3. Given the base and vertical angle, to construct the triangle. Now three conditions are necessary to fix the magnitude of a triangle or a parallelogram, and in general, three only are sufficient for the purpose; but here it will be observed that only two are given in each case. The precise triangle or parallelogram, viewed as peculiarly solving the Problem, cannot be separated from all the others, except by adding some third condition to the two already given. The side of the parallelogram in (1), and the vertex of the triangle in (2), opposite to the base, may be in any position in a certain line parallel to the base; and the vertex of the triangle in (3), may be at any point in the circumference of a segment of a certain circle. The parallel line in which the vertices of all the equal triangles are situated, in one case, and the arc of the circle in which the vertices of all the triangles having equal vertical angles are situated, are, each called the locus of the vertex of the triangle, since it occupies, in each case, all the places in which that vertex may be situated so as to fulfil the required conditions. In the same way, the parallel to the base is also the locus of all the positions in which the other two angular points of the parallelogram may be situated. These Problems are the simplest instances of that class which is called Local Problems; and their peculiar character is, that the data are one less than the number of conditions required by the nature of the Problem to restrict the quæsitum to a single or specified number of cases; as in these Problems the data consist of two conditions, while the exactly defining conditions must be three. Again, viewed as Theorems, they may be thus enunciated: 1. If the base and area of a parallelogram be given, the locus of the other angular points will be a straight line parallel to the base. 2. If the base and area of a triangle be given, the locus of its vertex is a straight line parallel to the base. 3. If the base and vertical angle of a triangle be given, the locus of the vertex will be an arc of a circle. In the original form of the propositions, the entire meaning, and that justified by Euclid's own reasoning, is that which would result from saying, "all triangles," "all parallelograms," &c. It will obviously be the case here, as in the Maxima and Minima, that the proposition may be enunciated either as a local theorem or as a local problem; and the circumstances will be similar as to the comparative brevity of enunciation and difficulty of the solution, when the proposition is given in the form of a Problem. The great use made of loci by the Ancient Geometers was in the construction of determinate Problems. If a problem relate to the determination of a single point, and the data be sufficient to determine the position of that point, the problem is determinate: but if one or more of the conditions be omitted, the data which remain may be sufficient for the determination of more points than one, each of which satisfies the condition of the problem; in that case the problem is indeterminate, and in general such points are found to be situated in some line, which satisfies the conditions of the problem. A certain number of data is required according to the nature of the problem for rendering the quæsitum determinate. The subject will be better illustrated by one or two examples. For instance, the locus of the centers of all the circles whose circumferences pass through two given points, A, B, is a straight line drawn perpendicular to AB at the point D, the point of bisection of the line AB; so that an indefinite number of circles may be described having their centers in the line perpendicular to AB drawn through D, and their circumferences passing through the two given points A, B. If a third point C be taken, but not in the same straight line with the first and second points A and B. Let A, C be joined and bisected in E, then the perpendicular to AC drawn through E the point of bisection of AC, will be the locus of the centers of the circles whose circumferences pass through the two points A and C. Hence the point F, the intersection of these two perpendiculars, will be the center of that circle whose circumference passes through the three given points A, B, C (Euc. IV. 5.), and is called the intersection of the two loci. Another example may be founded on the second and third theorems already noticed, which will take the following form : Given the base, the area, and the vertical angle of a triangle, to construct it. When the base and the area of a triangle are given, the locus of its vertex is a straight line which can be determined from these data; and when the base and vertical angle are given, the locus of the vertex is a portion of the circumference of a circle which can be determined from these data. Now the point or points of intersection of these loci, will fulfil both conditions, that the triangle shall have the given area, and the given vertical angle. To express the principle generally :let there be n conditions requisite for the determination of a point which either constitutes the solution, or upon which the solution of the problem depends. Find the locus of this point subject to (n − 1) of these conditions; and again, the locus of the point subject to any other (n - 1) of these conditions. The intersection of these two loci gives the point required. It may be observed that (n-2) of the data must be the same in determining the two loci, and no one of the n data must be a consequent of, or depend upon, the remaining (n - 1) data, in other words, the n data must separately express n independent conditions. There are however cases in which one datum is involved in another, and these are of two different kinds-essential