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CHAPTER XVI

GEOMETRY

There is certainly no royal road to geometry, but the way may be rendered easy and pleasant by timely preparations for the journey.

Without any previous knowledge of the country, or of its peculiar language, how can we expect that our young traveller should advance with facility or pleasure? We are anxious that our pupil should acquire a taste for accurate reasoning, and we resort to Geometry, as the most perfect, and the purest series of ratiocination which has been invented. Let us, then, sedulously avoid whatever may disgust him; let his first steps be easy, and successful; let them be frequently repeated until he can trace them without a guide.

We have recommended in the chapter upon Toys, that children should, from their earliest years, be accustomed to the shape of what are commonly called regular solids; they should also be accustomed to the figures in mathematical diagrams. To these should be added their respective names, and the whole language of the science should be rendered as familiar as possible.

Mr. Donne, an ingenious mathematician of Bristol, has published a prospectus of an Essay on Mechanical Geometry: he has executed, and employed with success, models in wood and metal for demonstrating propositions in geometry in a palpable manner. We have endeavoured, in vain, to procure a set of these models for our own pupils, but we have no doubt of their entire utility.

What has been acquired in childhood, should not be suffered to escape the memory. Dionysius¹⁹ had mathematical diagrams described upon the floors of his apartments, and thus recalled their demonstrations to his memory. The slightest addition that can be conceived, if it be continued daily, will imperceptibly, not only preserve what has been already acquired, but will, in a few years, amount to as large a stock of mathematical knowledge as we could wish. It is not our object to make mathematicians, but to make it easy to our pupil to become a mathematician, if his interest, or his ambition, make it desirable; and, above all, to habituate him to clear reasoning, and close attention. And we may here remark, that an early acquaintance with the accuracy of mathematical demonstration, does not, within our experience, contract the powers of the imagination. On the contrary, we think that a young lady of twelve years old, who is now no more, and who had an uncommon propensity to mathematical reasoning, had an imagination remarkably vivid and inventive.²⁰

We have accustomed our pupils to form in their minds the conception of figures generated from points and lines, and surfaces supposed to move in different directions, and with different velocities. It may be thought, that this would be a difficult occupation for young minds; but, upon trial, it will be found not only easy to them, but entertaining. In their subsequent studies, it will be of material advantage; it will facilitate their progress not only in pure mathematics, but in mechanics and astronomy, and in every operation of the mind which requires exact reflection.

To demand steady thought from a person who has not been trained to it, is one of the most unprofitable and dangerous requisitions that can be made in education.

 
"Full in the midst of Euclid dip at once,
And petrify a genius to a dunce."
 

In the usual commencement of mathematical studies, the learner is required to admit that a point, of which he sees the prototype, a dot before him, has neither length, breadth, nor thickness. This, surely, is a degree of faith not absolutely necessary for the neophyte in science. It is an absurdity which has, with much success, been attacked in "Observations on the Nature of Demonstrative Evidence," by Doctor Beddoes.

We agree with the doctor as to the impropriety of calling a visible dot, a point without dimensions. But, notwithstanding the high respect which the author commands by a steady pursuit of truth on all subjects of human knowledge, we cannot avoid protesting against part of the doctrine which he has endeavoured to inculcate. That the names point, radius, &c. are derived from sensible objects, need not be disputed; but surely the word centre can be understood by the human mind without the presence of any visible or tangible substance.

Where two lines meet, their junction cannot have dimensions; where two radii of a circle meet, they constitute the centre, and the name centre may be used for ever without any relation to a tangible or visible point. The word boundary, in like manner, means the extreme limit we call a line; but to assert that it has thickness, would, from the very terms which are used to describe it, be a direct contradiction. Bishop Berkely, Mr. Walton, Philathetes Cantabrigiensis, and Mr. Benjamin Robins, published several pamphlets upon this subject about half a century ago. No man had a more penetrating mind than Berkely; but we apprehend that Mr. Robins closed the dispute against him. This is not meant as an appeal to authority, but to apprize such of our readers as wish to consider the argument, where they may meet an accurate investigation of the subject. It is sufficient for our purpose, to warn preceptors not to insist upon their pupils' acquiescence in the dogma, that a point, represented by a dot, is without dimensions; and at the same time to profess, that we understand distinctly what is meant by mathematicians when they speak of length without breadth, and of a superfices without depth; expressions which, to our minds, convey a meaning as distinct as the name of any visible or tangible substance in nature, whose varieties from shade, distance, colour, smoothness, heat, &c. are infinite, and not to be comprehended in any definition.

