and the result will be thus carried back into direct contradiction to ̈ the original hypothesis. It may sometimes happen that our attempts thus to analyse a theorem may be carried on through a considerable number of successive steps, and yet no conclusive evidence of the truth or falsehood of the alleged theorem present itself. Nor can we ever judge, a priori, whether we should succeed by continuing the process further in any one particular direction. Under one aspect this may be considered an inconvenience; but even were it a real inconvenience, it is inevitable, and must so far be taken as a drawback upon the value of the method. The inconvenience is, however, more apparent than real; or, at least, the inconvenience is amply compensated by the advantages it otherwise confers, not indeed in reference to the demonstration of the proposed theorem, but in its extension of geometrical discovery. A mistake might occur in the synthetic deduction of a proposed theorem, or the theorem might be a mistaken inference from analogy, or from the contemplation of carefully drawn diagrams; but it does not often happen that a theorem is proposed for solution, of the truth of which the proposer has not satisfied himself. The probabilities then are greatly in favour of such proposition being correct. Now in this case, all the investigations which have been made with a view to the analysis of that theorem, will become so many synthetic demonstrations of the results which have been obtained during those unsuccessful attempts to analyse. It will in general be found, too, that they are of such a character as would scarcely have occurred to any Geometer to adopt with pure reference to synthetic purposes. There can, in fact, be little doubt that the greater part of the most profound and original theorems that are found in the writings of the greatest Geometers of ancient and of modern times, have originated in attempts to analyse some proposed theorem; and which have failed merely from the direction which was pursued, lying in that of the more recondite instead of the more simple order of truths connected with the proposed one. Such failures should therefore be always carefully preserved, till the proposition itself, from which they were deduced, be proved either to be false or true. If the course of analysing pursued in the first instance be not found to succeed, and if the conclusions become more and more elementary in their character, some other properties of the figure connected with the assumed truth should be tried in the same manner; and if this also fail to accomplish the immediate object, the investigations should be pursued as before. It has occasionally happened, though extremely seldom, that several such attempts have failed in succession. Yet some mode of deduction must necessarily become a true analysis of the theorem; which will in general result from adequate perseverance. All the results ob-. tained in the preceding efforts to analyse the theorem, will then constitute a number of truths, connected with each other through that one by which they were originally suggested; and often among truths so related, a general principle may be detected, that shall prove of the utmost value in the treatment of entire classes of propositions, which now stand in an uninteresting state of isolation from each other. Moreover, by systematising the propositions of Geometry, we simplify their didactic development; and by contemplating in connexion with each other, such attempted analyses of single theorems, great benefits may be conferred upon Geometrical Science and its practical applications. There is not the slightest difference between analysis and synthesis, as far as the course of consecutive deduction is concerned. Both are direct applications of the ordinary enthymeme; and both require the same specific habits of mind, and the same resources as regards truths already known. The only distinction consists, as far as mere reasoning is concerned, in the difference of the starting points of the investigation. In Analysis we start from the enunciated property as a truth temporarily admitted; and ultimately arrive at some property which we previously knew to be true of the hypothetical figure. We have only to reverse the order of the Syllogisms, and of the subject and predicate in each of them, to convert the analysis into the synthesis in one case, or the synthesis into the analysis in the other. They are so connected, in fact, that had the hypothesis of the proposed theorem been already proved by one process; the reversed process, the analysis of which we have spoken, would have become the synthesis of the theorem. It must now be obvious that the synthesis of the theorem can be at once formed from the analysis, by the reversal of the steps already described, and when this has been effected, the analysis may, if desirable, be altogether suppressed. On the other hand, for all the purposes of giving full and legitimate conviction of the truth of a theorem, the analysis is always sufficient, without adding the synthesis. It is, however, desirable, in a course of Geometrical study, to complete the formal draft of the