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tions which have been made of its researches to subjects of a physical nature, have rendered it necessary, for the purpose of gaining time, that the generality of students should omit those portions of the ancient geometry which are not now immediately requisite for, or capable of such applications: it has also been found convenient, in the propositions, relating to rectilineal solids, which are still retained, to diminish the length of Euclid's demonstrations; and, in those which relate to figures of revolution, to adopt principles which, without being less founded in truth, have not the strictness of those which are employed by the Greek geometer. Whatever licence, however, may be allowed for the sake of procuring a facility of investigation in propositions of a complex nature, no doubt can exist that the first steps in geometry should consist of propositions in the demonstrations of which the inferences are strictly drawn from assumptions explicitly stated, and from self-evident truths; for only by researches so conducted can the mind acquire habits of accurately deducing conclusions from the premises of a subject. Such habits being formed, processes of investigation founded on less rigorous principles may afterwards without impropriety be employed.
A desire to retain, in the demonstrations of the propositions, the exact logic of Euclid, as far as it can be done without occupying to an inconvenient extent the time of the student, has led to the adoption, in the following work, of the four first books, the sixth and part of the geometry of planes from Simson's edition of the "Elements." The order of the propositions has been preserved in the four first and the sixth books, and in the demonstrations of the propositions very few changes have been made; but the fifth, and all the books which follow the sixth, have been omitted or only partially followed; and it will be proper here to state the reason for adopting a method different from that of Euclid, in treating the subject of proportion among geometrical magnitudes.
Magnitudes in the abstract, when compared with one another in respect of their extent, are divided into those which are commensurable and those which are not so: these two conditions are defined at the commencement of the tenth book of Euclid, and in that book, the properties of magnitudes under both conditions are proved in considerable detail. If geometrical propriety, therefore, were studied, it is evident that such propositions of the tenth book as do not depend upon those which come after the fifth, should have formed part of the latter book. But the investigations relating to
incommensurable magnitudes, when conducted in the manner pursued by Euclid, have a degree of intricacy which renders them unfit to form a part of elementary instruction; and, as it is generally supposed to be impossible, by reasonings purely geometrical, to treat the subject of proportion with the requisite simplicity and brevity, it has become a frequent practice to employ algebraic processes alone in the investigations relating to that subject.
The principle on which Euclid's theory of proportion is founded, is laid down in the fifth definition of the fifth book; and Mr. Ivory, the distinguished author of the "Tract on Proportion," which is used at this Institution, observes, in his preface to that tract, that the definition, "though perfectly general, comprehending quantities that are not, as well as those which are, commensurable, and easily applied in the course of investigation, at least to subjects in the elementary geometry, is, on the other hand, very remote from common apprehension: the conditions of it," he adds, "are complex, and not easily reconciled to our first and most natural ideas on the subject." On this account he substituted for that definition one which may easily be comprehended by the generality of students, and which is therefore advantageous in rendering the theory of proportion more satisfactory to the minds of those who are entering upon that part of the elements. As the adoption of this principle permits, at the same time, the elementary propositions relating to proportion to be demonstrated in a geometrical manner, that tract is, in the present edition, put in place of the fifth book of Euclid. The definition alluded to is the tenth of those which precede the propositions in the tract on proportion; and being, in strictness, applicable to the comparison of commensurable magnitudes only, it becomes necessary to show how, by an approximation which may be carried to any degree of accuracy, the ratio between magnitudes which are incommensurable can be obtained, and this is the subject of the propositions A, B, and D, and of the first and second general theorems in the tract: the propositions A, B, and C contain the substance of the four first propositions in Euclid's tenth book. For the sake, however, of learners, who, postponing the study of proportion among geometrical magnitudes, may require immediately the corresponding theorems relating to numbers, an algebraic investigation is given after the geometrical demonstration of each proposition: those propositions which relate to compound proportion are investigated by algebraic processes only.
The first proposition of the sixth book is that which enters
into the comparison of all plane rectilineal figures with respect to magnitude, and its proof depends upon the general definition of proportion. Therefore, for the sake of preserving the demonstration as it stands in the text of Simson, the fifth (or general) definition in Euclid's fifth book has been retained among the definitions in the tract on proportion: there is given, however, in addition, a demonstration of that proposition in conformity to the tenth definition above-mentioned. The thirty-third proposition of the sixth book is also demonstrated according to the manner of Euclid, and again in conformity to the same definition.
In order that the geometry of plane figures might not be interrupted, as it is in "Euclid," by the propositions concerning solids, there are given, in the next place, the necessary propositions relating to the magnitudes of circles, and to the ratios between their circumferences, and between their areas or surfaces. This part of the "Elements" is followed by a tract containing the principal problems which relate to the tracing of lines and geometrical figures on the ground, and to the determination of inaccessible distances without instruments: subjects of considerable utility in one branch of the military art.
