THEOREM 2. If the one-fiftieth of the sum of the squares of the two sides of any square be deducted therefrom, the square root of the remaining can be extracted exactly.

9

THEOREM 3. In any right-angle triangle if a perpendicular be drawn from the right angle to the base the triangles on each side are sim· ilar to the whole triangle and to one another, and the perpendicular is a mean proportional between the segments of the base, and each of the sides of the triangle is a mean proportional between the whole base and the segment adjacent to that side.

THEOREM 4. If three straight lines are proportionals the rectangle contained by the extremes is equal to the square on the mean, and if the rectangle contained by the extremes is equal to the square on the mean the three straight lines are proportionals.

THEOREM 5. Any regular polygon inscribed in a circle is a mean proportional between the inscribed and circumscribed polygons of half the number of sides.

THEOREM 6. If two straight lines cut one another the opposite or vertrical angles are equal.

THEOREM 7. Triangles which have the same altitude are to each other as their bases, and their areas are to each other as the squares described on those bases.

Or, by trigonometry,

The cosine is to the sine as the radius is to the tangent, consequently the secant is to the tangent as the radius is to the sine, therefore the square of the secant is to the square of the tangent as the square of the radius is to the square of the sine.

The second class embraces those problems which either have not been fully demonstrated, or have been given up as impossible. Respecting these Mr. Todhunter says:

"There are three famous problems which are now admitted to be beyond the power of geometry, namely: To find a straight line equal in length to the circumference of a given circle, to trisect any given angle, and to find two mean proportionals between two given straight lines. The grounds upon which the geometrical solution of these problems is admitted to be impossible can not be explained without a knowledge of the higher parts of mathematics; the student of the elements may however be content with the fact that innumerable attempts have been made to obtain solutions, and that these attempts have been made in vain.

The first of these problems is usually referred to as the quadrature of the circle. For the history of it the student should consult the article in the English Cyclopedia under that head, and also a series of papers in the Atheneum for 1863 and subsequent years, entitled A Budget of Paradoxes, by Professor DeMorgan. For the approximate solutions of the problem we may refer to Davies' edition of Hutton's Course of Mathematics, Art. I, page 400; the Lady's and Gentleman's Diary for 1855, page 86, and the Philosophical Magazine for April, 1862. The third of the three problems is often referred to as the duplication of the cube. · See note on VI, 13, in Lardner's Euclid and a dissertation by C. H. Biering, entitled Historia Problematis Cubi Duplicandi—Hauniœ, 1844.

Under the head of "Geometrical Analysis" Mr. Todhunter says: "Geometrical analysis has sometimes been described in language which might lead to the expectation that directions could be given which would enable a student to proceed to the demonstration of any proposed theorem, or the solution of any proposed problem with confidence of success; but no such directions can be given. We will state the exact extent of these directions: Suppose that a new theorem is proposed for investigation, or a new problem for trial, assume the truth of the theorem, or the solution of the problem, and deduce consequences from this assumption combined with results which have already been established. If a consequence can be deduced which contradicts some result already established, this amounts to a demonstration that our assumption is inadmissible; that is the theorem is not true, or the problem can not be solved. If a consequence can be deduced which coincides with some result already established, we can not say that the assumption is not inadmissible; and it may happen that by starting from the consequence which we deduced, and retracing our steps, we can succeed in giving a synthetical demonstration of the theorem or a solution of the problem. These directions however are very vague, because no certain rule can be prescribed by which we are to combine our assumption with results already established; and, moreover, no test exists by which we can ascertain whether a valid consequence which we have drawn from an assumption will enable us to establish the assumption itself. That a proposition may be false and yet furnish consequences which are true, can be seen from a simple example. Suppose a theorem were proposed for investigation in the following words: one angle of a triangle is to another as the side opposite to the first angle is to the side opposite to the other. If this be assumed to be true we can immediately deduce Euclid's result in I, 19; but from Euclid's result in I, 19, we can not retrace our steps and establish the proposed theorem, and in fact the proposed theorem is false.

Thus the only definite statement in the directions respecting geometrical analysis is, that if a consequence can be deduced from an assumed proposition which contradicts a result already established, that assumed proposition must be false. We may mention, in particular, that a consequence would contradict results already established if we could show that it would lead to the solution of a problem already given up as impossible."

A brief review of the above quotations will be attempted, which, for the sake of convenience, I shall classify as follows:

1st. No directions can be given in language which would enable a student to proceed to the demonstration of any proposed theorem or the solution of any proposed problem with confidence of success.

