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The SERIES consists of the following Works :


Exercises. By J. D. MORELL, LL.D. Price 28. 6d.

This has already been in use for many years, and has been chiefly instrumental in placing the ordinary teaching of English Grammar upon a more intellectual footing than heretofore. Nearly half a million copies have been sold.

2. A Practical Introduction to ENGLISH COMPOSITION

on a New Plan, with Three Hundred Exercises. 28. 6d. Superior Edition, 3s. 6d.



RATURE, with Four Hundred Exercises and Lessons. 564 pp., 48. 6d. Superior Edition, 58. 6d., or in two parts at 28. 6d. each.



First Course in Reading was intended to teach the Symbol ; the Second Course to teach Fluency; and the present Manual is intended to teach Expression,-a quality of good Reading in which many Schools are still deficient.


5. A NEW GEOMETRY-being a Practical Introduction to Logic or Thought, applied to Space.


6. A COURSE of ENGLISH LESSONS, comprehending

a Brief History of the Language, a set of Critical Lessons, and

an Introduction to English Poetry. [In preparation. This last will probably be followed by a Series of small volumes of Belections from the best English Authors. Each volume will be a type of the literature of each of the great epochs, and will be furnished with an apparatus for teaching, a biography, critical notes questions, sets of examination papers, and exercises.




NO. 4.


Whilst every other branch of Mathematics has made the largest progress from discovery to discovery, ELEMENTARY GEOMETRY alone,-the earliest discovered, and therefore the oldest, theory in Mathematics,-has remained almost stationary for the last two thousand years. There is now, however, a new and strong desire stirring for a larger evolation of this Science, and for a wider development of it upon the old Euclidean lines.

The present work is an attempt to solve the difficult question, which is at present much agitated among teachers and students of this science: How can Geometry be best presented in a pure scientific form ? This question would have received a practical answer long ere this, but for the tendency natural to human beings to turn teachers into infallible authorities, and great men into deities. The help in geometrical science which Euclid gave, the aid in mental science afforded by Aristotle, became, in the course of ages, hardened into barriers against progress, and hindrances to the development of the scientific spirit in these subjects. But Aristotle is now better understood; and his spirit is better obeyed, even if there is less servile attachment to the letter of him. So likewise in the SCIENCE OF GEOMETRY, the time has now arrived when we can work in the true light of Euclid, and can receive from him the full benefit of his scientific spirit, without being trammelled by his heavier bonds. The application of Logic to Geometry still remains what Euclid made it; it is time that, led


by him, we should try to develop this application still further, and to follow this development trustingly to the goal whither it

leads us.

The present little work, accordingly, is presented to the public as a fuller development of Euclid's matter and Eaclid's method, and as embodying a Complete Exposition of Elementary Geometry. To keep faithful to him, it has declined to insert into Euclid's Elements the patchwork of modern formulæ, which give to many editions of Euclid a false look of Analytical Geometry, bat which are in every way fundamentally opposed to Euclid's method. The present work, like its prototype, the Greek Elements, is founded throughout upon pure reasoning.

If Euclid himself could live once more and re-edit his Elements, and give to them a form more in harmony with modern science, it is probable that a number of teachers would continue to prefer the accustomed bonds of the old Mummy to the living method of the new Master. But this has not been the feeling of the present writer. The spirit of Eaclid bas inspired him throughout this Book, if the letter of Euclid has not been closely adhered to. Whatever new developments are to be found in the

. book are due to Euclid himself and to his method; for they have been gained by following his example, and by immersing the old and well-known Elements in the waters of its original springthat is, LOGIC, from which all its life, old as well as new, has risen.

This volume contains only the PLANE GEOMETRY, divided into Four BOOKS :

Book the First: On Lines.
Book the Second: On Angles and Proportional Lines.
Book the Third: On Plane Figures.
Book the Fourth: On Areas.

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