investigation both in the analytic and synthetic form. THE ANALYSIS OF PROBLEMS. IN every Geometrical operation we perform in the construction of a Problem, we have in mind some precedent reason,--a knowledge of some properties of the figure, either assumed or proved, which would result from that operation, and a perception of its tendency towards accomplishing the object proposed in the Problem. Our processes for construction are founded on our knowledge of the properties of the figure, supposed to exist already, subjected to the conditions which are enunciated in the proposition itself. No Problem could be constructed (except by mere trial, and verification by mere instrumental experiments) antecedently to the admission of our knowledge of some properties of the figure which it is proposed to construct. The simple reason for the operations employed," is, that they collectively and ultimately fulfil the prescribed conditions; and their so fulfilling the conditions, is only known by previously reasoning upon the figure supposed already to be so constructed as to embody those conditions. Let any Problem be selected from Euclid, and at each step of the operation, let the question be asked, "Why that step is taken ?" It will in all cases be found that it is because of some known property of the figure required, either in its complete or intermediate states, of which the inventor of the construction must have been in possession. This antecedency of Theorems to all Geometrical construction in Scientific Geometry is universal and essential to its nature. Let the construction of Euc. v. 10 be taken in illustration of what has been stated. There are five operations specified in the construction: 1. Take any line AB. 2. Divide that line in C, so that, &c. 3. Describe the circle BDE with center A and radius AB. 4. Place BD in that circle, equal to AC. 5. Join the points A, D. Why should these operations be performed in order rather than any others? And what clue have we to enable us to foresee that the result of them will be such a triangle as was required? The demonstration affixed to the Proposition by Euclid, does undoubtedly prove that these operations must, in conjunction, produce such a triangle: but we are furnished in the Elements with no obvious reason for the adoption of these steps, unless we suppose them accidental. To suppose that all the constructions, even the simple ones, were the result of accident only, would be supposing more than could be shewn to be admissible. No construction of the problem could have been devised without a previous knowledge of some of the properties of the figure which was to be constituted. In fact, in directing the figure to be constructed, we assume the possibility of its existence; and we study the properties of such a figure on the hypothesis of its actual existence. It is this study of the properties of the figure that constitutes the Analysis of the Problem. Let then the existence of a triangle BAD (fig. Euc. IV. 10) be admitted which has each of the angles ABD, ADB double of the angle BAD, in order to ascertain any properties it may possess which would assist in the actual construction of such a triangle. Then, since the angle ADB is double of BAD, if we draw a line DC to bisect ADB and meet AB in C, the angle ADC will be equal to CAD; and hence (Euc. 1. 6) the side AC, is equal to CD. Again, there are three points A, C, D, not in the same straight line, let us examine the effect of describing a circle through them: that is, describe the circle ACD about the triangle ACD (Euc. IV. 5). Then, since the angle ADB has been bisected by DC, and since ADB is double of DAB, the angle CDB is equal to the angle DAC in the alternate segment of the circle; the line BD therefore coincides with a tangent to the circle at D (converse of Euc. II. 32). Whence it follows that the rectangle contained by AB, BC, is equal to the square on BD (Euc. 111. 36). But the angle BCD is equal to the two interior opposite angles CAD, CDA; or since these are equal to each other; BCD is the double of CAD, that is of BAD. And since ABD is also double of BAD, by the conditions of the triangle, the angles BCD, CBD are equal, 'and BD is equal to DC, that is, to AC. It has been proved that the rectangle AB, BC, is equal to the square on BD; and hence the point C in AB, found by the intersection of the bisecting line DC, is such, that the rectangle AB, BC is equal to the square on AC (Euc. II. 11). Finally, since the triangle ABD is isosceles, having each of the angles ABD, ADB double of BAD, the sides AB, AD are equal, and hence the points B, D, are in the circumference of the circle described about A with the radius AB. And since the magnitude of the triangle is not specified, the line AB may be of any length whatever. From this "Analysis of the Problem," which obviously is nothing more than an examination of the properties of such a figure supposed to exist already, it will be at once apparent, why those steps which are prescribed by Euclid for its construction, were adopted. The line AB is taken of any length, because the problem does not prescribe any specific magnitude to any of the sides of the