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Chapter 2 Mathematical Definitions and Teaching
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1. I should speak here of general definitions in mathematics; at least that is the title, but it will be impossible to confine myself to the subject as strictly1 as the rule of unity2 of action would require; I shall not be able to treat it without touching3 upon a few other related questions, and if thus I am forced from time to time to walk on the bordering flower-beds on the right or left, I pray you bear with me.

What is a good definition? For the philosopher or the scientist it is a definition which applies to all the objects defined, and only those; it is the one satisfying the rules of logic4. But in teaching it is not that; a good definition is one understood by the scholars.

How does it happen that so many refuse to understand mathematics? Is that not something of a paradox5? Lo and behold6! a science appealing only to the fundamental principles of logic, to the principle of contradiction, for instance, to that which is the skeleton, so to speak, of our intelligence, to that of which we can not divest7 ourselves without ceasing to think, and there are people who find it obscure! and they are even in the majority! That they are incapable8 of inventing may pass, but that they do not understand the demonstrations9 shown them, that they remain blind when we show them a light which seems to us flashing pure flame, this it is which is altogether prodigious11.

And yet there is no need of a wide experience with examinations to know that these blind men are in no wise exceptional beings. This is a problem not easy to solve, but which should engage the attention of all those wishing to devote themselves to teaching.

What is it, to understand? Has this word the same meaning for all the world? To understand the demonstration10 of a theorem, is that to examine successively each of the syllogisms composing it and to ascertain12 its correctness, its conformity13 to the rules of the game? Likewise, to understand a definition, is this merely to recognize that one already knows the meaning of all the terms employed and to ascertain that it implies no contradiction?

For some, yes; when they have done this, they will say: I understand.

For the majority, no. Almost all are much more exacting14; they wish to know not merely whether all the syllogisms of a demonstration are correct, but why they link together in this order rather than another. In so far as to them they seem engendered16 by caprice and not by an intelligence always conscious of the end to be attained17, they do not believe they understand.

Doubtless they are not themselves just conscious of what they crave18 and they could not formulate19 their desire, but if they do not get satisfaction, they vaguely20 feel that something is lacking. Then what happens? In the beginning they still perceive the proofs one puts under their eyes; but as these are connected only by too slender a thread to those which precede and those which follow, they pass without leaving any trace in their head; they are soon forgotten; a moment bright, they quickly vanish in night eternal. When they are farther on, they will no longer see even this ephemeral light, since the theorems lean one upon another and those they would need are forgotten; thus it is they become incapable of understanding mathematics.

This is not always the fault of their teacher; often their mind, which needs to perceive the guiding thread, is too lazy to seek and find it. But to come to their aid, we first must know just what hinders them.

Others will always ask of what use is it; they will not have understood if they do not find about them, in practise or in nature, the justification21 of such and such a mathematical concept. Under each word they wish to put a sensible image; the definition must evoke22 this image, so that at each stage of the demonstration they may see it transform and evolve. Only upon this condition do they comprehend and retain. Often these deceive themselves; they do not listen to the reasoning, they look at the figures; they think they have understood and they have only seen.

2. How many different tendencies! Must we combat them? Must we use them? And if we wish to combat them, which should be favored? Must we show those content with the pure logic that they have seen only one side of the matter? Or need we say to those not so cheaply satisfied that what they demand is not necessary?

In other words, should we constrain23 the young people to change the nature of their minds? Such an attempt would be vain; we do not possess the philosopher’s stone which would enable us to transmute24 one into another the metals confided25 to us; all we can do is to work with them, adapting ourselves to their properties.

Many children are incapable of becoming mathematicians26, to whom however it is necessary to teach mathematics; and the mathematicians themselves are not all cast in the same mold. To read their works suffices to distinguish among them two sorts of minds, the logicians like Weierstrass for example, the intuitives like Riemann. There is the same difference among our students. The one sort prefer to treat their problems ‘by analysis’ as they say, the others ‘by geometry.’

It is useless to seek to change anything of that, and besides would it be desirable? It is well that there are logicians and that there are intuitives; who would dare say whether he preferred that Weierstrass had never written or that there never had been a Riemann? We must therefore resign ourselves to the diversity of minds, or better we must rejoice in it.

3. Since the word understand has many meanings, the definitions which will be best understood by some will not be best suited to others. We have those which seek to produce an image, and those where we confine ourselves to combining empty forms, perfectly28 intelligible29, but purely30 intelligible, which abstraction has deprived of all matter.

I know not whether it be necessary to cite examples. Let us cite them, anyhow, and first the definition of fractions will furnish us an extreme case. In the primary schools, to define a fraction, one cuts up an apple or a pie; it is cut up mentally of course and not in reality, because I do not suppose the budget of the primary instruction allows of such prodigality31. At the Normal School, on the other hand, or at the college, it is said: a fraction is the combination of two whole numbers separated by a horizontal bar; we define by conventions the operations to which these symbols may be submitted; it is proved that the rules of these operations are the same as in calculating with whole numbers, and we ascertain finally that multiplying the fraction, according to these rules, by the denominator gives the numerator. This is all very well because we are addressing young people long familiarized with the notion of fractions through having cut up apples or other objects, and whose mind, matured by a hard mathematical education, has come little by little to desire a purely logical definition. But the débutant to whom one should try to give it, how dumfounded!

Such also are the definitions found in a book justly admired and greatly honored, the Foundations of Geometry by Hilbert. See in fact how he begins: We think three systems of things which we shall call points, straights and planes. What are these ‘things’?

We know not, nor need we know; it would even be a pity to seek to know; all we have the right to know of them is what the assumptions tell us; this for example: Two distinct points always determine a straight, which is followed by this remark: in place of determine, we may say the two points are on the straight, or the straight goes through these two points or joins the two points.

Thus ‘to be on a straight’ is simply defined as synonymous with ‘determine a straight.’ Behold a book of which I think much good, but which I should not recommend to a school boy. Yet I could do so without fear, he would not read much of it. I have taken extreme examples and no teacher would dream of going that far. But even stopping short of such models, does he not already expose himself to the same danger?

Suppose we are in a class; the professor dictates32: the circle is the locus33 of points of the plane equidistant from an interior point called the center. The good scholar writes this phrase in his note-book; the bad scholar draws faces; but neither understands; then the professor takes the chalk and draws a circle on the board. “Ah!” think the scholars, “why did he not say at once: a circle is a ring, we should have understood.” Doubtless the professor is right. The scholars’ definition would have been of no avail, since it could serve for no demonstration, since besides it would not give them the salutary habit of analyzing34 their conceptions. But one should show them that they do not comprehend what they think they know, lead them to be conscious of the roughness of their primitive35 conception, and of themselves to wish it purified and made precise.

4. I shall return to these examples; I only wished to show you the two opposed conceptions; they are in violent contrast. This contrast the history of science explains. If we read a book written fifty years ago, most of the reasoning we find there seems lacking in rigor36. Then it was assumed a continuous function can change sign only by vanishing; to-day we prove it. It was assumed the ordinary rules of calculation are applicable to incommensurable numbers; to-day we prove it. Many other things were assumed which sometimes were false.

We trusted to intuition; but intuition can not give rigor, nor even certainty; we see this more and more. It tells us for instance that every curve has a tangent, that is to say that every continuous function has a derivative37, and that is false. And as we sought certainty, we had to make less and less the part of intuition.

What has made necessary this evolution? We have not been slow to perceive that rigor could not be established in the reasonings, if it were not first put into the definitions.

The objects occupying mathematicians were long ill defined; we thought we knew them because we represented them with the senses or the imagination; but we had of them only a rough image and not a precise concept upon which reasoning could take hold. It is there that the logicians would have done well to direct their efforts.

So for the incommensurable number, the vague idea of continuity, which we owe to intuition, has resolved itself into a complicated system of inequalities bearing on whole numbers. Thus have finally vanished all those difficulties which frightened our fathers when they reflected upon the foundations of the infinitesimal calculus38. To-day only whole numbers are left in analysis, or systems finite or infinite of whole numbers, bound by a plexus of equalities and inequalities. Mathematics we say is arithmetized.

5. But do you think mathematics has attained absolute rigor without making any sacrifice? Not at all; what it has gained in rigor it has lost in objectivity. It is by separating itself from reality that it has acquired this perfect purity. We may freely run over its whole domain39, formerly40 bristling41 with obstacles, but these obstacles have not disappeared. They have only been moved to the frontier, and it would be necessary to vanquish42 them anew if we wished to break over this frontier to enter the realm of the practical.

