Introduction

The arts and sciences have become so extensive, that to facilitate their acquirement is of as much importance as to extend their boundaries. Illustration, if it does not shorten the time of study, will at least make it more agreeable. This work has a greater aim than mere illustration; we do not introduce colours for the purpose of entertainment, or to amuse by certain combinations of tint and form, but to assist the mind in its researches after truth, to increase the facilities of instruction, and to diffuse permanent knowledge. If we wanted authorities to prove the importance and usefulness of geometry, we might quote every philosopher since the days of Plato. Among the Greeks, in ancient, as in the school of Pestalozzi and others in recent times, geometry was adopted as the best gymnastic of the mind. In fact, Euclid’s Elements have become, by common consent, the basis of mathematical science all over the civilized globe. But this will not appear extraordinary, if we consider that this sublime science is not only better calculated than any other to call forth the spirit of inquiry, to elevate the mind, and to strengthen the reasoning faculties, but also it forms the best introduction to most of the useful and important vocations of human life. Arithmetic, land-surveying, mensuration, engineering, navigation, mechanics, hydrostatics, pneumatics, optics, physical astronomy, etc. are all dependent on the propositions of geometry.

Much however depends on the first communication of any science to a learner, though the best and most easy methods are seldom adopted. Propositions are placed before a student, who though having a sufficient understanding, is told just as much about them on entering at the very threshold of the science, as gives him a prepossession most unfavourable to his future study of this delightful subject; or “the formalities and paraphernalia of rigour are so ostentatiously put forward, as almost to hide the reality. Endless and perplexing repetitions, which do not confer greater exactitude on the reasoning, render the demonstrations involved and obscure, and conceal from the view of the student the consecution of evidence.” Thus an aversion is created in the mind of the pupil, and a subject so calculated to improve the reasoning powers, and give the habit of close thinking, is degraded by a dry and rigid course of instruction into an uninteresting exercise of the memory. To raise the curiosity, and to awaken the listless and dormant powers of younger minds should be the aim of every teacher; but where examples of excellence are wanting, the attempts to attain it are but few, while eminence excites attention and produces imitation. The object of this Work is to introduce a method of teaching geometry, which has been much approved of by many scientific men in this country, as well as in France and America. The plan here adopted forcibly appeals to the eye, the most sensitive and the most comprehensive of our external organs, and its pre-eminence to imprint it subject on the mind is supported by the incontrovertible maxim expressed in the well known words of Horace:—

Segnius irritant animos demissa per aurem
Quàm quæ sunt oculis subjecta fidelibus.

A feebler impress through the ear is made,
Than what is by the faithful eye conveyed.

All language consists of representative signs, and those signs are the best which effect their purposes with the greatest precision and dispatch. Such for all common purposes are the audible signs called words, which are still considered as audible, whether addressed immediately to the ear, or through the medium of letters to the eye. Geometrical diagrams are not signs, but the materials of geometrical science, the object of which is to show the relative quantities of their parts by a process of reasoning called Demonstration. This reasoning has been generally carried on by words, letters, and black or uncoloured diagrams but as the use of coloured symbols, signs, and diagrams in the linear arts and sciences, renders the process of reasoning more precise, and the attainment more expeditious, they have been in this instance accordingly adopted.

Such is the expedition of this enticing mode of communicating knowledge, that the Elements of Euclid can be acquired in less than one third the time usually employed, and the retention by the memory is much more permanent; these facts have been ascertained by numerous experiments made by the inventor, and several others who have adopted his plans. The particulars of which are few and obvious; the letters annexed to points, lines, or other parts of a diagram are in fact but arbitrary names, and represent them in the demonstration; instead of these, the parts being differently coloured, are made to name themselves, for their forms in corresponding colours represent them in the demonstration.

Triangle ABC A B C

In order to give a better idea of this system, and of the advantages gained by its adoption, let us take a right angled triangle, and express some of its properties both by colours and the method generally employed.

Some of the properties of the right angled triangle ABC, expressed by the method generally employed.

  1. The angle BAC, together with the angles BCA and ABC are equal to two right angles, or twice the angle ABC.
  2. The angle CAB added to the angle ACB will be equal to the angle ABC.
  3. The angle ABC is greater than either of the angles BAC or BCA.
  4. The angle BCA or the angle CAB is less than the angle ABC.
  5. If from the angle ABC, there be taken the angle BAC, the remainder will be equal to the angle ACB.
  6. The square of AC is equal to the sum of the squares of AB and BC.

The same properties expressed by colouring the different parts.

