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EUCLID EXAMINATION PAPERS,
FOUR HUNDRED STANDARD EXERCISES
ON EUC. (I.-IV.),
SET BY THE
Highest Mathematical Authorities,
RECENT GOVERNMENT EXAMINATIONS.
The Author of " Deductions from Euclid and How to
Work Them," " Matriculation Mathematics,” etc.
MOFFATT & PAIGE, 28, WARWICK LANE, PATERNOSTER Row.
J. T. AMNER, 127, LAUSANNE ROAD, PECKHAM, S.E.
Model Solutions-at full length, with diagrams of
numerous exercises on Euc. (Books I.-IV.), similar to
those contained in this book, will be found in “ Mathews'
Deductions from Euclid, and How to Work Them," of which
the exercises herein contained form the appendices.
London, Fanuary, 1879.
DEDUCTIONS FROM EUCLID.
REMARKS ON THE WORKING OF DEDUCTIONS.
BEFORE attempting to solve a deduction, it is absolutely necessary for the student to know thoroughly the Propositions of Euclid, their classification, and the relation which one bears to another. Should he fail to attend to this, he will be placed in the same position as a child attempting to work arithmetical problems without having first obtained a complete and satisfactory knowledge of the elementary rules of arithmetic.
Deductions may be given either in the form of Problems or Theorems.
I. A PROBLEM is something proposed to be done, e.g., the construction of a figure.
Every problem consists of two parts :-
It should also be noticed here that problems are either determinate' or 'indeterminate,' the former admitting of only one distinct proof, whereas the latter may be solved in different ways.
HINTS FOR THE SOLUTION OF PROBLEMS.
(a) Read carefully the problem given.
(6) Draw the necessary figure exactly in accordance with the enunciation.
(c) As every problem must have some dependence on the fundamental principles of geometry, a careful examination of the lines, angles, triangles, etc., used—as the case may bewill show the relation it bears to some known proposition in Euclid.
(d) In the case of indeterminate problems, the construction supplied by the data will be found insufficient. In such a case, add a secondary construction which will admit of the use of that proposition uppermost in the mind, by which the required quaesita may be obtained.
(e) Sometimes problems are purposely given susceptible of various meanings. In considering such, the student is advised to select one of these meanings, and trace it out to its legitimate end. By this means the inadmissible results may be eliminated, and the correct one found.
(f) The student should remember that the mastery of one difficulty leads to the mastery of others, and each attempt, successful or unsuccessful, will tend to add to his proficiency. He should not be discouraged if he does not readily obtain the required result, as, casually, others may be derived which will be of use in the future.
II. A THEOREM makes an assertion, the truth of which it proposes to prove.
Every theorem consists of a subject, or hypothesis, and a predicate, or conclusion, requiring demonstration.
Theorems are divided into two classes, direct and indirect.
When the proof of a proposition is directly inferred from the premisses naturally arising by a series of successive steps from the hypothesis, the theorem is said to be direct, e.g., Bk. I., Prop. 4.
When a proposition is proved to be true, by showing that any supposition to the contrary would lead to what logicians call a reductio ad absurdum, it is an instance of an indirect theorem, e.g., Bk. I., Prop. 6.
Deductions based upon theorems will, as a rule, test the powers of the student to a greater degree than those which
are problematical. Hence greater attention should be given to exercises of this class.
HINTS FOR PROVING THEOREMS.
(a) A theorem is generally proved by assuming the truth of its predicate, and deducing therefrom, by the aid of accepted geometrical truths, the successive steps necessary to obtain the hypothesis first assumed.
(6) The student should remember that the method of demonstrating theorems by Euclid is always based on some fixed principle previously proved or taken for granted; hence the importance of summarising his knowledge of Euclid in such a manner, that it can be readily applied to any hypothesis requiring demonstration.
(c) The student must be cautioned against false reasoning. A common error he will be likely to fall into is that of omitting essential parts of the premisses, e.g.,
All AS are contained by st. lines.
:: Parallelograms are As. Here we have an incomplete premiss, viz., “ All A8 are contained by st. lines,” instead of three st. lines, by omitting which we derive the above false deduction.
Again, the student is liable to error by neglecting suppressed premisses. Thus, he has no right to make use of such expressions as the following, Bk. I., Prop. 15
:: A is the centre of the circle BCD,
:: The st. line AB the st. line AC, without showing that he knows the suppressed premiss (Def. 15) by which the above conclusion can only be correctly obtained.
(d) After assuming the truth of the theorem, he should analyse the consequences arising therefrom. If these be found logically consistent and mathematically correct, he has arrived at a true conclusion; if inconsistent, a false one.