We had a vague notion, formed of incongruous elements, some a priori, others coming from experiences more or less digested; we thought we knew, by intuition, its principal properties. To-day we reject the empiric elements, retaining only the a priori; one of the properties serves as definition and all the others are deduced from it by rigorous reasoning. This is all very well, but it remains43 to be proved that this property, which has become a definition, pertains44 to the real objects which experience had made known to us and whence we drew our vague intuitive notion. To prove that, it would be necessary to appeal to experience, or to make an effort of intuition, and if we could not prove it, our theorems would be perfectly rigorous, but perfectly useless.

Logic sometimes makes monsters. Since half a century we have seen arise a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhaps continuity, but no derivatives45, etc. Nay46 more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appear except as particular case. There remains for them only a very small corner.

Heretofore when a new function was invented, it was for some practical end; to-day they are invented expressly to put at fault the reasonings of our fathers, and one never will get from them anything more than that.

If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum. If you do not do it, the logicians might say, you will achieve rigor only by stages.

6. Yes, perhaps, but we can not make so cheap of reality, and I mean not only the reality of the sensible world, which however has its worth, since it is to combat against it that nine tenths of your students ask of you weapons. There is a reality more subtile, which makes the very life of the mathematical beings, and which is quite other than logic.

Our body is formed of cells, and the cells of atoms; are these cells and these atoms then all the reality of the human body? The way these cells are arranged, whence results the unity of the individual, is it not also a reality and much more interesting?

A naturalist47 who never had studied the elephant except in the microscope, would he think he knew the animal adequately? It is the same in mathematics. When the logician27 shall have broken up each demonstration into a multitude of elementary operations, all correct, he still will not possess the whole reality; this I know not what which makes the unity of the demonstration will completely escape him.

In the edifices48 built up by our masters, of what use to admire the work of the mason if we can not comprehend the plan of the architect? Now pure logic can not give us this appreciation50 of the total effect; this we must ask of intuition.

Take for instance the idea of continuous function. This is at first only a sensible image, a mark traced by the chalk on the blackboard. Little by little it is refined; we use it to construct a complicated system of inequalities, which reproduces all the features of the primitive image; when all is done, we have removed the centering, as after the construction of an arch; this rough representation, support thenceforth useless, has disappeared and there remains only the edifice49 itself, irreproachable51 in the eyes of the logician. And yet, if the professor did not recall the primitive image, if he did not restore momentarily the centering, how could the student divine by what caprice all these inequalities have been scaffolded in this fashion one upon another? The definition would be logically correct, but it would not show him the veritable reality.

7. So back we must return; doubtless it is hard for a master to teach what does not entirely52 satisfy him; but the satisfaction of the master is not the unique object of teaching; we should first give attention to what the mind of the pupil is and to what we wish it to become.

Zoologists53 maintain that the embryonic54 development of an animal recapitulates55 in brief the whole history of its ancestors throughout geologic56 time. It seems it is the same in the development of minds. The teacher should make the child go over the path his fathers trod; more rapidly, but without skipping stations. For this reason, the history of science should be our first guide.

Our fathers thought they knew what a fraction was, or continuity, or the area of a curved surface; we have found they did not know it. Just so our scholars think they know it when they begin the serious study of mathematics. If without warning I tell them: “No, you do not know it; what you think you understand, you do not understand; I must prove to you what seems to you evident,” and if in the demonstration I support myself upon premises57 which to them seem less evident than the conclusion, what shall the unfortunates think? They will think that the science of mathematics is only an arbitrary mass of useless subtilities; either they will be disgusted with it, or they will play it as a game and will reach a state of mind like that of the Greek sophists.

Later, on the contrary, when the mind of the scholar, familiarized with mathematical reasoning, has been matured by this long frequentation, the doubts will arise of themselves and then your demonstration will be welcome. It will awaken58 new doubts, and the questions will arise successively to the child, as they arose successively to our fathers, until perfect rigor alone can satisfy him. To doubt everything does not suffice, one must know why he doubts.

8. The principal aim of mathematical teaching is to develop certain faculties59 of the mind, and among them intuition is not the least precious. It is through it that the mathematical world remains in contact with the real world, and if pure mathematics could do without it, it would always be necessary to have recourse to it to fill up the chasm60 which separates the symbol from reality. The practician will always have need of it, and for one pure geometer there should be a hundred practicians.

The engineer should receive a complete mathematical education, but for what should it serve him?

To see the different aspects of things and see them quickly; he has no time to hunt mice. It is necessary that, in the complex physical objects presented to him, he should promptly61 recognize the point where the mathematical tools we have put in his hands can take hold. How could he do it if we should leave between instruments and objects the deep chasm hollowed out by the logicians?

9. Besides the engineers, other scholars, less numerous, are in their turn to become teachers; they therefore must go to the very bottom; a knowledge deep and rigorous of the first principles is for them before all indispensable. But this is no reason not to cultivate in them intuition; for they would get a false idea of the science if they never looked at it except from a single side, and besides they could not develop in their students a quality they did not themselves possess.

For the pure geometer himself, this faculty62 is necessary; it is by logic one demonstrates, by intuition one invents. To know how to criticize is good, to know how to create is better. You know how to recognize if a combination is correct; what a predicament if you have not the art of choosing among all the possible combinations. Logic tells us that on such and such a way we are sure not to meet any obstacle; it does not say which way leads to the end. For that it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it the geometer would be like a writer who should be versed63 in grammar but had no ideas. Now how could this faculty develop if, as soon as it showed itself, we chase it away and proscribe64 it, if we learn to set it at naught65 before knowing the good of it.

And here permit a parenthesis66 to insist upon the importance of written exercises. Written compositions are perhaps not sufficiently67 emphasized in certain examinations, at the polytechnic68 school, for instance. I am told they would close the door against very good scholars who have mastered the course, thoroughly69 understanding it, and who nevertheless are incapable of making the slightest application. I have just said the word understand has several meanings: such students only understand in the first way, and we have seen that suffices neither to make an engineer nor a geometer. Well, since choice must be made, I prefer those who understand completely.

10. But is the art of sound reasoning not also a precious thing, which the professor of mathematics ought before all to cultivate? I take good care not to forget that. It should occupy our attention and from the very beginning. I should be distressed71 to see geometry degenerate72 into I know not what tachymetry of low grade and I by no means subscribe73 to the extreme doctrines74 of certain German Oberlehrer. But there are occasions enough to exercise the scholars in correct reasoning in the parts of mathematics where the inconveniences I have pointed75 out do not present themselves. There are long chains of theorems where absolute logic has reigned76 from the very first and, so to speak, quite naturally, where the first geometers have given us models we should constantly imitate and admire.

It is in the exposition of first principles that it is necessary to avoid too much subtility; there it would be most discouraging and moreover useless. We can not prove everything and we can not define everything; and it will always be necessary to borrow from intuition; what does it matter whether it be done a little sooner or a little later, provided that in using correctly premises it has furnished us, we learn to reason soundly.

11. Is it possible to fulfill77 so many opposing conditions? Is this possible in particular when it is a question of giving a definition? How find a concise78 statement satisfying at once the uncompromising rules of logic, our desire to grasp the place of the new notion in the totality of the science, our need of thinking with images? Usually it will not be found, and this is why it is not enough to state a definition; it must be prepared for and justified79.

What does that mean? You know it has often been said: every definition implies an assumption, since it affirms the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until one shall have proved that it involves no contradiction, neither in the terms, nor with the verities80 previously81 admitted.

But this is not enough; the definition is stated to us as a convention; but most minds will revolt if we wish to impose it upon them as an arbitrary convention. They will be satisfied only when you have answered numerous questions.

Usually mathematical definitions, as M. Liard has shown, are veritable constructions built up wholly of more simple notions. But why assemble these elements in this way when a thousand other combinations were possible?

Is it by caprice? If not, why had this combination more right to exist than all the others? To what need does it respond? How was it foreseen that it would play an important r?le in the development of the science, that it would abridge82 our reasonings and our calculations? Is there in nature some familiar object which is so to speak the rough and vague image of it?

This is not all; if you answer all these questions in a satisfactory manner, we shall see indeed that the new-born had the right to be baptized; but neither is the choice of a name arbitrary; it is needful to explain by what analogies one has been guided and that if analogous83 names have been given to different things, these things at least differ only in material and are allied84 in form; that their properties are analogous and so to say parallel.

At this cost we may satisfy all inclinations85. If the statement is correct enough to please the logician, the justification will satisfy the intuitive. But there is still a better procedure; wherever possible, the justification should precede the statement and prepare for it; one should be led on to the general statement by the study of some particular examples.