  1. Red angle + Yellow angle + Blue angle = 2 Yellow angle = Two right angles .

    That is, the red angle added to the yellow angle added to the blue angle, equal twice the yellow angle, equal two right angles.

  2. Red angle + Blue angle = Yellow angle .

    Or in words, the red angle added to the blue angle, equal the yellow angle.

  3. Yellow angle > Red angle or Blue angle .

    The yellow angle is greater than either the red or blue angle.

  4. Red angle or Blue angle < Yellow angle .

    Either the red or blue angle is less than the yellow angle.

  5. Yellow angle minus Blue angle = Red angle .

    In other terms, the yellow angle made less by the blue angle equal the red angle.

  6. Yellow line2 = Blue line2 + Red line2.

    That is, the square of the yellow line is equal to the sum of the squares of the blue and red lines.

In oral demonstrations we gain with colours this important advantage, the eye and the ear can be addressed at the same moment, so that for teaching geometry, and other linear arts and sciences, in classes, the system is the best ever proposed, this is apparent from the examples just given.

Whence it is evident that a reference from the text to the diagram is more rapid and sure, by giving the forms and colours of the parts, or by naming the parts and their colours, than naming the parts and letters on the diagram. Besides the superior simplicity, this system is likewise conspicuous for concentration, and wholly excludes the injurious though prevalent practice of allowing the student to commit the demonstration to memory; until reason, and fact, and proof only make impressions on the understanding.

Again, when lecturing on the principles or properties of figures, if we mention the colour of the part or parts referred to, as in saying, the red angle, the blue line, or lines, etc. the part or parts thus named will be immediately seen by all in the class at the same instant; not so if we say the angle ABC, the triangle PFQ, the figure EGKt, and so on; for the letters must be traced one by one before the students arrange in their minds the particular magnitude referred to, which often occasions confusion and error, as well as loss of time. Also if the parts which are given as equal, have the same colours in any diagram, the mind will not wander from the object before it; that is, such an arrangement presents an ocular demonstration of the parts to be proved equal, and the learner retains the data throughout the whole of the reasoning. But whatever may be the advantages of the present plan, if it be not substituted for, it can always be made a powerful auxiliary to the other methods, for the purpose of introduction, or of a more speedy reminiscence, or of more permanent retention by the memory.

The experience of all who have formed systems to impress facts on the understanding, agree in proving that coloured representations, as pictures, cuts, diagrams, etc. are more easily fixed in the mind than mere sentences unmarked by any peculiarity. Curious as it may appear, poets seem to be aware of this fact more than mathematicians; many modern poets allude to this visible system of communicating knowledge, one of them has thus expressed himself:

Sounds which address the ear are lost and die
In one short hour, but these which strike the eye,
Live long upon the mind, the faithful sight
Engraves the knowledge with a beam of light.

This perhaps may be reckoned the only improvement which plane geometry has received since the days of Euclid, and if there were any geometers of note before that time, Euclid’s success has quite eclipsed their memory, and even occasioned all good things of that kind to be assigned to him; like Æsop among the writers of Fables. It may also be worthy of remark, as tangible diagrams afford the only medium through which geometry and other linear arts and sciences can be taught to the blind, this visible system is no less adapted to the exigencies of the deaf and dumb.

Care must be taken to show that colour has nothing to do with the lines, angles, or magnitudes, except merely to name them. A mathematical line, which is length without breadth, cannot possess colour, yet the junction of the two colours on the same plane gives a good idea of what is meant by a mathematical line; recollect we are speaking familiarly, such a junction is to be understood and not the colour, when we say the black line, the red line or lines, etc.

Colours and coloured diagrams may at first appear a clumsy method to convey proper notations of the properties and parts of mathematical figures and magnitudes, however they will be found to afford a means more refined and extensive than any that has been hitherto proposed.

We shall here define a point, a line, and a surface, and demonstrate a proposition in order to show the truth of this assertion.

A point is that which has position, but not magnitude; or a point is position only, abstracted from the consideration of length, breadth, and thickness. Perhaps the following description is better calculated to explain the nature of a mathematical point to those who have not acquired the idea, than the above specious definition.

Circle divided into three equal slices

Let three colours meet and cover a portion of the paper, where they meet is not blue, nor is it yellow, nor is it red, as it occupies no portion of the plane, for if it did, it would belong to the blue, the red, or the yellow part; yet it exists, and has position without magnitude, so that with a little reflection, this junction of three colours on a plane gives a good idea of a mathematical point.