In fact, this is a dispute merely about words, and as the extension of the art of printing puts it in the power of every man to propose and to defend his opinions at length, and at leisure, the best friends may support different sides of a question with mutual regard, and the most violent enemies with civility and decorum. Can we believe that Tycho Brahe lost half his nose in a dispute with a Danish nobleman about a mathematical demonstration?

CHAPTER XVII

ON MECHANICS

Parents are anxious that children should be conversant with Mechanics, and with what are called the Mechanic Powers. Certainly no species of knowledge is better suited to the taste and capacity of youth, and yet it seldom forms a part of early instruction. Every body talks of the lever, the wedge, and the pulley, but most people perceive, that the notions which they have of their respective uses, are unsatisfactory, and indistinct; and many endeavour, at a late period of life, to acquire a scientific and exact knowledge of the effects that are produced by implements which are in every body's hands, or that are absolutely necessary in the daily occupations of mankind.

An itinerant lecturer seldom fails of having a numerous and attentive auditory; and if he does not communicate much of that knowledge which he endeavours to explain, it is not to be attributed either to his want of skill, or to the insufficiency of his apparatus, but to the novelty of the terms which he is obliged to use. Ignorance of the language in which any science is taught, is an insuperable bar to its being suddenly acquired; besides a precise knowledge of the meaning of terms, we must have an instantaneous idea excited in our minds whenever they are repeated; and, as this can be acquired only by practice, it is impossible that philosophical lectures can be of much service to those who are not familiarly acquainted with the technical language in which they are delivered; and yet there is scarcely any subject of human inquiry more obvious to the understanding, than the laws of mechanics. Only a small portion of geometry is necessary to the learner, if he even wishes to become master of the more difficult problems which are usually contained in a course of lectures, and most of what is practically useful, may be acquired by any person who is expert in common arithmetic.

But we cannot proceed a single step without deviating from common language; if the theory of the balance, or the lever, is to be explained, we immediately speak of space and time. To persons not versed in literature, it is probable that these terms appear more simple and unintelligible than they do to a man who has read Locke, and other metaphysical writers. The term space to the bulk of mankind, conveys the idea of an interval; they consider the word time as representing a definite number of years, days, or minutes; but the metaphysician, when he hears the words space and time, immediately takes the alarm, and recurs to the abstract notions which are associated with these terms; he perceives difficulties unknown to the unlearned, and feels a confusion of ideas which distracts his attention. The lecturer proceeds with confidence, never supposing that his audience can be puzzled by such common terms. He means by space, the distance from the place whence a body begins to fall, to the place where its motion ceases; and by time, he means the number of seconds, or of any determinate divisions of civil time which elapse from the commencement of any motion to its end; or, in other words, the duration of any given motion. After this has been frequently repeated, any intelligent person perceives the sense in which they are used by the tenour of the discourse; but in the interim, the greatest part of what he has heard, cannot have been understood, and the premises upon which every subsequent demonstration is founded, are unknown to him. If this be true, when it is affirmed of two terms only, what must be the situation of those to whom eight or ten unknown technical terms occur at the commencement of a lecture? A complete knowledge, such a knowledge as is not only full, but familiar, of all the common terms made use of in theoretic and practical mechanics, is, therefore, absolutely necessary before any person can attend public lectures in natural philosophy with advantage.

What has been said of public lectures, may, with equal propriety, be applied to private instruction; and it is probable, that inattention to this circumstance is the reason why so few people have distinct notions of natural philosophy. Learning by rote, or even reading repeatedly, definitions of the technical terms of any science, must undoubtedly facilitate its acquirement; but conversation, with the habit of explaining the meaning of words, and the structure of common domestic implements, to children, is the sure and effectual method of preparing the mind for the acquirement of science.