Still another thing: each of the parts of the statement of a definition has as aim to distinguish the thing to be defined from a class of other neighboring objects. The definition will be understood only when you have shown, not merely the object defined, but the neighboring objects from which it is proper to distinguish it, when you have given a grasp of the difference and when you have added explicitly86: this is why in stating the definition I have said this or that.

But it is time to leave generalities and examine how the somewhat abstract principles I have expounded87 may be applied89 in arithmetic, geometry, analysis and mechanics.
Arithmetic

12. The whole number is not to be defined; in return, one ordinarily defines the operations upon whole numbers; I believe the scholars learn these definitions by heart and attach no meaning to them. For that there are two reasons: first they are made to learn them too soon, when their mind as yet feels no need of them; then these definitions are not satisfactory from the logical point of view. A good definition for addition is not to be found just simply because we must stop and can not define everything. It is not defining addition to say it consists in adding. All that can be done is to start from a certain number of concrete examples and say: the operation we have performed is called addition.

For subtraction90 it is quite otherwise; it may be logically defined as the operation inverse91 to addition; but should we begin in that way? Here also start with examples, show on these examples the reciprocity of the two operations; thus the definition will be prepared for and justified.

Just so again for multiplication92; take a particular problem; show that it may be solved by adding several equal numbers; then show that we reach the result more quickly by a multiplication, an operation the scholars already know how to do by routine and out of that the logical definition will issue naturally.

Division is defined as the operation inverse to multiplication; but begin by an example taken from the familiar notion of partition and show on this example that multiplication reproduces the dividend93.

There still remain the operations on fractions. The only difficulty is for multiplication. It is best to expound88 first the theory of proportion; from it alone can come a logical definition; but to make acceptable the definitions met at the beginning of this theory, it is necessary to prepare for them by numerous examples taken from classic problems of the rule of three, taking pains to introduce fractional data.

Neither should we fear to familiarize the scholars with the notion of proportion by geometric images, either by appealing to what they remember if they have already studied geometry, or in having recourse to direct intuition, if they have not studied it, which besides will prepare them to study it. Finally I shall add that after defining multiplication of fractions, it is needful to justify94 this definition by showing that it is commutative, associative and distributive, and calling to the attention of the auditors95 that this is established to justify the definition.

One sees what a r?le geometric images play in all this; and this r?le is justified by the philosophy and the history of the science. If arithmetic had remained free from all admixture of geometry, it would have known only the whole number; it is to adapt itself to the needs of geometry that it invented anything else.
Geometry

In geometry we meet forthwith the notion of the straight line. Can the straight line be defined? The well-known definition, the shortest path from one point to another, scarcely satisfies me. I should start simply with the ruler and show at first to the scholar how one may verify a ruler by turning; this verification is the true definition of the straight line; the straight line is an axis96 of rotation97. Next he should be shown how to verify the ruler by sliding and he would have one of the most important properties of the straight line.

As to this other property of being the shortest path from one point to another, it is a theorem which can be demonstrated apodictically, but the demonstration is too delicate to find a place in secondary teaching. It will be worth more to show that a ruler previously verified fits on a stretched thread. In presence of difficulties like these one need not dread98 to multiply assumptions, justifying99 them by rough experiments.

It is needful to grant these assumptions, and if one admits a few more of them than is strictly necessary, the evil is not very great; the essential thing is to learn to reason soundly on the assumptions admitted. Uncle Sarcey, who loved to repeat, often said that at the theater the spectator accepts willingly all the postulates100 imposed upon him at the beginning, but the curtain once raised, he becomes uncompromising on the logic. Well, it is just the same in mathematics.

For the circle, we may start with the compasses; the scholars will recognize at the first glance the curve traced; then make them observe that the distance of the two points of the instrument remains constant, that one of these points is fixed102 and the other movable, and so we shall be led naturally to the logical definition.

The definition of the plane implies an axiom and this need not be hidden. Take a drawing board and show that a moving ruler may be kept constantly in complete contact with this plane and yet retain three degrees of freedom. Compare with the cylinder103 and the cone104, surfaces on which an applied straight retains only two degrees of freedom; next take three drawing boards; show first that they will glide105 while remaining applied to one another and this with three degrees of freedom; and finally to distinguish the plane from the sphere, show that two of these boards which fit a third will fit each other.

Perhaps you are surprised at this incessant106 employment of moving things; this is not a rough artifice107; it is much more philosophic108 than one would at first think. What is geometry for the philosopher? It is the study of a group. And what group? That of the motions of solid bodies. How define this group then without moving some solids?

Should we retain the classic definition of parallels and say parallels are two coplanar straights which do not meet, however far they be prolonged? No, since this definition is negative, since it is unverifiable by experiment, and consequently can not be regarded as an immediate109 datum110 of intuition. No, above all because it is wholly strange to the notion of group, to the consideration of the motion of solid bodies which is, as I have said, the true source of geometry. Would it not be better to define first the rectilinear translation of an invariable figure, as a motion wherein all the points of this figure have rectilinear trajectories111; to show that such a translation is possible by making a square glide on a ruler?

From this experimental ascertainment112, set up as an assumption, it would be easy to derive113 the notion of parallel and Euclid’s postulate101 itself.

Mechanics

I need not return to the definition of velocity114, or acceleration115, or other kinematic notions; they may be advantageously connected with that of the derivative.

I shall insist, on the other hand, upon the dynamic notions of force and mass.

I am struck by one thing: how very far the young people who have received a high-school education are from applying to the real world the mechanical laws they have been taught. It is not only that they are incapable of it; they do not even think of it. For them the world of science and the world of reality are separated by an impervious116 partition wall.

If we try to analyze117 the state of mind of our scholars, this will astonish us less. What is for them the real definition of force? Not that which they recite, but that which, crouching118 in a nook of their mind, from there directs it wholly. Here is the definition: forces are arrows with which one makes parallelograms. These arrows are imaginary things which have nothing to do with anything existing in nature. This would not happen if they had been shown forces in reality before representing them by arrows.

How shall we define force?

I think I have elsewhere sufficiently shown there is no good logical definition. There is the anthropomorphic definition, the sensation of muscular effort; this is really too rough and nothing useful can be drawn119 from it.

Here is how we should go: first, to make known the genus force, we must show one after the other all the species of this genus; they are very numerous and very different; there is the pressure of fluids on the insides of the vases wherein they are contained; the tension of threads; the elasticity120 of a spring; the gravity working on all the molecules121 of a body; friction122; the normal mutual123 action and reaction of two solids in contact.

This is only a qualitative124 definition; it is necessary to learn to measure force. For that begin by showing that one force may be replaced by another without destroying equilibrium125; we may find the first example of this substitution in the balance and Borda’s double weighing.

Then show that a weight may be replaced, not only by another weight, but by force of a different nature; for instance, Prony’s brake permits replacing weight by friction.

From all this arises the notion of the equivalence of two forces.

The direction of a force must be defined. If a force F is equivalent to another force F′ applied to the body considered by means of a stretched string, so that F may be replaced by F′ without affecting the equilibrium, then the point of attachment126 of the string will be by definition the point of application of the force F′, and that of the equivalent force F; the direction of the string will be the direction of the force F′ and that of the equivalent force F.

From that, pass to the comparison of the magnitude of forces. If a force can replace two others with the same direction, it equals their sum; show for example that a weight of 20 grams may replace two 10-gram weights.

Is this enough? Not yet. We now know how to compare the intensity127 of two forces which have the same direction and same point of application; we must learn to do it when the directions are different. For that, imagine a string stretched by a weight and passing over a pulley; we shall say that the tensor of the two legs of the string is the same and equal to the tension weight.

This definition of ours enables us to compare the tensions of the two pieces of our string, and, using the preceding definitions, to compare any two forces having the same direction as these two pieces. It should be justified by showing that the tension of the last piece of the string remains the same for the same tensor weight, whatever be the number and the disposition128 of the reflecting pulleys. It has still to be completed by showing this is only true if the pulleys are frictionless129.

Once master of these definitions, it is to be shown that the point of application, the direction and the intensity suffice to determine a force; that two forces for which these three elements are the same are always equivalent and may always be replaced by one another, whether in equilibrium or in movement, and this whatever be the other forces acting15.

It must be shown that two concurrent130 forces may always be replaced by a unique resultant; and that this resultant remains the same, whether the body be at rest or in motion and whatever be the other forces applied to it.

Finally it must be shown that forces thus defined satisfy the principle of the equality of action and reaction.

Experiment it is, and experiment alone, which can teach us all that. It will suffice to cite certain common experiments, which the scholars make daily without suspecting it, and to perform before them a few experiments, simple and well chosen.