A line is length without breadth. With the assistance of colours, nearly in the same manner as before, an idea of a line may be thus given:—

Two colors stacked

Let two colours meet and cover a portion of the paper; where they meet is not red, nor is it blue; therefore the junction occupies no portion of the plane, and therefore it cannot have breadth but only length: from which we can readily form an idea of what is meant by a mathematical line. For the purpose of illustration, one colour differing from the colour of the paper, or plane upon which it is drawn, would have been sufficient; hence in future, if we say the red line, the blue line, or lines, etc. it is the junctions with the plane upon which they are drawn are to be understood.

Surface is that which has length and breadth without thickness.

Figure PQ P R S Q

When we consider a solid body (PQ), we preceive at once that it has three dimensions, namely:—length, breadth, and thickness; suppose one part of this solid (PS) to be red, and the other part (QR) yellow, and that the colours be distinct without commingling, the blue surface (RS) which separates these parts, or which is the same thing, that which divides the solid without loss of material, must be without thickness, and only possesses length and breadth; this plainly appears from reasoning, similar to that just employed in defining, or rather describing a point and a line.

Proposition 5 figure A B C D E

The proposition which we have selected to elucidate the manner in which the principles are applied is the fifth of the first Book.

In an isosceles triangle ABC, the internal angles at the base ABC, ACB are equal, and when the sides AB, AC are produced, the external angles at the base BCE, CBD are also equal.

Produce Red line and Red line
make Yellow line = Yellow line
Draw Blue line = Blue line (B. 1. pr. 3.)
in Left triangle and Right triangle
we have Left red and yellow lines = Right red and yellow lines
Red line = Red line and Black angle common:
∴ Blue line = Blue line , Left blue and yellow angles = Right blue and yellow angles
and Left red angle = Right red angle (B. 1. pr. 4.)

Again in Lower left triangle and Lower right triangle ,
Yellow line = Yellow line ,
Blue line = Blue line ,
and Left red angle = Right red angle ;
∴ Left yellow angle plus remaining angle = Right yellow angle plus remaining angle
and Left yellow angle = Right yellow angle (B. 1. pr. 4).

But Left blue and yellow angles = Right blue and yellow angles ,
∴ Left blue angle = Right blue angle .

Q. E. D.

By annexing Letters to the Diagram.

Let the equal sides AB and AC be produced through the extremities BC, of the third side, and in the produced part BD of either, let any point D be assumed, and from the other let AE be cut off equal to AD (B. 1. pr. 3). Let the points E and D, so taken in the produced sides, be connected by straight lines DC and BE with the alternate extremities of the third side of the triangle.

In the triangles DAC and EAB the sides DA and AC are respectively equal to EA and AB, and the included angle A is common to both triangles. Hence (B. 1. pr. 4.) the line DC is equal to BE, the angle ADC to the angle AEB, and the angle ACD to the angle ABE; if from the equal lines AD and AE the equal sides AB and AC be taken, the remainders BD and CE will be equal. Hence in the triangles BDC and CEB, the sides BD and DC are respectively equal to CE and EB, and the angles D and E included by those sides are also equal. Hence (B. 1. pr. 4.) the angles DBC and ECB, which are those included by the third side BC and the productions of the equal sides AB and AC are equal. Also the angles DCB and EBC are equal if those equals be taken from the angles DCA and EBA before proved equal, the remainders, which are the angles ABC and ACB opposite to the equal sides, will be equal.

Therefore in an isosceles triangle, etc.

Q. E. D.

Our object in this place being to introduce the system rather than to teach any particular set of propositions, we have therefore selected the foregoing out of the regular course. For schools and other public places of instruction, dyed chalks will answer to describe diagrams, etc. for private use coloured pencils will be found very convenient.

We are happy to find that the Elements of Mathematics now forms a considerable part of every sound female education, therefore we call the attention of those interested or engaged in the education of ladies to this very attractive mode of communicating knowledge, and to the succeeding work for its future development.

We shall for the present conclude by observing, as the senses of sight and hearing can be so forcibly and instantaneously addressed alike with one thousand as with one, the million might be taught geometry and other branches of mathematics with great ease, this would advance the purpose of education more than any thing that might be named, for it would teach the people how to think, and not what to think; it is in this particular the great error of education originates.

Books.

  1. Proposition 47 figure
    Book I. Basic plane geometry
  2. Proposition 7 figure
    Book II. Geometric algebra
  3. Proposition 21 figure 1
    Book III. Circles and angles
  4. Proposition 4 figure
    Book IV. Regular polygons
  5. Proposition 24 figure
    Book V. Ratios and proportions
  6. Proposition 3 figure
    Book VI. Geometric proportions