is a mere delusion to believe that boys or men have got a logical training from the study of Euclid if they are unable to solve a problem or work a deduction. Their recollection of Euclid may be perfect, but the test of training is the power it has given; and if boys can produce no original work, the training is assuredly worth very little."* This latter remark applies with equal force to the study of Geometry apart from Euclid, except that the peculiar fault of Euclid in mixing up problems and theorems and implying that the one is essential to the other, increases the difficulty of the learner both in remembering which are essential and which are not, and in applying the necessary properties of similar figures to various independent exercises. Many a teacher is aware "how unavailable for problems a boy's knowlede of Euclid generally is. Yet this is the true test of geometrical knowledge." The theorems treat of the science of Geometry, and are theoretical and general ; whilst problems, as a rule, are practical and special, and treat of the application of the science to the art of drawing geometrical figures and to other mechanical processes. "There is no doubt that great light may be thrown upon, and additional interest imparted to, the theorems by the problems; but this advantage is counterbalanced by the disadvantages attending Euclid's treatment. The strictly logical form adopted by him induces, and was intended to induce, the belief that each individual proposition is essential to all that follow. Hence an inevitable confusion arises in the mind of the reader between that which is possible theoretically and conceivably, and that which is possible in relation to the instruments to which Euclid chooses to restrict himself, i.e., the compasses and ruler."+

A New Syllabus of Plane Geometry.

Hence we need not wonder that various attempts have been made to introduce a better classified and more compact system of Geometry and to supersede Euclid as a text-book. The report of the Schools Enquiry Commission had already indicated the necessity of this in 1868. One witness who had conversed with French, German, and Swiss teachers on the subject says: "When I asked why Euclid was an unfit text-book to teach Geometry from, I was told that Euclid's propositions were drawn out with a view to meet all possible cavils, and not with a view of developing geometrical ideas in the most lucid and natural manner." French opinion on English teaching may be summed up in the words: "La géométrie is alourdie par d'interminable rédites, et en voulant faire de la précision on fait du verbiage." Head masters of some of the great public schools have met together and have urged that the universities and teaching bodies generally should allow and encourage greater freedom in the

J. M. Wilson, M.A.

H. W. Watson, M.A.

use of geometrical text-books; and an "Association for the Improvement of Geometrical Teaching, consisting of many eminent mathematicians and mathematical masters, has at length been formed, and has already issued a Syllabus of Plane Geometry (1s., Macmillan, 1877). This Syllabus corresponds to the subjects of Euclid, Books I.—VI., and is prepared with the design of " securing an approximate, if not an absolute, uniformity of sequence in the book or books to be substituted for Euclid. This is necessary to protect the public against Geometries written by totally incompetent persons, and to render possible the conducting of Examinations in Geometry.'

The only book at present published, so far as we are aware, which strictly follows the above "Syllabus" is Wilson's Elementary Geometry (3rd edit., 3s. 6d., Macmillan), containing the subjects of Euclid's First Four Books. We would earnestly urge Matriculation candidates to use this clear, concise, and admirable text-book. In the first and second editions, Proportion-the subject of Euclid's Fifth and Sixth Books-was included; but it was not treated in accordance with the "Syllabus," for this was not published then. These editions are now out of print; but if the candidate can get a second-hand copy of the second edition, or of the part in which Proportion was also separately published, it might answer his purpose. We regret to learn that it may be some time before the author prepares a work on Proportion according with the "Syllabus." Surely we shall not have long to wait before either he, or some other member of the Association equally competent, shall furnish us with a suitable textbook, even though the Association itself may feel that it is not within its province to publish, or to make itself responsible for, any such book.

The candidate must not imagine from what we have said that this Association has anything to do with the University of London, or that the University has sanctioned the use of the "Syllabus." The University simply prescribes the subject of examination, and leaves the candidate to choose his own method of studying it. It merely tests the result of his study by what he has put on the examination-paper. He may therefore please himself whether he studies the "Relations and Properties of Similar Rectilinear Figures" in Euclid, or in some modern treatise independent of Euclid and either based upon the "Syllabus" or otherwise.

The order of Propositions in the "Syllabus," and their relation to Euclid.

A glance at the system or sequence of theorems adopted by the Association will be useful in showing what is regarded as essential

* J. M. Wilson, M.A.

to a system, and in what way the theorems may be classified and be made to depend on one another in that system. The main points of the "Syllabus," for the First B.A. candidate, are the following:

In Book IV., the Definitions and Theorems of Ratio and Proportion contained in Section 1, so far only as those of the other sections are dependent on them--although with most of these theorems the student will be already well acquainted through the treatment of Ratio and Proportion given in his text-books of Arithmetic and Algebra; next, all the five "Fundamental Geometrical Propositions" which are presented and partially demonstrated in Section 2.

In Book V., the whole of Sections 1, 2, and 3, on "Similar Figures," "Areas," "Loci and Problems."

We cannot give them here. The student and teachers generally had better consult the "Syllabus" itself, observing that, for those who understand the doctrine of proportion in reference to incommensurable magnitudes, Book V. alone is sufficient.