The ancients, in learning this species of knowledge, had an advantage of which we are deprived: many of their terms of science were the common names of familiar objects. How few do we meet who have a distinct notion of the words radius, angle, or valve. A Roman peasant knew what a radius or a valve meant, in their original signification, as well as a modern professor; he knew that a valve was a door, and a radius a spoke of a wheel; but an English child finds it as difficult to remember the meaning of the word angle, as the word parabola. An angle is usually confounded, by those who are ignorant of geometry and mechanics, with the word triangle, and the long reasoning of many a laborious instructer has been confounded by this popular mistake. When a glass pump is shown to an admiring spectator, he is desired to watch the motion of the valves: he looks "above, about, and underneath;" but, ignorant of the word valve, he looks in vain. Had he been desired to look at the motion of the little doors that opened and shut, as the handle of the pump was moved up and down, he would have followed the lecturer with ease, and would have understood all his subsequent reasoning. If a child attempts to push any thing heavier than himself, his feet slide away from it, and the object can be moved only at intervals, and by sudden starts; but if he be desired to prop his feet against the wall, he finds it easy to push what before eluded his little strength. Here the use of a fulcrum, or fixed point, by means of which bodies may be moved, is distinctly understood. If two boys lay a board across a narrow block of wood, or stone, and balance each other at the opposite ends of it, they acquire another idea of a centre of motion. If a poker is rested against a bar of a grate, and employed to lift up the coals, the same notion of a centre is recalled to their minds. If a boy, sitting upon a plank, a sofa, or form, be lifted up by another boy's applying his strength at one end of the seat, whilst the other end of the seat rests on the ground, it will be readily perceived by them, that the point of rest, or centre of motion, or fulcrum, is the ground, and that the fulcrum is not, as in the first instance, between the force that lifts, and the thing that is lifted; the fulcrum is at one end, the force which is exerted acts at the other end, and the weight is in the middle. In trying, these simple experiments, the terms fulcrum, centre of motion, &c. should be constantly employed, and in a very short time they would be as familiar to a boy of eight years old as to any philosopher. If for some years the same words frequently recur to him in the same sense, is it to be supposed that a lecture upon the balance and the lever would be as unintelligible to him as to persons of good abilities, who at a more advanced age hear these terms from the mouth of a lecturer? A boy in such circumstances would appear as if he had a genius for mechanics, when, perhaps, he might have less taste for the science, and less capacity, than the generality of the audience. Trifling as it may at first appear, it will not be found a trifling advantage, in the progress of education, to attend to this circumstance. A distinct knowledge of a few terms, assists a learner in his first attempts; finding these successful, he advances with confidence, and acquires new ideas without difficulty or disgust. Rousseau, with his usual eloquence, has inculcated the necessity of annexing ideas to words; he declaims against the splendid ignorance of men who speak by rote, and who are rich in words amidst the most deplorable poverty of ideas. To store the memory of his pupil with images of things, he is willing to neglect, and leave to hazard, his acquirement of language. It requires no elaborate argument to prove that a boy, whose mind was stored with accurate images of external objects, of experimental knowledge, and who had acquired habitual dexterity, but who was unacquainted with the usual signs by which ideas are expressed, would be incapable of accurate reasoning, or would, at best, reason only upon particulars. Without general terms, he could not abstract; he could not, until his vocabulary was enlarged, and familiar to him, reason upon general topics, or draw conclusions from general principles: in short, he would be in the situation of those who, in the solution of difficult and complicated questions relative to quantity, are obliged to employ tedious and perplexed calculations, instead of the clear and comprehensive methods that unfold themselves by the use of signs in algebra.