The above quotation will apply with more force to the demonstration of a new theorem or the solution of a new problem; for it would be necessary to know something of the nature of a theorem and the construction of a problem before directions could be given for the demonstration of the one or the solution of the other; but if a theorem is proposed for demonstration, or a problem for solution, which have certain parts in common with similar theorems or problems before known and demonstrated, the demonstration of such theorems and the solution of such problems would be rendered easy in the same proportion as they contained those parts in common. The mathematician, therefore, who should venture to give explicit directions for the demonstration of any theorem or the solution of any problem whatever-such for instance as the quadrature of the circle, the trisection of the angle, or the duplication of the cube-would certainly pretend to a superior knowledge concerning those things in which the ablest and wisest mathematicians have failed; for the discovery of a new method for the demonstration of the above theorems or the solution of the above problems might involve new laws which no one knows anything of except the author who makes the discovery.

2d. "Assume the truth of the problem, and deduce consequences from this assumption combined with results already established.* If a consequence can be deduced which contradicts some result already established, this amounts to a demonstration that our assumption is inadmissible; that is, the theorem is not true or the problem can not be solved."

NOTE. A proposition consists of various parts; we have first the general enunciation of the problem or theorem, as for example: To describe an equilateral triangle on a given finite straight line, or any two angles of a triangle are together less than two right angles. After the general enunciation follows the discussion of the proposition. First, the enunciation is repeated and applied to the particular figure which is to be considered, as for example: Let A B be the given straight line; it is required to describe an equilateral triangle on A B. The construction then usually follows straight lines and circles, which must be drawn in order to constitute the solution of the problem, or to furnish assistance in the demonstration of the theorem. Lastly, we have the demonstration itself, which shows that the problem has been solved or that the theorem is truė.

Sometimes however, no construction is required, and sometimes the construction and demonstration are combined. The demonstration is a process of reasoning, in which we draw infer

With reference to point 2nd, it must be confessed that so far as those problems are concerned, for which solutions have been already obtained and which are universally received as final, it may be true that an assumed proposition would be false which would contradict a result so well established and so universally admitted to be correct; but it does not follow as a necessary consequence that an assumed proposition would be false which would give a result different from the one already established with respect to those problems which are admitted to be impossible--such, for example, as the quadrature of the circle, etc.; for as long as the final solution of this problem is not obtained it is difficult to determine which solution is true and which false as long as they are confined within certain known limits. Thus it is mathematically certain that the ratio of the circumference to the diameter is either 34 exactly or very near it, and any ratio which claims to be either much above or below it can not be true. For example, suppose a circle is described with a radius equal to the square root of two, and cosine of the given arc is 7, which is not quite though very near the true radius, and the sine of the same arc is, which is not quite though very near the 4 of the true circumference, now, the double of the cosine 714, is not quite the true diameter though very near it, and the double of the sine, is not quite the part of the true circumference though very near it, being the chord of double the arc, whose sine is. If then we assume 14 to be the true diameter, and 44 to be the true circumference, and they are very near it, then dividing 44 44 the circumference by the diameter we have by cancellation 5 5

finity, so that if

44

14,

5

22

which is equal to 3.142857, 142857, or 34 to in

were the true circumference and

14
5

44 the true diam5 eter, the true ratio of the circumference to the diameter, would be

3.142857 or 34.

3rd. "These directions are very vague because no certain rule can be prescribed by which we are to combine our assumption with results already estab

ences from results already obtained. (These results consist partly of truths established in former propositions, or are admitted as obvious in commencing the subject; and partly of truths which follow from the construction that has been made, or which are given in the supposition of the proposition itself. The word hypothesis is used in the same sense as supposition.)

lished, and no test exists by which we can ascertain whether a valid consequence, which we have drawn from an assumption, will enable us to establish the assumption itself."

A theorem is a truth which becomes evident by a train of reasoning called a demonstration; the demonstration proceeds from the premises. by a regular deduction; a deduction is the logical consequences which follow an assumption; an assumption is based upon certain truths which are found in the construction of a figure, or which result from a combination of other truths. Every combination of truths and every construction of a figure necessarily contain within themselves the logical reasons for their demonstration; and if a supposition is made in conformity with the truths developed in the construction of a figure, or an assumption be made which is based upon truths which are the necessary consequence of a combination of other truths, this assumption will lead to a valid consequence which may result in the demonstration of the truth involved. But if, from a misconception of those truths, a false assumption is made, a valid consequence can not be drawn and the assumption will necessarily lead to an absurdity. It is also next to impossible to prove the truth of one method by another, or of combining our results with those already established; it is only when the two methods both contain certain principles in common that they can be compared; that is when they contain elements that may be reduced to the same denomination, to the same dimensions, the same weight, or the same measure.

儡

In reference to point 4th, it is admitted that so far as those problems are concerned which have received a final solution that is universally admitted to be correct, it may be true that an assumed proposition would be false if it should contradict a result already established; but with respect to those problems which are admitted to be impossible it does not follow that an assumed proposition would be false which contradicts a result already established; for it is impossible to know which is true, or which is false, until a final solution is obtained. Consequently point 5th may very readily be acquiesced in, for the author

says:

We may mention in particular, that a consequence would contradict a result already established if we could show that it would lead to the solution of a problem already given up as impossible."

It is asserted by apologists and advocates of the present quadrature