and accidental. To illustrate this distinction, let the following Problems be taken : Given one angle of a triangle a right angle, the base, and the difference of the squares on the hypotenuse and the perpendicular, to construct the triangle. Given the base, the area, and the perpendicular drawn from the vertex to the base of the triangle, to construct it. Given the base, the vertical angle and the sum of the other two angles at the base of the triangle, to construct it. Now in each of these problems, the third datum is absolutely determined and invariable, in consequence of its essential dependence on the two previous ones. This dependence is universal and essential. Again, suppose the problem were: Given the base of a triangle and a circle in magnitude and position, and likewise the vertical angle, to construct the triangle which shall have its vertex in the circumference of the given circle. In this case, the given circle will generally be a different one from that which forms the locus of the vertical angle, and in that case, the intersections, or the point of contact, of the two circles will give either two solutions or one solution of the Problem. But on the other hand, the given circle may coincide with the locus, and thus again render the Problem indeterminate in this particular case. Generally the construction is possible, and only accidentally it becomes indeterminate. The distinction between these two cases is very important. As Problems are generally constructed by the intersections of loci, it is easy to imagine cases and conditions that shall give loci which can never meet. For instance, in the problem just stated, the two circles may never meet; and in the preceding one, the straight line and circle may never meet. In all such cases a problem is impossible with given conditions, when these conditions are incompatible with each other in their nature, or in their magnitude and position, or with the coexistence of that which constitutes the quæsitum. The importance of the distinction alluded to, when one datum is contained in another, arises from its constituting the foundation of another Class of Propositions. These are called the Porisms. Whenever the quæsitum is a point, the problem on being rendered indeterminate, becomes a locus, whether the deficient datum be of the essential or of the accidental kind. When the quæsitum is a straight line or a circle, (which were the only two loci admitted into the ancient Elementary Geometry) the problem may admit of an accidentally indeterminate case; but will not invariably or even very frequently do so. This will happen when the line or circle shall be so far arbitrary in its position, as depends upon the deficiency of a single condition to fix it perfectly: that is, (for instance) one point in the line, or two points in the circle, may be determined from the given conditions, but the remaining one is indeterminate from the accidental relations among the data of the problem. Determinate Problems become indeterminate by the merging of some one datum in the results of the remaining ones. This may arise in three different ways; first, from the coincidence of two points; secondly, from that of two straight lines; and thirdly, from that of two circles. These, moreover, are the only three ways in which the accidental coincidence of data can produce this indeterminateness in the problem. pro There is a large class of indeterminate Problems which involve loci, and satisfy certain defined conditions. Every indeterminate blem containing a locus may be made to assume the form of a porism, but the converse of this does not hold. Porisms are of a more general nature than indeterminate problems which involve a locus. In conclusion, it may be observed that the Ancient Geometers appear to have undertaken the solution of Problems with a scrupulous and minute attention, which would scarcely allow any of the collateral truths to escape their observation. They never considered a Problem as solved till they had distinguished all its varieties, and evolved separately every different case that could occur, carefully distinguishing whatever change might arise in the construction from any change that was supposed to take place among the magnitudes which were given. This cautious method of proceeding would lead them to see that from circumstances in the data, the solution of some Problems would be impossible: that some would be determinate while others would be indeterminate: that some would admit · f a maximum or a minimum: that some would involve a locus: and lastly, that some would assume the form of a Porism. THE ANCIENT GEOMETRICAL ANALYSIS. THE terms Analysis and Synthesis are usually understood, the former to signify the separation of any whole thing into its constituent parts, for the purpose of examining them separately: the latter to signify the composition, or the putting together of the several parts of any thing for the purpose of constituting the whole of it. These terms used in their strict etymological sense, are not exactly in accordance with the use made of them in Geometry, where they are employed to indicate the direct and reverse order of a construction or demonstration. Synthesis, or the method of composition, is a mode of reasoning in geometry, which commences with the data of a Problem, or the hypothesis of a Theorem, and proceeding regularly by a process of construction and reasoning, ends with the quæsitum of the Problem or with the proof of the predicate of the Theorem. The Synthetic method is pursued in the demonstrations of the Propositions in Euclid's Elements: certain principles are assumed or admitted, and on these principles are founded the construction of Problems and the demonstration of Theorems. This may be termed