It is after having passed through all these meanders131 that one may represent forces by arrows, and I should even wish that in the development of the reasonings return were made from time to time from the symbol to the reality. For instance it would not be difficult to illustrate132 the parallelogram of forces by aid of an apparatus133 formed of three strings134, passing over pulleys, stretched by weights and in equilibrium while pulling on the same point.

Knowing force, it is easy to define mass; this time the definition should be borrowed from dynamics135; there is no way of doing otherwise, since the end to be attained is to give understanding of the distinction between mass and weight. Here again, the definition should be led up to by experiments; there is in fact a machine which seems made expressly to show what mass is, Atwood’s machine; recall also the laws of the fall of bodies, that the acceleration of gravity is the same for heavy as for light bodies, and that it varies with the latitude136, etc.

Now, if you tell me that all the methods I extol137 have long been applied in the schools, I shall rejoice over it more than be surprised at it. I know that on the whole our mathematical teaching is good. I do not wish it overturned; that would even distress70 me. I only desire betterments slowly progressive. This teaching should not be subjected to brusque oscillations under the capricious blast of ephemeral fads138. In such tempests its high educative value would soon founder139. A good and sound logic should continue to be its basis. The definition by example is always necessary, but it should prepare the way for the logical definition, it should not replace it; it should at least make this wished for, in the cases where the true logical definition can be advantageously given only in advanced teaching.

Understand that what I have here said does not imply giving up what I have written elsewhere. I have often had occasion to criticize certain definitions I extol to-day. These criticisms hold good completely. These definitions can only be provisory. But it is by way of them that we must pass.

