TEXT-BOOKS ON THE BASIS OF EUCLID.

Euclid, Book V., treats on Proportion, and may be regarded as an introduction to Book VI., which consists of the application of the theory of proportion in order chiefly to show the relations and to establish the properties of Similiar Rectilinear Figures. But "no teacher dreams of taking his pupils through Euclid's Fifth Book ;" the reasoning is "artificial and remote from practical common-sense notions on the subject." "The unanswerable objection to Euclid's treatment of Ratio and Proportion is that it is practically disregarded." If the candidate will, nevertheless, adhere to Euclid, and to Euclid only, we will do our best to advise him in the choice of a text-book; but we would by no means recommend him to do so. His Euclidian conservatism will, we believe, be an obstacle, though not an impassable barrier, to him in the University which examines not in Euclid but in the Geometry of specified subjects. Let him compare the examination-papers of the last few years with the form and statement of propositions in Euclid and he will observe, as we have already indicated, a difference of language and a scope in questioning which must be sorely puzzling to the candidate who has read Euclid and nothing but Euclid.

Either Potts's Euclid (4s. 6d., Longmans), with Notes, Questions, Geometrical Exercises, and Hints; or Todhunter's (3s. 6d., Macmillan), with Notes, an Appendix, and Exercises, will suffice, and will render excellent aid.

But we would most decidedly urge the student to use in preference either Hamblin Smith's Elements of Geometry (3s. 6d., Rivington), or Cuthbertson's Euclidian Geometry (4s. 6d., Macmillan), because whilst true to the general spirit and order of the ancient geometer, these authors have adapted his Elements to the altered character of the times and the improved methods of geometrical demonstration. The former book especially is edited on this principle, and the result is a neat, clear, and exceedingly serviceable edition of Euclid, in which Mr. Smith has not hesitated to sacrifice much of the verbiage and ambiguity of Euclid whilst conserving his essential points. "To preserve Euclid's order, to supply omissions, to remove defects, to give short notes of explanation and simpler methods of proof in cases of acknowledged difficulty, such are the main objects of this edition of the Elements." Euclid's Fifth Book is here presented in a more compact, simple, and systematic form than ever we have seen before, and a method of notation is adopted which shortens and simplifies the proofs considerably. A separate section is given of the propositions most frequently referred to in Book VI., and the student need only read this section and the notes necessary to it. The notes throughout are excellent, always to the point, and just where they are wanted. The examples are also a

good feature of the work, placed where we like to see them, classified under the head of the several propositions, annexed to them where possible, with miscellaneous exercises to test the pupil more generally. An extensive use of allowable symbols is exemplified in this admirable text-book.

Mr. Cuthbertson's work is also eminently satisfactory, and one that may safely be used by the candidate who is in unchanging love with Euclid and does not like to depart far from Euclid's method. It has been prepared with great care, attention to system, and simplicity of form; and the student will be sure to like the neatness, lucidity, and pleasant readable appearance of the book. "With a view to facilitating the student's progress it has been arranged that, as a rule, each proposition should have a page to itself, and that those of greater length should occupy pages facing each other; so also that propositions and the corresponding converse ones should be placed on opposite pages. Moreover, Definitions and Axioms, instead of being all at the beginning of the book, have been introduced when required." The chief characteristic of the book is that, whilst all the propositions required at the Universities have been demonstrated, and the Problems separated from the Theorems and placed at the end of the various sections, and whilst both are well classified and proved in the simplest manner possible, yet the author has been "careful in all cases to retain Euclid's order as far as is necessary to allow of the proofs given being substituted for those of Euclid in the examinations. The great objection entertained by those in authority at the Universities towards modern text-books on Geometry is grounded on the fact that enormous inconvenience would arise in conducting examinations with no recognised sequence of propositions. This objection" the author believes he has "effectually dealt with by giving demonstrations which depend on propositions occupying a prior position not only in this work but in Euclid." Important Tables are given showing the Classific tion of Subjects, and for Comparison with Euclid. In the author's "Scheme for Examination on Geometry" he indicates how desirable it is in any given proof to refer to the Definitions and the Test of Proportion upon which the system of Geometry we may use is based. In his treatment of Ratio and Proportion he has adopted a method which 'possesses the combined advantages of being rigorous, geometrical, simple, and short." Altogether we consider Mr. Cuthbertson's work one of the best adaptations of Euclid we have seen, and we cordially recommend its use to the Euclidian student.

*This "objection" has no reference to the University of London.

II. THE ELEMENTARY PROPERTIES OF THE PLANE, INCLUDING THOSE OF THE ANGLES MADE BY PLANES WITH RIGHT LINES AND WITH EACH OTHER.