Most of our readers will have a copy of Euclid; and to them we may usefully indicate the essential propositions of the "Syllabus" so far as they are found in Euclid, Book VI., together with the order in which they are given in the "Syllabus," where in many cases they are far more simply worded, and in terms too of wider meaning: Euclid, Book VI., Propositions 2 (1st part), 10, 2 (2nd part), 1, 33, 21, 4, 6, 5, 7, 20 (1st part), 8, 8 (Cor.), C, A, 16, 17, B, D, 23, 23, 19, 20 (2nd part), 20 (Cor. 1), 22, 31, 10, 10, 9, 12, 13, 18, 25.

On this statement we may remark, that we have not been able to represent three or four of the theorems and corollaries in the "Syllabus" because they are not distinctively found in Euclid; and that many of the formal theorems and problems in Euclid are either implied in those of the "Syllabus or are left to be deduced and demonstrated as exercises. The first five numbers of the above statement denote the five "Fundamental Propositions" before referred to. The reader will observe that "10" is given three times; it represents a theorem and two problems, which stand

thus:

A given finite straight line can be divided internally into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality; and in each case there is only one such point of division.

To divide a straight line similarly to a given divided straight line. To divide a straight line internally and externally in a given ratio. By "20 (1st part)" there is represented the corollary only to the following peculiar theorem in the "Syllabus ":

If two similar rectilineal figures are placed so as to have their corresponding sides parallel, all the straight lines joining the angular points of the one to the corresponding angular points of the other are

Jarallel, or meet in a point; and the distances from that point along any straight line to the points where it meets corresponding sides of the figures are in the ratio of the corresponding sides of the figures.

Cor. Similar rectilinear figures may be divided into the same number of similar figures.

The following are some of the other theorems of the "Syllabus " which specially vary from Euclid in expression and more general significance :

2 (1st part)." If two straight lines are cut by three parallel straight lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other.

"7." If two triangles have one angle of the one equal to one angle of the other, and the sides about one other angle in each proportional, so that the sides opposite the equal angles are homologous, the triangles have their third angles either equal or supplementary, and in the former case the triangles are similar.

"C." If from any angle of a triangle a straight line is drawn perpendicular to the base, the diameter of the circle circumscribing the triangle is a fourth proportional to the perpendicular and the sides of the triangle which contain that angle.

"B." If two chords of a circle intersect either within or without a circle, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

This is the same as saying: "If a secant is drawn through a fixed point in the plane of a circle, the rectangle contained by the distances of that point from the two points of intersection with the circumference is constant, and conversely."

Hence, also, either as a separate theorem or as a corollary to "B:" "If from a point without a circle a tangent and a secant be drawn to it, the square on the tangent will be equal to the rectangle contained by the whole secant and the part of it without the circle, and conversely."

"D." The rectangle contained by the diagonals of a quadrilateral is less than the sum of the rectangles contained by opposite sides, unless a circle can be circumscribed about the quadrilateral, in which case it is equal to that sum.

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23." If two triangles or parallelograms have one angle of the one equal to one angle of the other, their areas have to one another the ratio compounded of the ratios of the including sides of the first to the including sides of the second.

"23." Triangles and parallelograms have to one another the ratio compounded of the ratios of their bases and of their altitudes.

We may observe that Prop. 7, which in ordinary editions of Euclid is insufferably long and tedious, may be worded thus, and that in this form its proof need not occupy one-third the space taken up in Euclid :

If two triangles have the sides about an angle of the one triangle proportional to the sides about an angle of the other, and have also the angle opposite that which is not the less of the two sides of the one equal to the corresponding angle of the other, these triangles will be similar.

Some Important Theorems and Problems.

We now add a few propositions of which the student should be quick to catch the meaning when differently worded, and to give some simple proof. They depend upon well-known theorems. Others will be found in Euclid, and among the exercises of different text-books.

1. The perimeters of similar polygons are to one another in the ratio of the homologous sides of the polygon.

2. The locus of a point whose distances from two fixed points are to each other in a given ratio is a certain circle.

3. If a line be drawn parallel to one side of a triangle it will form, with the remaining sides, or those sides produced, a second triangle similar to the first.

4. If there be two similar polygons, and two points so situated that when the first point is joined to the extremities of a side of the one polygon, and the second to the extremities of the side homologous to it of the other, the triangles thus formed are similar, and similarly situated with respect to each polygon; then all the triangles formed by joining each of these points with the extremities of any pair of homologous sides, one from each polygon respectively, shall be also similar, and similarly situated.

The above suggests an elegant proof of the theorem which might also be given as a corollary to it: Similar polygons (or “similar rectilineal figures") can be divided into the same number of similar figures; in which case the vertices of the two (or more) sets of triangles would be homologous points.

5. If A B C D E and F G H K L be a pair of similar pentagons in which A B and F G are a pair of homologous sides, and if M and N, P and Q, be two pairs of homologous points, one pair within each pentagon respectively, so that the lines MN and PQ are a pair of homologous lines, then shall MN: PQ = AB: FG; and, further, if R and S be another pair of homologous points, then shall the angle MNR = the angle PQS.

6. All regular polygons which have the same number of sides are similar.

7. The three lines drawn from the angular points of a triangle to bisect the opposite sides will intersect in one point.

8. To construct a triangle such that the three perpendiculars from the angles upon the opposite sides may be respectively equal to three given straight lines.