The tract on the intersection of planes, and that on the properties of prisms and pyramids, remain nearly the same as in the editions which have hitherto been in use at this Institution: they consist of propositions taken from the eleventh and twelfth books of Euclid's "Elements," and of others which were added by Mr. Ivory, in order to render the geometry of solids bounded by plane surfaces more complete. But the investigations in the tract on circles, and in that on solid figures of revolution (cylinders, cones, and spheres), differ from those which, in the editions just mentioned, and in that of Dr. Simson, are employed in propositions relating to such figures: and, without departing from the spirit of the ancient geometry, an effort has been made to render this part of the "Elements" less tedious for the learner. It will therefore, in this place, be proper to show, in a general way, the manner in which the subject of figures of revolution was treated by the ancients.
The direct methods used in the elementary geometry to determine the magnitudes of plane figures bounded by straight lines, or of solids bounded by planes only, and to prove the relations between them, fail when it is required to investigate the magnitudes and relations of figures bounded by curve lines or surfaces; and the Greek geometers found it necessary to introduce into the "Elements" a principle
by which the investigations relating to polygons, pyramids and prisms might be rendered applicable to circles, and to solids of revolution. This principle consists in considering one magnitude to be equivalent to another, when it can be proved that they do not differ from each other, either in excess or defect, by any assignable quantity: also, that two ratios are equal to one another when it can be proved that, in one ratio, the antecedent is not to the consequent, as, in the other ratio, the antecedent is to any term which differs from the consequent, in excess or defect, by an assignable quantity. The processes used in consequence of the introduction of this principle constitute the "Method of Exhaustions;" it is supposed to have been invented by Euclid, but of this there is no proof; and, as it is not probable that the properties demonstrated by it were unknown before the time of Euclid and Archimedes, it may be inferred that the method is of higher antiquity than the age in which these matheticians lived.
The principle is applied by Euclid in the tenth proposition of the twelfth book, where it is demonstrated that a cylinder is equal in volume to three times the volume of a cone having an equal base and an equal altitude; and the process may be generalised in the following manner. Let the cylinder be represented by A, and the cone by B; and let there be inscribed in A a series of rectilineal figures (prisms), having altitudes equal to that of A, and continually increasing in volume till, at length, agreeably to the first proposition of the tenth book, there is obtained one (x) which differs, in defect, from a by a magnitude less than any magnitude by which a may be supposed to exceed the multiple of B: it will follow that x is greater than that multiple of B; but the altitudes of A and B being equal, and the base of x being less than that of A or B, X is less than that multiple of B (because it is equal to the same multiple of a pyramid inscribed in B); that is, x is both greater and less than the multiple of B, which is absurd. Therefore A is not greater than the multiple of B. Then, inverting the process, assuming that B exceeds the like submultiple of A by an assignable magnitude, and reasoning as before, it may be shown that A is not less than the multiple of B: hence the equivalence is inferred. It is evident, however, that this reasoning, instead of strictly proving the equivalence of the magnitudes, shows only that a does not differ from the assumed multiple of B, either in excess or defect, by a magnitude so great as the least that can be assigned. It does not follow that A may not differ from the multiple of в by a magnitude less than that by which it
differs from x: this difference is, indeed, supposed to be less than the least magnitude that can be assigned, but it is not nothing.
This is called the method of exhaustions, because there are supposed to be inscribed in a figure of revolution all the different rectilineal figures that can be so inscribed; or the number of such figures is exhausted.
Again, Euclid's application of the method of exhaustions in finding the ratio between two figures of revolution (x and Y) of the same kind, consists in inscribing in Y a series of rectilineal figures continually increasing in magnitude, till, at length, there is found one (s) which shall differ from it by a magnitude less than that by which an assumed magnitude (M) differs, in defect, from Y; so that s is greater than M : there is then inscribed in x a rectilineal figure similar to s. Now, if the known ratio of the two inscribed rectilineal figures be represented by that of A to B, and it be required to prove that X and Y are to one another in the same ratio, let it be assumed first that a may have to в the same ratio as x has to M (which is less than Y); then it will follow, from equality of ratios, and because X is greater than the figure inscribed in it, that M is greater than s: but, as above, it is less, which is impossible; therefore a is not to B as X is to a magnitude less than Y. On inverting the process it may be proved in like manner that A is not to B, as x is to a magnitude greater than Y; and hence the equality of the ratios of A to B and of x to Y is inferred.
In strictness, however, it is only proved that A is to B as one of the figures (x) of revolution is to a magnitude which does not differ from the other figure (Y) either in excess or defect, by a quantity so great as the least that can be assigned. It does not follow that the magnitude may not differ from Y by a quantity less than that by which the figure inscribed in Y differs from Y: this difference is certainly less than that which is, by supposition, less than the least magnitude that can be assigned, but it is not nothing.
The method which, in this edition, is employed for determining the magnitudes and relations of figures formed by revolution, is founded upon the inscription, in a circle, of a regular polygon, each of whose sides is less than any line that can be assigned, and upon the description, about the circle, of a similar polygon, whose sides are also less than any assignable line. It is afterwards proved that the perimeters, and the areas of these polygons differ from the circumference, and from the area of the circle by lines and areas less than the least that can be assigned; also that the