Up to this point the student has been dealing with the Geometry of Lines and Surfaces; he has now to approach the somewhat more abstruse subject of the GEOMETRY OF SPACE, in which his powers of conception and of reasoning in the abstract will be more severely tested and applied in new directions. But we do not think the student who hitherto has made his footing sure, and must by this time have trained his geometrical imagination, will find his task relatively more perplexing than in previous studies. In our judgment Solid Geometry is not so difficult to understand when we read it in our text-books, as it is difficult to remember and to reproduce on the spur of the moment in an examination. Clearly to conceive and represent lines, planes, and figures in space requires a more special exercise of the mental powers, and more practised skill in the drawing and manipulation of diagrams, than in elementary Geometry. However, this only means that the student must frequently go over the same ground, and rehearse* his work, until its lines and elements are deeply cut into his memory, and the seed sown by his reading begins to spring up and branch out in new and original applications.

The above subject, the earlier parts of which form an easy introduction to Solid Geometry at large, is usually regarded as synonymous with part of the Eleventh Book of Euclid. In previous Regulations it was stated thus: "The Eleventh Book of Euclid to Prop. 21, or the subjects thereof." These subjects, comprehensively considered, are Straight Lines and Points in a Plane; Straight Lines Perpendicular, Parallel, or Oblique to a Plane or Planes; Intersection of Planes; Planes Parallel, Perpendicular, or Oblique to one another; Lines in Space; the Properties of Dihedral, Trihedral, and Polyhedral Angles. It is very important that the student should clearly understand the nature of a dihedral angle, its mode of measurement, division into parts, and the general idea of its magnitude and relations as illustrated by reference to plane angles.

The few notes we give on this part of the examination are mainly with a view to indicate its scope, and to enable the candidate to recognise, under divergent forms and expressions, some of those theorems which are variously worded or which are not distinctively set forth in each text-book. The main elements of the subject are found in every good manual treating of Solid Geometry in its theoretical aspects. Thither also the reader must resort for exercises which are to be found in abundance, and which alone can adequately

* Literally, "To harrow over again."

familiarise his mind with all the deductions and problems of which the subject is capable.

The Determination of a Plane.

1. From the elementary theorems it will be evident that a plane is completely determined—

(1) By a given straight line and a given point not in that line. (2) By three points not collinear, i.e., not in the same straight line. (3) By two intersecting straight lines.

(4) By two parallel straight lines.

(5) By one point and the normal (or perpendicular) to the plane at that point; for, The locus of straight lines which cut a given straight line at right angles at a given point is a plane. Or, in other words: Through a point of a straight line as many perpendiculars as are desired can be drawn to the straight line, all of which will be necessarily in one and the same plane perpendicular to the given straight line.

(6) By one point and the directions of two lines in the plane. This follows from the proof of the proposition: Through a given point one plane, and only one plane, can be drawn parallel to two given lines.

It may be well to note here, that although in representing planes on paper we seem to suppose them limited, yet they ought always to be conceived of as unlimited or capable of infinite extension.

Pairs of Identical Propositions.

2. The two propositions in each of the following groups are practically the same :—

(1) It is always possible to find one plane, and only one plane, containing a given straight line and a given point not in that straight line. Or:

Through three given points, not in the same straight line, one plane, and only one plane, can be drawn.

(2) If two straight lines meet one another a plane can be drawn to contain both, and every plane containing both must coincide with the aforesaid plane. Or :

Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.

NOTE. The second group is essentially identical with the first group. We may prove one proposition of either group first, and then the two propositions of the other group may be deduced or written down as corollaries. This is a good illustration of the way in which theorems may be made to depend upon one another, according to the order of proof we may adopt; also of the way in which a theorem

in one author may be found as a corollary in another. Whichever form is given in an examination-paper we must prove it as an original proposition.

(3) If two straight lines be parallel, the straight line drawn from any point in one to any point in the other is in the same plane with the parallels. Or:

It is always possible to draw one plane, and only one plane, containing two given parallel straight lines.

(4) There cannot be two perpendiculars drawn to a given plane from a point either in the plane or without it. Or :

Through a given point only one perpendicular can be drawn to a plane.

NOTE.-See also Euclid XI., 13. The following differently expressed proposition, and the deduction from it, shows how the proof may be so varied as to involve other considerations and lead to other inferences according to the purpose in view or the method adopted :

Theorem : From a given point outside a given plane it is always possible to draw one straight line, and only one straight line, perpendicular to the given plane, and the straight line so drawn is shorter than any other straight line that can be drawn from the given point to the given plane."

Deduction: "If a straight line be drawn from a point in a given plane parallel to a perpendicular to that plane from a point without it, then the former line will be also perpendicular to the given plane; and therefore one line, and only one line, may always be drawn from a given point in a given plane perpendicular to that plane."

(5) If two straight lines be at right angles to the same plane they shall be parallel to one another. If two straight lines be parallel, and one of them be at right angles to a plane, the other also shall be at right angles to the same plane. (Euclid XI., 6 and 8. Or :

"If a straight line be perpendicular to a given plane it shall be also perpendicular to every straight line parallel to the given plane; and if a straight line be perpendicular to a given plane every straight line perpendicular to the given straight line shall be parallel to the given plane." Deduction: If a given straight line be perpendicular to a given plane, every straight line parallel to the given line will be also perpendicular to the given plane."

The following proposition, which is the first part of (5), is also a deduction from the theorem given in the last note: If two straight lines be each of them perpendicular to a given plane they will be parallel to one another. For if they meet they will lie in one plane, and