9. In a given triangle to inscribe a triangle similar to a given triangle.

10. If a quadrilateral figure be described about a circle, and the points of contact of opposite sides be joined, prove that these lines and the diagonals of the quadrilateral figure all intersect in one point.

11. To make a square which shall be to a given square in a given ratio. Show that the same construction and proof are applicable to any polygon.

12. To construct a triangle similar to a given triangle, on a straight line homologous to a given side of the triangle, and hence show that a polygon can be constructed similiar to any given polygon on a straight line homologous to a given side of the polygon.

13. Any two parallels are cut proportionally by a series of secants drawn through the same point, and the ratio of any two corresponding parts is equal to the ratio of the parts of any secant contained between the point and the parallels.

14. If two parallelograms are equal in area, and have an angle of the one equal to an angle of the other, then the sides which contain the angle of the first are the extremes of a proportion of which the sides containing the equal angle of the second are the means.

Compare this with Euclid VI., 14, 15, 16. The proof of the above will, of course, apply to rectangles, for they are equiangular parallelograms.

15. Two regular polygons of the same name are similar and the ratio of their perimeters is equal to the ratio of the radii of the circumscribed or inscribed circles.

16. Similar rectilinear figures inscribed in circles are to each other as the squares of the diameters of those circles.

A mere glance at the above propositions, which are all found in text-books which treat on the "Relations and Properties of Similar Rectilinear Figures," will show the candidate the wide scope of the subject, and how essential is a thorough and patient study of it. He should be so familiar with all the main theorems and problems, that he may be able to recognise them in a moment, both when set in different phraseology and when required in new applications or original deductions. In reading each demonstration in his text-book, let him follow up the hint we gave on Algebra concerning the possible variations of each proposition and the inventing of new exercises. Let him draw fresh figures and apply his proofs to altered constructions; let him modify his premises and test the wider bearings of the principles under consideration: he will thus familiarise himself with each proposition and detect its meaning and use when presented in another guise.

The Sequence of Theorems.

It is essential that, whichever text-book of Geometry the student may use, he should rigidly adhere to the order of the propositions in that book, with a view to preserve the chain of reasoning and keep a firm grasp of it in working his exercises and in answering questions at the examination. If he have another book on the subject--which it is often desirable for the private student to have-he should only consult it for the purpose of comparison and illustration when he wants to clear up any particular point or perceive the more general bearings of the problem in hand.

The Test of Proportion.

Another caution may be given: Observe the fundamental basis or principle on which the author proceeds, and examine your own work, to see whether it is firmly grounded on the same principle. Some authors proceed on the well-known definition of the proportionality of four magnitudes given by Euclid, and then, like him, simply apply the definition to the proof. This is done in the "Syllabus," where the definition is applied in the less verbose and handier form :

The ratio of A to B is equal to that of P to Q, when m A>=<n B according as mP>=<nQ, whatever whole numbers m and n may be. And then follows as an "immediate consequence" its application to incommensurable magnitudes :

The ratio of A to B is equal to that of P to Q, when, m being any number whatever, and n another number determined so that mA is between n B and (n + 1) B or is equal to n B, according as mA is between n B and (n+1) B or is equal to n B, so is m P between n Q and (n+1) Q or equal to n Q.

The following is given as a simpler form of the definition: The ratio of A to B is equal to that of P to Q when the multiples of A are distributed among those of B in the same manner as the multiples of P are among those of Q.

This principle is quite sufficient to apply in the proof of a proposition if the doctrine of "limits" be thoroughly understood. But the subject requires care.

To show how various are the methods of applying this principle of Geometrical Proportion in reference to all kinds of magnitudes, commensurable and incommensurable, we may refer to three or four authors whose manuals we shall presently describe.

Mr. Aldis puts the principle in this form: If A, B, P, Q be four magnitudes, and the process of finding the Greatest Common Measure of A and B is identical in form with that for finding the G. C. M. of P and Q, then the four magnitudes are proportional, or A: B=P: Q. In this form he gives it a very simple and concise application to such a proposition as that "Triangles of the same altitude are to one

another as their bases," and that "Straight lines are cut proportionately by parallel straight lines."

Mr. Wilson first proves the theorem: If A, B, P, Q be four magnitudes, such that B and Q always contain the same aliquot part of A and P respectively, the same number of times, however great the number of times into which A and P are divided, then A: B=P: Q. This he rigorously proves by reasoning applicable to both commensurable and incommensurable magnitudes, and then concisely applies to the two propositions given above.

Both Mr. Wormell and Mr. Watson fully illustrate the principle on which the definition of proportion is based, but make their footing sure by briefly applying it in two ways: (1) to supposed commensurable, (2) to supposed incommensurable magnitudes in the same proposition. Mr. Wormell begins his application with the usual proposition, Mr. Watson with the following: "In equal circles angles at the centres are to one another in the same ratio as the circumferences by which they are subtended."

These illustrations of the different methods of approaching the subject and of commencing the links in the chain of geometrical proof, will again confirm our view of the wide scope of this part of the examination, and of the necessity of adhering to one and the same system or order of propositions throughout.