triangle: the circle BDE is described about A with the distance AB, because the triangle is to be isosceles, having AB for one side, and therefore the other extremity of the base is in the circumference of that circle: the 1 line AB is divided in C so that the rectangle AB, BC shall be equal to the square on AC, because the base of the triangle must be equal to the segment AC: and the line AD is drawn, because it completes the triangle, two of whose sides AB, BD are already drawn. A careful examination of this process will point out the true character of the method by which the construction of all problems (except perhaps a few simple ones which involve but very few and very obvious steps) have been invented: although the actual analysis itself has been suppressed or concealed, as, amongst the ancient Geometers, appears to have been the general practice. It will be inferred at once, that the use of the Analysis in reference to the construction of problems, is altogether indispensable in its actual form, where the problem requires several steps for its construction; as it has been shown to be virtually (though the operations may in certain simple problems be carried on mentally and almost unsuspectedly) essential to the construction of all problems whatever. When we have reduced the construction to depend upon problems which have been already constructed, our analysis may be terminated; as was the case when, in the preceding example, we arrived at the division of the line AB in C; this problem having been already constructed as the eleventh of the second book. From the nature of the subject, it must be at once obvious that no general rules can be prescribed which will be found applicable to all cases, and lead to the solution of every problem. The conditions of problems must suggest what constructions may be possible; and the consequences which follow from these constructions and the assumed solution, will shew the possibility or impossibility of arriving at some known property consistent with the data of the Problem. In the following exercises, many will be found to be of so simple a character, (being obvious deductions from the Elements) as scarcely to admit of the principle of the Geometrical Analysis being applied, in their solution. A clear and exact knowledge of first principles must necessarily precede any intelligent application of them. Indistinctness or defectiveness of understanding with respect to these, will be a perpetual source of error and confusion. The learner is therefore recommended to understand the principles of the Science, and their connexion, clearly, before he attempt the application of them. The following directions may assist him in his proceedings. 1. In general, any given problem will be found to depend on several problems and theorems, and these ultimately on some problem or theorem in Euclid. 2. Describe the diagram as directed in the enunciation, and suppose the solution of the problem effected. 3. Examine the relations of the lines, angles, triangles, &c. in the diagram, and find the dependence of the assumed solution on some theorem or problem in the Elements. 4. If such cannot be found, draw other lines parallel or perpendicular as the case may require, join given points, or points assumed in the solution, and describe circles if need be: and then proceed to trace the dependence of the assumed solution on some theorem or problem in Euclid. 5. Let not the first unsuccessful attempts at the solution of a Problem be considered as of no value; such attempts may lead to the solution of other problems, or to the discovery of new geometrical truths. GEOMETRICAL EXERCISES ON BOOK I. PROPOSITION I. PROBLEM. To trisect a given straight line. ANALYSIS. Let AB be the given straight line, and suppose it divided into three equal parts in the points D, E. On DE describe an equilateral triangle DEF, Then because AD is equal to DF, therefore the angle AFD is equal to the angle DAF, and the two angles DĂF, DFA are double of one of them DAF. But the angle FDE is equal to the angles DAF, DFA, and the angle FDE is equal to DAC, each being an angle of an equilateral triangle; therefore the angle DAC is double the angle DAF; wherefore the angle DAC is bisected by AF. Also because the angle FAC is equal to the angle FAD, therefore the angle CAF is equal to the alternate angle AFD: Synthesis. Upon AB describe an equilateral triangle ABC, bisect the angles at A and B by the straight lines AF, BF, meeting in F; through Fdraw FD parallel to AC, and FE parallel to BC. Then AB is trisected in the points D, E. For since AC is parallel to FD, and FA meets them, But the angle FDE is equal to the angle CAB, therefore the remaining angle DFE is equal to the remaining angle, ACB. Hence the three sides of the triangle DFE, are equal to one another, and DF has been shown to be equal to DA; therefore AD, DE, EB are equal to one another. Hence the following theorem : If the angles at the base of an equilateral triangle be bisected by two lines which meet at a point within the triangle; the two lines drawn from this point parallel to the sides of the triangle, divide the base into three equal parts. |