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1 strictly GtNwe     
adv.严厉地,严格地;严密地
参考例句:
  • His doctor is dieting him strictly.他的医生严格规定他的饮食。
  • The guests were seated strictly in order of precedence.客人严格按照地位高低就座。
2 unity 4kQwT     
n.团结,联合,统一;和睦,协调
参考例句:
  • When we speak of unity,we do not mean unprincipled peace.所谓团结,并非一团和气。
  • We must strengthen our unity in the face of powerful enemies.大敌当前,我们必须加强团结。
3 touching sg6zQ9     
adj.动人的,使人感伤的
参考例句:
  • It was a touching sight.这是一幅动人的景象。
  • His letter was touching.他的信很感人。
4 logic j0HxI     
n.逻辑(学);逻辑性
参考例句:
  • What sort of logic is that?这是什么逻辑?
  • I don't follow the logic of your argument.我不明白你的论点逻辑性何在。
5 paradox pAxys     
n.似乎矛盾却正确的说法;自相矛盾的人(物)
参考例句:
  • The story contains many levels of paradox.这个故事存在多重悖论。
  • The paradox is that Japan does need serious education reform.矛盾的地方是日本确实需要教育改革。
6 behold jQKy9     
v.看,注视,看到
参考例句:
  • The industry of these little ants is wonderful to behold.这些小蚂蚁辛勤劳动的样子看上去真令人惊叹。
  • The sunrise at the seaside was quite a sight to behold.海滨日出真是个奇景。
7 divest 9kKzx     
v.脱去,剥除
参考例句:
  • I cannot divest myself of the idea.我无法消除那个念头。
  • He attempted to divest himself of all responsibilities for the decision.他力图摆脱掉作出该项决定的一切责任。
8 incapable w9ZxK     
adj.无能力的,不能做某事的
参考例句:
  • He would be incapable of committing such a cruel deed.他不会做出这么残忍的事。
  • Computers are incapable of creative thought.计算机不会创造性地思维。
9 demonstrations 0922be6a2a3be4bdbebd28c620ab8f2d     
证明( demonstration的名词复数 ); 表明; 表达; 游行示威
参考例句:
  • Lectures will be interspersed with practical demonstrations. 讲课中将不时插入实际示范。
  • The new military government has banned strikes and demonstrations. 新的军人政府禁止罢工和示威活动。
10 demonstration 9waxo     
n.表明,示范,论证,示威
参考例句:
  • His new book is a demonstration of his patriotism.他写的新书是他的爱国精神的证明。
  • He gave a demonstration of the new technique then and there.他当场表演了这种新的操作方法。
11 prodigious C1ZzO     
adj.惊人的,奇妙的;异常的;巨大的;庞大的
参考例句:
  • This business generates cash in prodigious amounts.这种业务收益丰厚。
  • He impressed all who met him with his prodigious memory.他惊人的记忆力让所有见过他的人都印象深刻。
12 ascertain WNVyN     
vt.发现,确定,查明,弄清
参考例句:
  • It's difficult to ascertain the coal deposits.煤储量很难探明。
  • We must ascertain the responsibility in light of different situtations.我们必须根据不同情况判定责任。
13 conformity Hpuz9     
n.一致,遵从,顺从
参考例句:
  • Was his action in conformity with the law?他的行动是否合法?
  • The plan was made in conformity with his views.计划仍按他的意见制定。
14 exacting VtKz7e     
adj.苛求的,要求严格的
参考例句:
  • He must remember the letters and symbols with exacting precision.他必须以严格的精度记住每个字母和符号。
  • The public has been more exacting in its demands as time has passed.随着时间的推移,公众的要求更趋严格。
15 acting czRzoc     
n.演戏,行为,假装;adj.代理的,临时的,演出用的
参考例句:
  • Ignore her,she's just acting.别理她,她只是假装的。
  • During the seventies,her acting career was in eclipse.在七十年代,她的表演生涯黯然失色。
16 engendered 9ea62fba28ee7e2bac621ac2c571239e     
v.产生(某形势或状况),造成,引起( engender的过去式和过去分词 )
参考例句:
  • The issue engendered controversy. 这个问题引起了争论。
  • The meeting engendered several quarrels. 这次会议发生了几次争吵。 来自《简明英汉词典》
17 attained 1f2c1bee274e81555decf78fe9b16b2f     
(通常经过努力)实现( attain的过去式和过去分词 ); 达到; 获得; 达到(某年龄、水平、状况)
参考例句:
  • She has attained the degree of Master of Arts. 她已获得文学硕士学位。
  • Lu Hsun attained a high position in the republic of letters. 鲁迅在文坛上获得崇高的地位。
18 crave fowzI     
vt.渴望得到,迫切需要,恳求,请求
参考例句:
  • Many young children crave attention.许多小孩子渴望得到关心。
  • You may be craving for some fresh air.你可能很想呼吸呼吸新鲜空气。
19 formulate L66yt     
v.用公式表示;规划;设计;系统地阐述
参考例句:
  • He took care to formulate his reply very clearly.他字斟句酌,清楚地做了回答。
  • I was impressed by the way he could formulate his ideas.他陈述观点的方式让我印象深刻。
20 vaguely BfuzOy     
adv.含糊地,暖昧地
参考例句:
  • He had talked vaguely of going to work abroad.他含糊其词地说了到国外工作的事。
  • He looked vaguely before him with unseeing eyes.他迷迷糊糊的望着前面,对一切都视而不见。
21 justification x32xQ     
n.正当的理由;辩解的理由
参考例句:
  • There's no justification for dividing the company into smaller units. 没有理由把公司划分成小单位。
  • In the young there is a justification for this feeling. 在年轻人中有这种感觉是有理由的。
22 evoke NnDxB     
vt.唤起,引起,使人想起
参考例句:
  • These images are likely to evoke a strong response in the viewer.这些图像可能会在观众中产生强烈反响。
  • Her only resource was the sympathy she could evoke.她以凭借的唯一力量就是她能从人们心底里激起的同情。
23 constrain xpCzL     
vt.限制,约束;克制,抑制
参考例句:
  • She tried to constrain herself from a cough in class.上课时她竭力忍住不咳嗽。
  • The study will examine the factors which constrain local economic growth.这项研究将考查抑制当地经济发展的因素。
24 transmute KmWwy     
vt.使变化,使改变
参考例句:
  • We can transmute water power into electrical power.我们能将水力变成电力。
  • A radioactive atom could transmute itself into an entirely different kind of atom.放射性原子本身能嬗变为性质完全不同的另一种原子。
25 confided 724f3f12e93e38bec4dda1e47c06c3b1     
v.吐露(秘密,心事等)( confide的过去式和过去分词 );(向某人)吐露(隐私、秘密等)
参考例句:
  • She confided all her secrets to her best friend. 她向她最要好的朋友倾吐了自己所有的秘密。
  • He confided to me that he had spent five years in prison. 他私下向我透露,他蹲过五年监狱。 来自《简明英汉词典》
26 mathematicians bca28c194cb123ba0303d3afafc32cb4     
数学家( mathematician的名词复数 )
参考例句:
  • Do you suppose our mathematicians are unequal to that? 你以为我们的数学家做不到这一点吗? 来自英汉文学
  • Mathematicians can solve problems with two variables. 数学家们可以用两个变数来解决问题。 来自哲学部分
27 logician 1ce64af885e87536cbdf996e79fdda02     
n.逻辑学家
参考例句:
  • Mister Wu Feibai is a famous Mohist and logician in Chinese modern and contemporary history. 伍非百先生是中国近、现代著名的墨学家和逻辑学家。 来自互联网
28 perfectly 8Mzxb     
adv.完美地,无可非议地,彻底地
参考例句:
  • The witnesses were each perfectly certain of what they said.证人们个个对自己所说的话十分肯定。
  • Everything that we're doing is all perfectly above board.我们做的每件事情都是光明正大的。
29 intelligible rbBzT     
adj.可理解的,明白易懂的,清楚的
参考例句:
  • This report would be intelligible only to an expert in computing.只有计算机运算专家才能看懂这份报告。
  • His argument was barely intelligible.他的论点不易理解。
30 purely 8Sqxf     
adv.纯粹地,完全地
参考例句:
  • I helped him purely and simply out of friendship.我帮他纯粹是出于友情。
  • This disproves the theory that children are purely imitative.这证明认为儿童只会单纯地模仿的理论是站不住脚的。
31 prodigality f35869744d1ab165685c3bd77da499e1     
n.浪费,挥霍
参考例句:
  • Laughter is easier minute by minute, spilled with prodigality. 笑声每时每刻都变得越来越容易,毫无节制地倾泻出来。 来自辞典例句
  • Laughter is easier minute by minute, spilled with prodigality, tipped out at a cheerful word. 笑声每时每刻都变得越来越容易,毫无节制地倾泻出来,只要一句笑话就会引起哄然大笑。 来自英汉文学 - 盖茨比
32 dictates d2524bb575c815758f62583cd796af09     
n.命令,规定,要求( dictate的名词复数 )v.大声讲或读( dictate的第三人称单数 );口授;支配;摆布
参考例句:
  • Convention dictates that a minister should resign in such a situation. 依照常规部长在这种情况下应该辞职。 来自《简明英汉词典》
  • He always follows the dictates of common sense. 他总是按常识行事。 来自《简明英汉词典》
33 locus L0zxF     
n.中心
参考例句:
  • Barcelona is the locus of Spanish industry.巴塞罗那是西班牙工业中心。
  • Thereafter,the military remained the locus of real power.自此之后,军方一直掌握着实权。
34 analyzing be408cc8d92ec310bb6260bc127c162b     
v.分析;分析( analyze的现在分词 );分解;解释;对…进行心理分析n.分析
参考例句:
  • Analyzing the date of some socialist countries presents even greater problem s. 分析某些社会主义国家的统计数据,暴露出的问题甚至更大。 来自辞典例句
  • He undoubtedly was not far off the mark in analyzing its predictions. 当然,他对其预测所作的分析倒也八九不离十。 来自辞典例句
35 primitive vSwz0     
adj.原始的;简单的;n.原(始)人,原始事物
参考例句:
  • It is a primitive instinct to flee a place of danger.逃离危险的地方是一种原始本能。