In reference to incommensurable magnitudes let the student grasp the following principle, and he will have no further difficulty with them In the case of two incommensurable magnitudes (i.e., straight lines, triangles, parallelograms, &c.) it is conceivable that we may always find a magnitude differing from one of them by a magnitude less than any assignable magnitude (i.e., differing from it by as small a magnitude as we please) and at the same time commensurable with the other magnitude; or we may always find two magnitudes differing respectively from the two incommensurable magnitudes by magnitudes less than any assignable magnitude (ie., differing from them by as small magnitudes as we please) and at the same time commensurable with one another. In the limit, the ratio of these two commensurable magnitudes is called the ratio of the two incommensurable magnitudes themselves.

TEXT-BOOKS IN ACCORDANCE WITH MODERN METHODS.

Perhaps the simplest treatment of Proportion for the beginner, or for the candidate who loves to look at the practical applications of his studies in order to feel more personal interest in them, is in Dr. Wormell's Modern Geometry: A New Elementary Course of Plane Geometry (2s., Solutions to Exercises, 2s. 6d., Thomas Murby). The work includes all the subjects of Euclid's First Six Books; but, as its name implies, it is written independently of Euclid, and is specially

adapted to the use of candidates for the numerous examinations in which the demonstrations of Euclid are no longer a sine quâ non. The plan of the book is to make the explanatory and illustrative hints on the concrete applications of the science introductory to the strictly logical demonstrations of the science itself. These explanatory hints are very full and clear in the chapters on Ratio and Proportion, which treat on the Relation of Number and Geometrical Magnitude, Commensurable and Incommensurable Magnitudes, Arithmetical Proportion, Theorems on Similar Rectilinear Figures, Problems on Proportion, and Proportionals in the Circle. A very important feature of the work will be found in the arrangement of the numerous exercises for solution. These are at first very simple, and increase in difficulty just as rapidly as is needful to concentrate the attention of the pupil at each successive stage. Complete solutions of all these exercises are published separately." Considerable interest attaches to the Appendix, which contains a complete analysis of the propositions of the book as compared with those of Euclid. "When a proposition in Euclid has no corresponding one in this work, the page on which it might be logically inserted as an additional exercise is indicated. It must be borne in mind, however, that, as the order of arrangement and mutual dependence of the propositions are different in the two cases, the demonstrations of corresponding propositions are seldom identical." Some propositions which were essential links in the chain of reasoning adopted by Euclid are not necessary with modern methods, nevertheless they furnish interesting additional problems for solution by the student.' By the judicious use of this appendix, and in being directed by it to those propositions of Euclid which should be worked as exercises but have not been adopted in this book, the candidate may feel himself in a safer position, and have the entire subject comprehensively before him. We sincerely commend the work as likely to be of great service to those who are entering on this part of their studies for the first time, or feeling their need of clear exposition and practical illustrations.

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The essential points of the "Relations and Properties of Similar Figures" are very neatly and clearly set forth in Aldis's Text-Book of Geometry, Part II., Proportion (2s., Bell and Sons). It is a compact little work, where, in about 40 pages, the student will find a simple and natural demonstration of the chief propositions in Geometrical Proportion, supplemented by numerous exercises attached to and immediately following the theorems, so that "the pupil may learn at once the value and use of what he studies." These exercises are very practical and suggestive, and will enable the reader to test his knowledge of each proposition and its implications. Notes and explanations are, as a rule, avoided; but the treatise is not one difficult to understand. It is certainly much easier than

Euclid, and if used as a text-book to be independently studied, and then supplemented by a reference to and proof of the Euclidian propositions not included in the above work, it will in every way serve the candidate's purpose. We would, however, decidedly advise him to shorten the labour of writing out his proofs by using all the symbols and abbreviations which are fairly allowable to candidates who take care to preserve the "chain of reasoning" and to indicate it in words whenever absolutely necessary to show the processes of the demonstration. This remark applies to both this and the following treatise, in neither of which are abbreviations used to the extent desirable for University candidates, whose time at the examination is exceedingly limited for the work that has to be done. Watson's Elements of Plane and Solid Geometry (3s. 6d., Longmans) is one of the admirable series of Text-Books in Science recently published by the above firm. It embraces a thorough and painstaking treatment of all the parts of Euclid's First Six Books, and the part of the Eleventh Book required at the Universities generally; but, whilst agreeing with Euclid in retaining the syllogistic form throughout, in its arrangement and clear exposition of the subject it is independent of Euclid, and is fully in accordance with modern methods of teaching Geometry. It is, perhaps, more systematic and comprehensive than either of the two works already mentioned, and cannot fail to ground the student firmly and intelligibly in the whole subject together with all its inter-dependencies and logical relations. "The properties of Ratio and Proportion are, in the first place, explained and proved in a few simple propositions with reference to commensurable magnitudes, and they are afterwards extended by the simple application of the method of limits to incommensurable magnitudes." The whole of Book V. should be read as a definition and introduction of the subject; and then the whole of Book VI.-except the section on Regular Polygons, very little of which is needed-should be completely mastered for, the Miscellaneous Propositions, the theorems on Similar Rectilinear Figures, and the first five problems of construction connected with Ratio and Proportion are the very essence of what is needed by the candidate, who should by no means omit the working of the useful examples at the end of every section. The author very appositely remarks with regard to the abundant explanations given: "As it is desirable that an elementary treatise should be as independent as possible of external aid, an attempt has been made to assist the self-taught student by means of notes; these notes are inserted from time to time in the course of the work, because they are intended to be read, and not grouped together at the end, in which case they would certainly be neglected." We have said enough to show that it is a reliable and scholarly work.