It is not necessary, in teaching children the technical language of any art or science, that we should pursue the same order that is requisite in teaching the science itself. Order is required in reasoning, because all reasoning is employed in deducing propositions from one another in a regular series; but where terms are employed merely as names, this order may be dispensed with. It is, however, of great consequence to seize the proper time for introducing a new term; a moment when attention is awake, and when accident has produced some particular interest in the object. In every family, opportunities of this sort occur without any preparation, and such opportunities are far preferable to a formal lecture and a splendid apparatus for the first lessons in natural philosophy and chemistry. If the pump belonging to the house is out of order, and the pump-maker is set to work, an excellent opportunity presents itself for variety of instruction. The centre pin of the handle is taken out, and a long rod is drawn up by degrees, at the end of which a round piece of wood is seen partly covered with leather. Your pupil immediately asks the name of it, and the pump-maker prevents your answer, by informing little master that it is called a sucker. You show it to the child, he handles it, feels whether the leather is hard or soft, and at length discovers that there is a hole through it which is covered with a little flap or door. This, he learns from the workmen, is called a clack. The child should now be permitted to plunge the piston (by which name it should now be called) into a tub of water; in drawing it backwards and forwards, he will perceive that the clack, which should now be called the valve, opens and shuts as the piston is drawn backwards and forwards. It will be better not to inform the child how this mechanism is employed in the pump. If the names sucker and piston, clack and valve, are fixed in his memory, it will be sufficient for his first lesson. At another opportunity, he should be present when the fixed or lower valve of the pump is drawn up; he will examine it, and find that it is similar to the valve of the piston; if he sees it put down into the pump, and sees the piston put into its place, and set to work, the names that he has learned will be fixed more deeply in his mind, and he will have some general notion of the whole apparatus. From time to time these names should be recalled to his memory on suitable occasions, but he should not be asked to repeat them by rote. What has been said, is not intended as a lesson for a child in mechanics, but as a sketch of a method of teaching which has been employed with success.

Whatever repairs are carried on in a house, children should be permitted to see: whilst every body about them seems interested, they become attentive from sympathy; and whenever action accompanies instruction, it is sure to make an impression. If a lock is out of order, when it is taken off, show it to your pupil; point out some of its principal parts, and name them; then put it into the hands of a child, and let him manage it as he pleases. Locks are full of oil, and black with dust and iron; but if children have been taught habits of neatness, they may be clock-makers and white-smiths, without spoiling their clothes, or the furniture of a house. Upon every occasion of this sort, technical terms should be made familiar; they are of great use in the every-day business of life, and are peculiarly serviceable in giving orders to workmen, who, when they are spoken to in a language that they are used to, comprehend what is said to them, and work with alacrity.

An early use of a rule and pencil, and easy access to prints of machines, of architecture, and of the implements of trades, are of obvious use in this part of education. The machines published by the Society of Arts in London; the prints in Desaguliers, Emerson, le Spectacle de la Nature, Machines approuvées par l'Académie, Chambers's Dictionary, Berthoud sur l'Horlogerie, Dictionaire des Arts et des Métiers, may, in succession, be put into the hands of children. The most simple should be first selected, and the pupils should be accustomed to attend minutely to one print before another is given to them. A proper person should carefully point out and explain to them the first prints that they examine; they may afterwards be left to themselves.

To understand prints of machines, a previous knowledge of what is meant by an elevation, a profile, a section, a perspective view, and a (vue d'oiseau) bird's eye view, is necessary. To obtain distinct ideas of sections, a few models of common furniture, as chests of drawers, bellows, grates, &c. may be provided, and may be cut asunder in different directions. Children easily comprehend this part of drawing, and its uses, which may be pointed out in books of architecture; its application to the common business of life, is so various and immediate, as to fix it for ever in the memory; besides, the habit of abstraction, which is acquired by drawing the sections of complicated architecture or machinery, is highly advantageous to the mind. The parts which we wish to express, are concealed, and are suggested partly by the elevation or profile of the figure, and partly by the connection between the end proposed in the construction of the building, machine, &c. and the means which are adapted to effect it.

A knowledge of perspective, is to be acquired by an operation of the mind directly opposite to what is necessary in delineating the sections of bodies; the mind must here be intent only upon the objects that are delineated upon the retina, exactly what we see; it must forget or suspend the knowledge which it has acquired from experience, and must see with the eye of childhood, no further than the surface. Every person, who is accustomed to drawing in perspective, sees external nature, when he pleases, merely as a picture: this habit contributes much to form a taste for the fine arts; it may, however, be carried to excess. There are improvers who prefer the most dreary ruin to an elegant and convenient mansion, and who prefer a blasted stump to the glorious foliage of the oak.

