourth, a special link in the chain connects (4) and (10).

11. The Eleventh Proposition depends on the Eighth, therefore link (11) is hung from link (8); but as the construction requires that from the greater of two given straight lines a part be cut off equal to the less (Prop. III.), a special link connects those marked (11) and (3).

12. In the construction of the Twelfth, it is necessary to bisect a straight line; therefore, in the chain (12) is linked to (10). It is demonstrated by the Eighth; therefore a special link connects those marked (8) and (12).

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QUESTIONS ON THE FOREGOING TREATISE.

To those learners who find that the explanations and questions are more than they require for a thorough understanding of Euclid's reasoning, the writer has but to repeat what he said in the preface to the first edition. Let learners do with these explanations and questions as they did with their swimming-belts when learning to swim. As soon as each one found he could swim alone, he flung away his swimmingbelt, and dashed off into deep waters. So let him do with these explanations and questions. When he feels that he has learned how to learn Euclid, let him throw these helps aside, and take to his Euclid. The glorious six books of Euclid will afford him plenty of deep water to revel in.

INTRODUCTORY.

1. How long is it since Euclid wrote his treatise on geometry?

2. In what language did he write it?

3. And where?

4. What are the parts of Euclid commonly read in schools in England?

5. Distinguish between plane and solid geometry.

6. Which books of Euclid treat of plane geometry, and which of solid geometry?

On Points, Lines, Surfaces.

7. How many dimensions must a body have, which can be held in the hand?

8. What are they called?

9. Suppose the body is a wooden brick, when its thickness has been taken away, what two dimensions yet remain ?

10. And what name does Euclid give to that which has the two remaining dimensions after the thickness has been taken away?

11. Now suppose the breadth to be also taken away, what name does Euclid give to that which has the single remaining dimension of length ?

12. Suppose now the remaining dimension of length taken away, what name does Euclid give to that corner dot, the remains of the wooden brick, after its thickness, breadth, and length have been successively taken away?