  • His book describes the march of the civilization of a primitive society.他的著作描述了一个原始社会的开化过程。
36 rigor as0yi     
n.严酷,严格,严厉
参考例句:
  • Their analysis lacks rigor.他们的分析缺乏严谨性。||The crime will be treated with the full rigor of the law.这一罪行会严格依法审理。
37 derivative iwXxI     
n.派(衍)生物;adj.非独创性的,模仿他人的
参考例句:
  • His paintings are really quite derivative.他的画实在没有创意。
  • Derivative works are far more complicated.派生作品更加复杂。
38 calculus Is9zM     
n.微积分;结石
参考例句:
  • This is a problem where calculus won't help at all.对于这一题,微积分一点也用不上。
  • After studying differential calculus you will be able to solve these mathematical problems.学了微积分之后,你们就能够解这些数学题了。
39 domain ys8xC     
n.(活动等)领域,范围;领地,势力范围
参考例句:
  • This information should be in the public domain.这一消息应该为公众所知。
  • This question comes into the domain of philosophy.这一问题属于哲学范畴。
40 formerly ni3x9     
adv.从前,以前
参考例句:
  • We now enjoy these comforts of which formerly we had only heard.我们现在享受到了过去只是听说过的那些舒适条件。
  • This boat was formerly used on the rivers of China.这船从前航行在中国内河里。
41 bristling tSqyl     
a.竖立的
参考例句:
  • "Don't you question Miz Wilkes' word,'said Archie, his beard bristling. "威尔克斯太太的话,你就不必怀疑了。 "阿尔奇说。他的胡子也翘了起来。
  • You were bristling just now. 你刚才在发毛。
42 vanquish uKTzU     
v.征服,战胜;克服;抑制
参考例句:
  • He tried to vanquish his fears.他努力克服恐惧心理。
  • It is impossible to vanquish so strong an enemy without making an extensive and long-term effort.现在要战胜这样一个强敌,非有长期的广大的努力是不可能的。
43 remains 1kMzTy     
n.剩余物,残留物;遗体,遗迹
参考例句:
  • He ate the remains of food hungrily.他狼吞虎咽地吃剩余的食物。
  • The remains of the meal were fed to the dog.残羹剩饭喂狗了。
44 pertains 9d46f6a676147b5a066ced3cf626e0cc     
关于( pertain的第三人称单数 ); 有关; 存在; 适用
参考例句:
  • When one manages upward, none of these clear and unambiguous symbols pertains. 当一个人由下而上地管理时,这些明确无误的信号就全都不复存在了。
  • Her conduct hardly pertains to a lady. 她的行为与女士身份不太相符。
45 derivatives f75369b9e0ef2282b4d10e367e4ee2a9     
n.衍生性金融商品;派生物,引出物( derivative的名词复数 );导数
参考例句:
  • Many English words are derivatives of Latin words. 许多英语词来自拉丁语。 来自《简明英汉词典》
  • These compounds are nitrosohydroxylamine derivatives. 这类合成物是亚硝基羟胺衍生物。 来自辞典例句
46 nay unjzAQ     
adv.不;n.反对票,投反对票者
参考例句:
  • He was grateful for and proud of his son's remarkable,nay,unique performance.他为儿子出色的,不,应该是独一无二的表演心怀感激和骄傲。
  • Long essays,nay,whole books have been written on this.许多长篇大论的文章,不,应该说是整部整部的书都是关于这件事的。
47 naturalist QFKxZ     
n.博物学家(尤指直接观察动植物者)
参考例句:
  • He was a printer by trade and naturalist by avocation.他从事印刷业,同时是个博物学爱好者。
  • The naturalist told us many stories about birds.博物学家给我们讲述了许多有关鸟儿的故事。
48 edifices 26c1bcdcaf99b103a92f85d17e87712e     
n.大建筑物( edifice的名词复数 )
参考例句:
  • They complain that the monstrous edifices interfere with television reception. 他们抱怨说,那些怪物般的庞大建筑,干扰了电视接收。 来自辞典例句
  • Wealthy officials and landlords built these queer edifices a thousand years ago. 有钱的官吏和地主在一千年前就修建了这种奇怪的建筑物。 来自辞典例句
49 edifice kqgxv     
n.宏伟的建筑物(如宫殿,教室)
参考例句:
  • The American consulate was a magnificent edifice in the centre of Bordeaux.美国领事馆是位于波尔多市中心的一座宏伟的大厦。
  • There is a huge Victorian edifice in the area.该地区有一幢维多利亚式的庞大建筑物。
50 appreciation Pv9zs     
n.评价;欣赏;感谢;领会,理解;价格上涨
参考例句:
  • I would like to express my appreciation and thanks to you all.我想对你们所有人表达我的感激和谢意。
  • I'll be sending them a donation in appreciation of their help.我将送给他们一笔捐款以感谢他们的帮助。
51 irreproachable yaZzj     
adj.不可指责的,无过失的
参考例句:
  • It emerged that his past behavior was far from irreproachable.事实表明,他过去的行为绝非无可非议。
  • She welcomed her unexpected visitor with irreproachable politeness.她以无可指责的礼仪接待了不速之客。
52 entirely entirely     
ad.全部地,完整地;完全地,彻底地
参考例句:
  • The fire was entirely caused by their neglect of duty. 那场火灾完全是由于他们失职而引起的。
  • His life was entirely given up to the educational work. 他的一生统统献给了教育工作。
53 zoologists f4b4b0086bc1410e2fe80f76b127c27e     
动物学家( zoologist的名词复数 )
参考例句:
  • Zoologists refer barnacles to Crustanceans. 动物学家把螺蛳归入甲壳类。
  • It is now a source of growing interest for chemists and zoologists as well. 它现在也是化学家和动物学家愈感兴趣的一个所在。
54 embryonic 58EyK     
adj.胚胎的
参考例句:
  • It is still in an embryonic stage.它还处于萌芽阶段。
  • The plan,as yet,only exists in embryonic form.这个计划迄今为止还只是在酝酿之中。
55 recapitulates f272e8c2838db63f165d40e29bc32ee6     
n.总结,扼要重述( recapitulate的名词复数 )v.总结,扼要重述( recapitulate的第三人称单数 )
参考例句:
  • A player building a SimCity recapitulates Will Wright's sequence in inventing it. 一位《虚拟城市》玩家总结了威尔发明这个游戏的顺序。 来自互联网
  • He hesitates, he recapitulates. 他犹豫不决,颠三倒四。 来自互联网
56 geologic dg3x9     
adj.地质的
参考例句:
  • The Red Sea is a geologic continuation of the valley.红海就是一个峡谷在地质上的继续发展。
  • Delineation of channels is the first step of geologic evaluation.勾划河道的轮廓是地质解译的第一步。
57 premises 6l1zWN     
n.建筑物,房屋
参考例句:
  • According to the rules,no alcohol can be consumed on the premises.按照规定,场内不准饮酒。
  • All repairs are done on the premises and not put out.全部修缮都在家里进行,不用送到外面去做。
58 awaken byMzdD     
vi.醒,觉醒;vt.唤醒,使觉醒,唤起,激起
参考例句:
  • Old people awaken early in the morning.老年人早晨醒得早。
  • Please awaken me at six.请于六点叫醒我。
59 faculties 066198190456ba4e2b0a2bda2034dfc5     
n.能力( faculty的名词复数 );全体教职员;技巧;院
参考例句:
  • Although he's ninety, his mental faculties remain unimpaired. 他虽年届九旬,但头脑仍然清晰。
  • All your faculties have come into play in your work. 在你的工作中,你的全部才能已起到了作用。 来自《简明英汉词典》
60 chasm or2zL     
n.深坑,断层,裂口,大分岐,利害冲突
参考例句:
  • There's a chasm between rich and poor in that society.那社会中存在着贫富差距。
  • A huge chasm gaped before them.他们面前有个巨大的裂痕。
61 promptly LRMxm     
adv.及时地,敏捷地
参考例句:
  • He paid the money back promptly.他立即还了钱。
  • She promptly seized the opportunity his absence gave her.她立即抓住了因他不在场给她创造的机会。
62 faculty HhkzK     
n.才能;学院,系;(学院或系的)全体教学人员
参考例句:
  • He has a great faculty for learning foreign languages.他有学习外语的天赋。
  • He has the faculty of saying the right thing at the right time.他有在恰当的时候说恰当的话的才智。
63 versed bffzYC     
adj. 精通,熟练
参考例句:
  • He is well versed in history.他精通历史。
  • He versed himself in European literature. 他精通欧洲文学。
64 proscribe WRsx2     
v.禁止;排斥;放逐,充军;剥夺公权
参考例句:
  • They are proscribed by federal law from owning guns.根据联邦法律的规定,他们不准拥有枪支。
  • The sale of narcotics is proscribed by law.法律禁止贩卖毒品。
65 naught wGLxx     
n.无,零 [=nought]
参考例句:
  • He sets at naught every convention of society.他轻视所有的社会习俗。
  • I hope that all your efforts won't go for naught.我希望你的努力不会毫无结果。
66 parenthesis T4MzP     
n.圆括号,插入语,插曲,间歇,停歇
参考例句:
  • There is no space between the function name and the parenthesis.函数名与括号之间没有空格。
  • In this expression,we do not need a multiplication sign or parenthesis.这个表达式中,我们不需要乘号或括号。
67 sufficiently 0htzMB     
adv.足够地,充分地
参考例句:
  • It turned out he had not insured the house sufficiently.原来他没有给房屋投足保险。
  • The new policy was sufficiently elastic to accommodate both views.新政策充分灵活地适用两种观点。
68 polytechnic g1vzw     
adj.各种工艺的,综合技术的;n.工艺(专科)学校;理工(专科)学校
参考例句:
  • She was trained as a teacher at Manchester Polytechnic.她在曼彻斯特工艺专科学校就读,准备毕业后做老师。
  • When he was 17,Einstein entered the Polytechnic Zurich,Switzerland,where he studied mathematics and physics.17岁时,爱因斯坦进入了瑞士苏黎士的专科学院,学习数学和物理学。
69 thoroughly sgmz0J     
adv.完全地,彻底地,十足地
参考例句:
  • The soil must be thoroughly turned over before planting.一定要先把土地深翻一遍再下种。
  • The soldiers have been thoroughly instructed in the care of their weapons.士兵们都系统地接受过保护武器的训练。
70 distress 3llzX     