Perspective is not, however, recommended merely as a means of improving the taste, but as it is useful in facilitating the knowledge of mechanics. When once children are familiarly acquainted with perspective, and with the representations of machines by elevations, sections, &c. prints will supply them with an extensive variety of information; and when they see real machines, their structure and uses will be easily comprehended. The noise, the seeming confusion, and the size of several machines, make it difficult to comprehend and combine their various parts, without much time, and repeated examination; the reduced size of prints lays the whole at once before the eye, and tends to facilitate not only comprehension, but contrivance. Whoever can delineate progressively as he invents, saves much labour, much time, and the hazard of confusion. Various contrivances have been employed to facilitate drawing in perspective, as may be seen in "Cabinet de Servier, Memoires of the French Academy, Philosophical Transactions, and lately in the Repertory of Arts." The following is simple, cheap, and portable.

PLATE 1. FIG. 1

A B C, three mahogany boards, two, four, and six inches long, and of the same breadth respectively, so as to double in the manner represented.

PLATE 1. FIG. 2

The part A is screwed, or clamped to a table of a convenient height, and a sheet of paper, one edge of which is put under the piece A, will be held fast to the table.

The index P is to be set (at pleasure) with it sharp point to any part of an object which the eye sees through E, the eye-piece.

The machine is now to be doubled as in Fig. 2, taking care that the index be not disturbed; the point, which was before perpendicular, will then approach the paper horizontally, and the place to which it points on the paper, must be marked with a pencil. The machine must be again unfolded, and another point of the object is to be ascertained in the same manner as before; the space between these points may be then connected with a line; fresh points should then be taken, marked with a pencil, and connected with a line; and so on successively, until the whole object is delineated.

Besides the common terms of art, the technical terms of science should, by degrees, be rendered familiar to our pupils. Amongst these the words Space and Time occur, as we have observed, the soonest, and are of the greatest importance. Without exact definitions, or abstract reasonings, a general notion of the use of these terms may be inculcated by employing them frequently in conversation, and by applying them to things and circumstances which occur without preparation, and about which children are interested, or occupied. "There is a great space left between the words in that printing." The child understands, that space in this sentence means white paper between black letters. "You should leave a greater space between the flowers which you are planting" – he knows that you mean more ground. "There is a great space between that boat and the ship" – space of water. "I hope the hawk will not be able to catch that pigeon, there is a great space between them" – space of air. "The men who are pulling that sack of corn into the granary, have raised it through half the space between the door and the ground." A child cannot be at any loss for the meaning of the word space in these or any other practical examples which may occur; but he should also be used to the word space as a technical expression, and then he will not be confused or stopped by a new term when employed in mechanics.

The word time may be used in the same manner upon numberless occasions to express the duration of any movement which is performed by the force of men, or horses, wind, water, or any mechanical power.

"Did the horses in the mill we saw yesterday, go as fast as the horses which are drawing the chaise?" "No, not as fast as the horses go at present on level ground; but they went as fast as the chaise-horses do when they go up hill, or as fast as horses draw a waggon."

"How many times do the sails of that wind-mill go round in a minute? Let us count; I will look at my watch; do you count how often the sails go round; wait until that broken arm is uppermost, and when you say now, I will begin to count the time; when a minute has past, I will tell you."

After a few trials, this experiment will become easy to a child of eight or nine years old; he may sometimes attend to the watch, and at other times count the turns of the sails; he may easily be made to apply this to a horse-mill, or to a water-mill, a corn-fan, or any machine that has a rotatory motion; he will be entertained with his new employment; he will compare the velocities of different machines; the meaning of this word will be easily added to his vocabulary.

"Does that part of the arms of the wind-mill which is near the axle-tree, or centre, I mean that part which has no cloth or sail upon it, go as fast as the ends of the arms that are the farthest from the centre?"

"No, not near so fast."

"But that part goes as often round in a minute as the rest of the sail."

"Yes, but it does not go as fast."

"How so?"

"It does not go so far round."

"No, it does not. The extremities of the sails go through more space in the same time than the part near the centre."