a direct process, as it leads from principles to their consequences. Analysis, or the method of resolution, is a process, the reverse of Synthesis, which commences with assuming the quæsitum of the Problem as found, or the predicate of the Theorem as proved, and by a process of construction and reasoning, terminates in the data of the Problem or the hypothesis of the Theorem. This may be considered an indirect process, a method of reasoning from consequences to principles. Hence, "Analysis presents the medium of invention; while Synthesis naturally directs the course of instruction." The Synthesis begins where the Analysis ends; the last step of the Analysis becomes the first step of the Synthesis, and the other steps of the Analysis are retraced in order to the first, which constitutes the last step of the Synthesis. THE ANALYSIS OF THEOREMS. It may have been remarked, that in the Elements, Euclid frequently uses the indirect method of demonstration:—that is, of proving the truth of a theorem by demonstrating that a contrary conclusion is incompatible with the hypothesis of that theorem. To effect this, he supposes the enunciated property to be false; and its contrary to be true. He reasons from the assumed truth of this false property, till he arrives at a conclusion dependent upon that assumption, which is contrary to the original hypothesis; and thence it is inferred that the assumption being incompatible in its consequences with the original conditions of the theorem, those conditions and that assumption cannot coexist. If, then, all the alternatives of the alleged property be thus examined, and shown to be incompatible with the original. hypothesis, it will necessarily follow, that this property itself is true. Thus, in Euc. I. 25, where one included angle BAC is alleged in the enunciation to be greater than the other EDF, under the hypothesis of BA, AC, being respectively equal to ED, DF, but BC greater than EF: instead of proving the assertion itself, he admits, that in the first place the angle BAC is equal to the angle EDF, and in the second, that it is less. The consequences of these admissions are both shewn to be incompatible with the hypothesis, and hence it is inferred that the angle BAC can neither be equal to EDF, nor less than it. Wherefore as these are the only alternatives to the truth of the enunciation, and both these are false, it follows that the alleged relation of the angles BAC, EDF is true. This method of proof occurs frequently in the first and third books of Euclid; and it may be remarked generally, that it occurs more often in the outset of the development of a system of truths than in the more advanced parts, or in the more recondite theorems. It must naturally have occurred to Geometers, who were familiar with the use of this mode of assumption, to inquire: "What would be the effect of supposing the alleged theorem to be true, instead of false ?" He who first asked this question made the first step in the Geometrical Analysis. He would see at once that the conclusion ought to be consistent with the hypothesis, and with all previously known properties of the hypothetical figure. He may, indeed, find it of little convenience, often of none, in suggesting a direct proof of a very elementary theorem, but as he would be of course led to try its efficacy in more complex cases, he would be gradually impressed with the facts:—that in many cases his steps were merely the reversal of the steps which he had employed in the hypothetic demonstration of the theorem; and that in all cases, a reversal in the order of the steps of his analysis would constitute a synthetic demonstration, though perhaps different from any one previously known to him. He would then have discovered the true principle of the Geometrical Analysis of Theorems; and he would require but little additional skill to reduce the whole process to a complete system. It is probable that his discovery might lead to some such rules as the following: 1. Assume that the theorem is true. 2. Proceed to examine any consequences that result from this admission, by the aid of other truths respecting the figure, which have been already proved. 3. Examine whether any of these consequences be themselves such as are already known to be true, or to be false. 4. If any one of them be false, we have arrived at a reductio ad absurdum, which proves that the theorem itself is false, as in Euc. 1. 25. 5. If none of the consequences so deduced be known to be either true or false, proceed to deduce other consequences from all or any of these, as in (2). 6. Examine these results, and proceed as in (3) and (4); and if still without any conclusive indications of the truth or falsehood of the alleged theorem, proceed still further, until such are obtained. In the case of the theorem being false, we shall ultimately arrive at some result contradictory either to the original hypothesis, or to some truth depending upon it. Euclid's indirect demonstrations always end with a contradiction to the immediate hypothesis; but as the propositions to which he applies the method-are so extremely elementary, this could scarcely happen otherwise, as, so far, deductions would be made from the hypothesis by direct steps. Where, however, we find a contradiction in our results to any one of the consequences of the hypothesis, our conclusion, that the theorem is false, is as legitimate as though the contradiction had immediately been of the hypothesis itself. Nevertheless, if it should be imposed as a rule, that the contradiction shall be that of the hypothesis itself, it is only requisite to reverse that consequence of the hypothesis which is so contradicted, and to employ the contradiction instead of the conclusion of that consequence ; U |