n.苦恼,痛苦,不舒适;不幸;vt.使悲痛
参考例句:
  • Nothing could alleviate his distress.什么都不能减轻他的痛苦。
  • Please don't distress yourself.请你不要忧愁了。
71 distressed du1z3y     
痛苦的
参考例句:
  • He was too distressed and confused to answer their questions. 他非常苦恼而困惑,无法回答他们的问题。
  • The news of his death distressed us greatly. 他逝世的消息使我们极为悲痛。
72 degenerate 795ym     
v.退步,堕落;adj.退步的,堕落的;n.堕落者
参考例句:
  • He didn't let riches and luxury make him degenerate.他不因财富和奢华而自甘堕落。
  • Will too much freedom make them degenerate?太多的自由会令他们堕落吗?
73 subscribe 6Hozu     
vi.(to)订阅,订购;同意;vt.捐助,赞助
参考例句:
  • I heartily subscribe to that sentiment.我十分赞同那个观点。
  • The magazine is trying to get more readers to subscribe.该杂志正大力发展新订户。
74 doctrines 640cf8a59933d263237ff3d9e5a0f12e     
n.教条( doctrine的名词复数 );教义;学说;(政府政策的)正式声明
参考例句:
  • To modern eyes, such doctrines appear harsh, even cruel. 从现代的角度看,这样的教义显得苛刻,甚至残酷。 来自《简明英汉词典》
  • His doctrines have seduced many into error. 他的学说把许多人诱入歧途。 来自《现代汉英综合大词典》
75 pointed Il8zB4     
adj.尖的,直截了当的
参考例句:
  • He gave me a very sharp pointed pencil.他给我一支削得非常尖的铅笔。
  • She wished to show Mrs.John Dashwood by this pointed invitation to her brother.她想通过对达茨伍德夫人提出直截了当的邀请向她的哥哥表示出来。
76 reigned d99f19ecce82a94e1b24a320d3629de5     
vi.当政,统治(reign的过去式形式)
参考例句:
  • Silence reigned in the hall. 全场肃静。 来自《现代汉英综合大词典》
  • Night was deep and dead silence reigned everywhere. 夜深人静,一片死寂。 来自《现代汉英综合大词典》
77 fulfill Qhbxg     
vt.履行,实现,完成;满足,使满意
参考例句:
  • If you make a promise you should fulfill it.如果你许诺了,你就要履行你的诺言。
  • This company should be able to fulfill our requirements.这家公司应该能够满足我们的要求。
78 concise dY5yx     
adj.简洁的,简明的
参考例句:
  • The explanation in this dictionary is concise and to the point.这部词典里的释义简明扼要。
  • I gave a concise answer about this.我对于此事给了一个简要的答复。
79 justified 7pSzrk     
a.正当的,有理的
参考例句:
  • She felt fully justified in asking for her money back. 她认为有充分的理由要求退款。
  • The prisoner has certainly justified his claims by his actions. 那个囚犯确实已用自己的行动表明他的要求是正当的。
80 verities e8cae4271fa3f5fdf51cd6c5be5c935f     
n.真实( verity的名词复数 );事实;真理;真实的陈述
参考例句:
  • the eternal verities of life 生命永恒的真理
81 previously bkzzzC     
adv.以前,先前(地)
参考例句:
  • The bicycle tyre blew out at a previously damaged point.自行车胎在以前损坏过的地方又爆开了。
  • Let me digress for a moment and explain what had happened previously.让我岔开一会儿,解释原先发生了什么。
82 abridge XIUyG     
v.删减,删节,节略,缩短
参考例句:
  • They are going to abridge that dictionary.他们将要精简那本字典。
  • He decided to abridge his stay here after he received a letter from home.他接到家信后决定缩短在这里的逗留时间。
83 analogous aLdyQ     
adj.相似的;类似的
参考例句:
  • The two situations are roughly analogous.两种情況大致相似。
  • The company is in a position closely analogous to that of its main rival.该公司与主要竞争对手的处境极为相似。
84 allied iLtys     
adj.协约国的;同盟国的
参考例句:
  • Britain was allied with the United States many times in history.历史上英国曾多次与美国结盟。
  • Allied forces sustained heavy losses in the first few weeks of the campaign.同盟国在最初几周内遭受了巨大的损失。
85 inclinations 3f0608fe3c993220a0f40364147caa7b     
倾向( inclination的名词复数 ); 倾斜; 爱好; 斜坡
参考例句:
  • She has artistic inclinations. 她有艺术爱好。
  • I've no inclinations towards life as a doctor. 我的志趣不是行医。
86 explicitly JtZz2H     
ad.明确地,显然地
参考例句:
  • The plan does not explicitly endorse the private ownership of land. 该计划没有明确地支持土地私有制。
  • SARA amended section 113 to provide explicitly for a right to contribution. 《最高基金修正与再授权法案》修正了第123条,清楚地规定了分配权。 来自英汉非文学 - 环境法 - 环境法
87 expounded da13e1b047aa8acd2d3b9e7c1e34e99c     
论述,详细讲解( expound的过去式和过去分词 )
参考例句:
  • He expounded his views on the subject to me at great length. 他详细地向我阐述了他在这个问题上的观点。
  • He warmed up as he expounded his views. 他在阐明自己的意见时激动起来了。
88 expound hhOz7     
v.详述;解释;阐述
参考例句:
  • Why not get a diviner to expound my dream?为什么不去叫一个占卜者来解释我的梦呢?
  • The speaker has an hour to expound his views to the public.讲演者有1小时时间向公众阐明他的观点。
89 applied Tz2zXA     
adj.应用的;v.应用,适用
参考例句:
  • She plans to take a course in applied linguistics.她打算学习应用语言学课程。
  • This cream is best applied to the face at night.这种乳霜最好晚上擦脸用。
90 subtraction RsJwl     
n.减法,减去
参考例句:
  • We do addition and subtraction in arithmetic.在算术里,我们作加减运算。
  • They made a subtraction of 50 dollars from my salary.他们从我的薪水里扣除了五十美元。
91 inverse GR6zs     
adj.相反的,倒转的,反转的;n.相反之物;v.倒转
参考例句:
  • Evil is the inverse of good.恶是善的反面。
  • When the direct approach failed he tried the inverse.当直接方法失败时,他尝试相反的做法。
92 multiplication i15yH     
n.增加,增多,倍增;增殖,繁殖;乘法
参考例句:
  • Our teacher used to drum our multiplication tables into us.我们老师过去老是让我们反覆背诵乘法表。
  • The multiplication of numbers has made our club building too small.会员的增加使得我们的俱乐部拥挤不堪。
93 dividend Fk7zv     
n.红利,股息;回报,效益
参考例句:
  • The company was forced to pass its dividend.该公司被迫到期不分红。
  • The first quarter dividend has been increased by nearly 4 per cent.第一季度的股息增长了近 4%。
94 justify j3DxR     
vt.证明…正当(或有理),为…辩护
参考例句:
  • He tried to justify his absence with lame excuses.他想用站不住脚的借口为自己的缺席辩解。
  • Can you justify your rude behavior to me?你能向我证明你的粗野行为是有道理的吗?
95 auditors 7c9d6c4703cbc39f1ec2b27542bc5d1a     
n.审计员,稽核员( auditor的名词复数 );(大学课程的)旁听生
参考例句:
  • The company has been in litigation with its previous auditors for a full year. 那家公司与前任审计员已打了整整一年的官司。
  • a meeting to discuss the annual accounts and the auditors' report thereon 讨论年度报表及其审计报告的会议
96 axis sdXyz     
n.轴,轴线,中心线;坐标轴,基准线
参考例句:
  • The earth's axis is the line between the North and South Poles.地轴是南北极之间的线。
  • The axis of a circle is its diameter.圆的轴线是其直径。
97 rotation LXmxE     
n.旋转;循环,轮流
参考例句:
  • Crop rotation helps prevent soil erosion.农作物轮作有助于防止水土流失。
  • The workers in this workshop do day and night shifts in weekly rotation.这个车间的工人上白班和上夜班每周轮换一次。
98 dread Ekpz8     
vt.担忧,忧虑;惧怕,不敢;n.担忧,畏惧
参考例句:
  • We all dread to think what will happen if the company closes.我们都不敢去想一旦公司关门我们该怎么办。
  • Her heart was relieved of its blankest dread.她极度恐惧的心理消除了。
99 justifying 5347bd663b20240e91345e662973de7a     
证明…有理( justify的现在分词 ); 为…辩护; 对…作出解释; 为…辩解(或辩护)
参考例句:
  • He admitted it without justifying it. 他不加辩解地承认这个想法。
  • The fellow-travellers'service usually consisted of justifying all the tergiversations of Soviet intenal and foreign policy. 同路人的服务通常包括对苏联国内外政策中一切互相矛盾之处进行辩护。
100 postulates a2e60978b0d3ff36cce5760c726afc83     
v.假定,假设( postulate的第三人称单数 )
参考例句:
  • They proclaimed to be eternal postulates of reason and justice. 他们宣称这些原则是理性和正义的永恒的要求。 来自辞典例句
  • The school building programme postulates an increase in educational investment. 修建校舍的计画是在增加教育经费的前提下拟定的。 来自辞典例句
101 postulate oiwy2     
n.假定,基本条件;vt.要求,假定
参考例句:
  • Let's postulate that she is a cook.我们假定她是一位厨师。
  • Freud postulated that we all have a death instinct as well as a life instinct.弗洛伊德曾假定我们所有人都有生存本能和死亡本能。
102 fixed JsKzzj     
adj.固定的,不变的,准备好的;(计算机)固定的
参考例句:
  • Have you two fixed on a date for the wedding yet?你们俩选定婚期了吗?
  • Once the aim is fixed,we should not change it arbitrarily.目标一旦确定,我们就不应该随意改变。
103 cylinder rngza     
n.圆筒,柱(面),汽缸
参考例句:
  • What's the volume of this cylinder?这个圆筒的体积有多少?
  • The cylinder is getting too much gas and not enough air.汽缸里汽油太多而空气不足。
104 cone lYJyi     
n.圆锥体,圆锥形东西,球果
参考例句:
  • Saw-dust piled up in a great cone.锯屑堆积如山。
  • The police have sectioned off part of the road with traffic cone.警察用锥形路标把部分路面分隔开来。
105 glide 2gExT     
n./v.溜,滑行;(时间)消逝
参考例句:
  • We stood in silence watching the snake glide effortlessly.我们噤若寒蝉地站着,眼看那条蛇逍遥自在地游来游去。
  • So graceful was the ballerina that she just seemed to glide.那芭蕾舞女演员翩跹起舞,宛如滑翔。
106 incessant WcizU     
adj.不停的,连续的
参考例句:
  • We have had incessant snowfall since yesterday afternoon.从昨天下午开始就持续不断地下雪。
  • She is tired of his incessant demands for affection.她厌倦了他对感情的不断索取。
107 artifice 3NxyI     
n.妙计,高明的手段;狡诈,诡计
参考例句:
  • The use of mirrors in a room is an artifice to make the room look larger.利用镜子装饰房间是使房间显得大一点的巧妙办法。
  • He displayed a great deal of artifice in decorating his new house.他在布置新房子中表现出富有的技巧。
108 philosophic ANExi     
adj.哲学的,贤明的
参考例句:
  • It was a most philosophic and jesuitical motorman.这是个十分善辩且狡猾的司机。
  • The Irish are a philosophic as well as a practical race.爱尔兰人是既重实际又善于思想的民族。
109 immediate aapxh     
adj.立即的;直接的,最接近的;紧靠的
参考例句:
  • His immediate neighbours felt it their duty to call.他的近邻认为他们有责任去拜访。
  • We declared ourselves for the immediate convocation of the meeting.我们主张立即召开这个会议。
110 datum JnvzF     
n.资料;数据;已知数
参考例句:
  • The author has taught foreigners Chinese manyand gathered rich language and datum.作者长期从事对外汉语教学,积累了丰富的语言资料。
  • Every theory,datum,or fact is generated by purpose.任何理论,资料、事实都来自于一定的目的。
111 trajectories 5c5d2685e0c45bbfa4a80b6d43c087fa     
n.弹道( trajectory的名词复数 );轨道;轨线;常角轨道
参考例句:
  • To answer this question, we need to plot trajectories of principal stresses. 为了回答这个问题,我们尚须画出主应力迹线图。 来自辞典例句
  • In the space program the theory is used to determine spaceship trajectories. 在空间计划中,这个理论用于确定飞船的轨道。 来自辞典例句
112 ascertainment 2efb1e114e03f7d913d11272cebdd6bb     
n.探查,发现,确认
参考例句:
  • Part 1 introduces the ascertainment of key stuff in state-owned commercial banks. 第1部分介绍了国有商业银行核心员工的界定。 来自互联网
  • IV The judicial ascertainment and criminal liability of involuntary dangerous crime. 过失危险犯的司法认定及刑事责任。 来自互联网
113 derive hmLzH     
v.取得;导出;引申;来自;源自;出自
参考例句:
  • We derive our sustenance from the land.我们从土地获取食物。
  • We shall derive much benefit from reading good novels.我们将从优秀小说中获得很大好处。
114 velocity rLYzx     
n.速度,速率
参考例句:
  • Einstein's theory links energy with mass and velocity of light.爱因斯坦的理论把能量同质量和光速联系起来。
  • The velocity of light is about 300000 kilometres per second.光速约为每秒300000公里。
115 acceleration ff8ya     
n.加速,加速度
参考例句:
  • All spacemen must be able to bear acceleration.所有太空人都应能承受加速度。
  • He has also called for an acceleration of political reforms.他同时呼吁加快政治改革的步伐。
116 impervious 2ynyU     
adj.不能渗透的,不能穿过的,不易伤害的
参考例句:
  • He was completely impervious to criticism.他对批评毫不在乎。
  • This material is impervious to gases and liquids.气体和液体都透不过这种物质。
117 analyze RwUzm     
vt.分析,解析 (=analyse)
参考例句:
  • We should analyze the cause and effect of this event.我们应该分析这场事变的因果。
  • The teacher tried to analyze the cause of our failure.老师设法分析我们失败的原因。
118 crouching crouching     
v.屈膝,蹲伏( crouch的现在分词 )
参考例句:
  • a hulking figure crouching in the darkness 黑暗中蹲伏着的一个庞大身影
  • A young man was crouching by the table, busily searching for something. 一个年轻人正蹲在桌边翻看什么。 来自汉英文学 - 散文英译
119 drawn MuXzIi     
v.拖,拉,拔出;adj.憔悴的,紧张的
参考例句:
  • All the characters in the story are drawn from life.故事中的所有人物都取材于生活。
  • Her gaze was drawn irresistibly to the scene outside.她的目光禁不住被外面的风景所吸引。
120 elasticity 8jlzp     
n.弹性,伸缩力
参考例句:
  • The skin eventually loses its elasticity.皮肤最终会失去弹性。
  • Every sort of spring has a definite elasticity.每一种弹簧都有一定的弹性。
121 molecules 187c25e49d45ad10b2f266c1fa7a8d49     
分子( molecule的名词复数 )
参考例句:
  • The structure of molecules can be seen under an electron microscope. 分子的结构可在电子显微镜下观察到。
  • Inside the reactor the large molecules are cracked into smaller molecules. 在反应堆里,大分子裂变为小分子。
122 friction JQMzr     
n.摩擦,摩擦力
参考例句:
  • When Joan returned to work,the friction between them increased.琼回来工作后,他们之间的摩擦加剧了。
  • Friction acts on moving bodies and brings them to a stop.摩擦力作用于运动着的物体,并使其停止。
123 mutual eFOxC     
adj.相互的,彼此的;共同的,共有的
参考例句:
  • We must pull together for mutual interest.我们必须为相互的利益而通力合作。
  • Mutual interests tied us together.相互的利害关系把我们联系在一起。
124 qualitative JC4yi     
adj.性质上的,质的,定性的
参考例句:
  • There are qualitative differences in the way children and adults think.孩子和成年人的思维方式有质的不同。
  • Arms races have a quantitative and a qualitative aspects.军备竞赛具有数量和质量两个方面。
125 equilibrium jiazs     
n.平衡,均衡,相称,均势,平静
参考例句:
  • Change in the world around us disturbs our inner equilibrium.我们周围世界的变化扰乱了我们内心的平静。
  • This is best expressed in the form of an equilibrium constant.这最好用平衡常数的形式来表示。
126 attachment POpy1     
n.附属物,附件;依恋;依附
参考例句:
  • She has a great attachment to her sister.她十分依恋她的姐姐。
  • She's on attachment to the Ministry of Defense.她现在隶属于国防部。
127 intensity 45Ixd     
n.强烈,剧烈;强度;烈度
参考例句:
  • I didn't realize the intensity of people's feelings on this issue.我没有意识到这一问题能引起群情激奋。
  • The strike is growing in intensity.罢工日益加剧。
128 disposition GljzO     
n.性情,性格;意向,倾向;排列,部署
参考例句:
  • He has made a good disposition of his property.他已对财产作了妥善处理。
  • He has a cheerful disposition.他性情开朗。
129 frictionless tiTxY     
adj.没有摩擦力的
参考例句:
  • The suspension of the mirrors must be very frictionless, but strongly damped. 反射镜的悬挂既要无摩擦,但又要有强阻尼。
  • There is a frictionless hinge at C. C点是无摩擦铰。
130 concurrent YncyG     
adj.同时发生的,一致的
参考例句:
  • You can't attend two concurrent events!你不能同时参加两项活动!
  • The twins had concurrent birthday. 双胞胎生日在同一天。
131 meanders 7964da4b1e5447a140417a4f8c3af48b     
曲径( meander的名词复数 ); 迂回曲折的旅程
参考例句:
  • The stream meanders slowly down to the sea. 这条小河弯弯曲曲缓慢地流向大海。
  • A brook meanders through the meadow. 一条小溪从草地中蜿蜒流过。
132 illustrate IaRxw     
v.举例说明,阐明;图解,加插图
参考例句:
  • The company's bank statements illustrate its success.这家公司的银行报表说明了它的成功。
  • This diagram will illustrate what I mean.这个图表可说明我的意思。
133 apparatus ivTzx     
n.装置,器械;器具,设备
参考例句:
  • The school's audio apparatus includes films and records.学校的视听设备包括放映机和录音机。
  • They had a very refined apparatus.他们有一套非常精良的设备。
134 strings nh0zBe     
n.弦
参考例句:
  • He sat on the bed,idly plucking the strings of his guitar.他坐在床上,随意地拨着吉他的弦。
  • She swept her fingers over the strings of the harp.她用手指划过竖琴的琴弦。
135 dynamics NuSzQq     
n.力学,动力学,动力,原动力;动态
参考例句:
  • In order to succeed,you must master complicated knowledge of dynamics.要取得胜利,你必须掌握很复杂的动力学知识。
  • Dynamics is a discipline that cannot be mastered without extensive practice.动力学是一门不做大量习题就不能掌握的学科。
136 latitude i23xV     
n.纬度,行动或言论的自由(范围),(pl.)地区
参考例句:
  • The latitude of the island is 20 degrees south.该岛的纬度是南纬20度。
  • The two cities are at approximately the same latitude.这两个城市差不多位于同一纬度上。
137 extol ImzxY     
v.赞美,颂扬
参考例句:
  • We of the younger generation extol the wisdom of the great leader and educator.我们年轻一代崇拜那位伟大的引路人和教育家的智慧。
  • Every day I will praise you and extol your name for ever and ever. 我要天天称颂你,也要永永远远赞美你的名。
138 fads abecffaa52f529a2b83b6612a7964b02     
n.一时的流行,一时的风尚( fad的名词复数 )
参考例句:
  • It was one of the many fads that sweep through mathematics regularly. 它是常见的贯穿在数学中的许多流行一时的风尚之一。 来自辞典例句
  • Lady Busshe is nothing without her flights, fads, and fancies. 除浮躁、时髦和幻想外,巴歇夫人一无所有。 来自辞典例句
139 Founder wigxF     
n.创始者,缔造者
参考例句:
  • He was extolled as the founder of their Florentine school.他被称颂为佛罗伦萨画派的鼻祖。
  • According to the old tradition,Romulus was the founder of Rome.按照古老的传说,罗穆卢斯是古罗马的建国者。


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