By conversations like these, the technical meaning of the word velocity may be made quite familiar to a child much younger than what has been mentioned; he may not only comprehend that velocity means time and space considered together, but if he is sufficiently advanced in arithmetic, he may be readily taught how to express and compare in numbers velocities composed of certain portions of time and space. He will not inquire about the abstract meaning of the word space; he has seen space measured on paper, on timber, on the water, in the air, and he perceives distinctly that it is a term equally applicable to all distances that can exist between objects of any sort, or that he can see, feel, or imagine.

Momentum, a less common word, the meaning of which is not quite so easy to convey to a child, may, by degrees, be explained to him: at every instant he feels the effect of momentum in his own motions, and in the motions of every thing that strikes against him; his feelings and experience require only proper terms to become the subject of his conversation. When he begins to inquire, it is the proper time to instruct him. For instance, a boy of ten years old, who had acquired the meaning of some other terms in science, this morning asked the meaning of the word momentum; he was desired to explain what he thought it meant.

He answered, "Force."

"What do you mean by force?"

"Effort."

"Of what?"

"Of gravity."

"Do you mean that force by which a body is drawn down to the earth?"

"No."

"Would a feather, if it were moving with the greatest conceivable swiftness or velocity, throw down a castle?"

"No."²¹

"Would a mountain torn up by the roots, as fabled in Milton, if it moved with the least conceivable velocity, throw down a castle?"

"Yes, I think it would."

The difference between an uniform, and an uniformly accelerated motion, the measure of the velocity of falling bodies, the composition of motions communicated to the same body in different directions at the same time, and the cause of the curvilinear track of projectiles, seem, at first, intricate subjects, and above the capacity of boys of ten or twelve years old; but by short and well-timed lessons, they may be explained without confounding or fatiguing their attention. We tried another experiment whilst this chapter was writing, to determine whether we had asserted too much upon this subject. After a conversation between two boys upon the descent of bodies towards the earth, and upon the measure of the increasing velocity with which they fall, they were desired, with a view to ascertain whether they understood what was said, to invent a machine which should show the difference between an uniform and an accelerated velocity, and in particular to show, by occular demonstration, "that if one body moves in a given time through a given space, with an uniform motion, and if another body moves through the same space in the same time with an uniformly accelerated motion, the uniform motion of the one will be equal to half the accelerated motion of the other." The eldest boy, H – , thirteen years old, invented and executed the following machine for this purpose:

Plate I, Fig. 3. b is a bracket 9 inches by 5, consisting of a back and two sides of hard wood: two inches from the back two slits are made in the sides of the bracket half an inch deep, and an eighth of an inch wide, to receive the two wire pivots of a roller; which roller is composed of a cylinder, three inches long and half an inch diameter; and a cone three inches long and one inch diameter in its largest part or base. The cylinder and cone are not separate, but are turned out of one piece; a string is fastened to the cone at its base a, with a bullet or any other small weight at the other end of it; and another string and weight are fastened to the cylinder at c; the pivot p of wire is bent into the form of a handle; if the handle is turned either way, the strings will be respectively wound up upon the cone and cylinder; their lengths should now be adjusted, so that when the string on the cone is wound up as far as the cone will permit, the two weights may be at an equal distance from the bottom of the bracket, which bottom we suppose to be parallel with the pivots; the bracket should now be fastened against a wall, at such a height as to let the weights lightly touch the floor when the strings are unwound: silk or bobbin is a proper kind of string for this purpose, as it is woven or plaited, and therefore is not liable to twist. When the strings are wound up to their greatest heights, if the handle be suddenly let go, both the weights will begin to fall at the same moment; but the weight 1, will descend at first but slowly, and will pass through but small space compared with the weight 2. As they descend further, No. 2 still continues to get before No. 1; but after some time, No. 1 begins to overtake No. 2, and at last they come to the ground together. If this machine is required to show exactly the space that a falling body would describe in given times, the cone and cylinder must have grooves cut spirally upon their circumference, to direct the string with precision. To describe these spiral lines, became a new subject of inquiry. The young mechanics were again eager to exert their powers of invention; the eldest invented a machine upon the same principle as that which is used by the best workmen for cutting clock fusees; and it is described in Berthoud. The youngest invented the engine delineated, Plate 1, Fig. 4.