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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strictly
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adv.严厉地,严格地;严密地 | |
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2
unity
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n.团结,联合,统一;和睦,协调 | |
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3
touching
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adj.动人的,使人感伤的 | |
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4
logic
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n.逻辑(学);逻辑性 | |
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5
paradox
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n.似乎矛盾却正确的说法;自相矛盾的人(物) | |
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6
behold
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v.看,注视,看到 | |
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7
divest
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v.脱去,剥除 | |
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8
incapable
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adj.无能力的,不能做某事的 | |
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9
demonstrations
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证明( demonstration的名词复数 ); 表明; 表达; 游行示威 | |
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10
demonstration
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n.表明,示范,论证,示威 | |
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11
prodigious
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adj.惊人的,奇妙的;异常的;巨大的;庞大的 | |
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12
ascertain
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vt.发现,确定,查明,弄清 | |
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13
conformity
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n.一致,遵从,顺从 | |
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14
exacting
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adj.苛求的,要求严格的 | |
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15
acting
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n.演戏,行为,假装;adj.代理的,临时的,演出用的 | |
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16
engendered
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v.产生(某形势或状况),造成,引起( engender的过去式和过去分词 ) | |
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17
attained
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(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) | |
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18
crave
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vt.渴望得到,迫切需要,恳求,请求 | |
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19
formulate
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v.用公式表示;规划;设计;系统地阐述 | |
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20
vaguely
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adv.含糊地,暖昧地 | |
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21
justification
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n.正当的理由;辩解的理由 | |
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22
evoke
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vt.唤起,引起,使人想起 | |
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23
constrain
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vt.限制,约束;克制,抑制 | |
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24
transmute
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vt.使变化,使改变 | |
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25
confided
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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) | |
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26
mathematicians
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数学家( mathematician的名词复数 ) | |
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27
logician
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n.逻辑学家 | |
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28
perfectly
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adv.完美地,无可非议地,彻底地 | |
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29
intelligible
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adj.可理解的,明白易懂的,清楚的 | |
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30
purely
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adv.纯粹地,完全地 | |
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31
prodigality
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n.浪费,挥霍 | |
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32
dictates
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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 | |
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33
locus
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n.中心 | |
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34
analyzing
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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 | |
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35
primitive
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adj.原始的;简单的;n.原(始)人,原始事物 | |
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36
rigor
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n.严酷,严格,严厉 | |
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37
derivative
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n.派(衍)生物;adj.非独创性的,模仿他人的 | |
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38
calculus
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n.微积分;结石 | |
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39
domain
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n.(活动等)领域,范围;领地,势力范围 | |
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40
formerly
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adv.从前,以前 | |
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41
bristling
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a.竖立的 | |
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42
vanquish
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v.征服,战胜;克服;抑制 | |
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43
remains
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n.剩余物,残留物;遗体,遗迹 | |
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44
pertains
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关于( pertain的第三人称单数 ); 有关; 存在; 适用 | |
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45
derivatives
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n.衍生性金融商品;派生物,引出物( derivative的名词复数 );导数 | |
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46
nay
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adv.不;n.反对票,投反对票者 | |
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47
naturalist
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n.博物学家(尤指直接观察动植物者) | |
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48
edifices
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n.大建筑物( edifice的名词复数 ) | |
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49
edifice
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n.宏伟的建筑物(如宫殿,教室) | |
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50
appreciation
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n.评价;欣赏;感谢;领会,理解;价格上涨 | |
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51
irreproachable
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adj.不可指责的,无过失的 | |
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52
entirely
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ad.全部地,完整地;完全地,彻底地 | |
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53
zoologists
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动物学家( zoologist的名词复数 ) | |
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54
embryonic
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adj.胚胎的 | |
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55
recapitulates
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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 ) | |
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56
geologic
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adj.地质的 | |
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57
premises
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n.建筑物,房屋 | |
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58
awaken
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vi.醒,觉醒;vt.唤醒,使觉醒,唤起,激起 | |
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59
faculties
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n.能力( faculty的名词复数 );全体教职员;技巧;院 | |
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60
chasm
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n.深坑,断层,裂口,大分岐,利害冲突 | |
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61
promptly
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adv.及时地,敏捷地 | |
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62
faculty
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n.才能;学院,系;(学院或系的)全体教学人员 | |
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63
versed
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adj. 精通,熟练 | |
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64
proscribe
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v.禁止;排斥;放逐,充军;剥夺公权 | |
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65
naught
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n.无,零 [=nought] | |
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66
parenthesis
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n.圆括号,插入语,插曲,间歇,停歇 | |
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67
sufficiently
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adv.足够地,充分地 | |
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68
polytechnic
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adj.各种工艺的,综合技术的;n.工艺(专科)学校;理工(专科)学校 | |
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69
thoroughly
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adv.完全地,彻底地,十足地 | |
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70
distress
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n.苦恼,痛苦,不舒适;不幸;vt.使悲痛 | |
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71
distressed
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痛苦的 | |
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72
degenerate
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v.退步,堕落;adj.退步的,堕落的;n.堕落者 | |
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73
subscribe
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vi.(to)订阅,订购;同意;vt.捐助,赞助 | |
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74
doctrines
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n.教条( doctrine的名词复数 );教义;学说;(政府政策的)正式声明 | |
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75
pointed
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adj.尖的,直截了当的 | |
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76
reigned
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vi.当政,统治(reign的过去式形式) | |
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77
fulfill
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vt.履行,实现,完成;满足,使满意 | |
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78
concise
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adj.简洁的,简明的 | |
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79
justified
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a.正当的,有理的 | |
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80
verities
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n.真实( verity的名词复数 );事实;真理;真实的陈述 | |
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81
previously
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adv.以前,先前(地) | |
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82
abridge
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v.删减,删节,节略,缩短 | |
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83
analogous
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adj.相似的;类似的 | |
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84
allied
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adj.协约国的;同盟国的 | |
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85
inclinations
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倾向( inclination的名词复数 ); 倾斜; 爱好; 斜坡 | |
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86
explicitly
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ad.明确地,显然地 | |
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87
expounded
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论述,详细讲解( expound的过去式和过去分词 ) | |
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88
expound
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v.详述;解释;阐述 | |
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89
applied
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adj.应用的;v.应用,适用 | |
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90
subtraction
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n.减法,减去 | |
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91
inverse
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adj.相反的,倒转的,反转的;n.相反之物;v.倒转 | |
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92
multiplication
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n.增加,增多,倍增;增殖,繁殖;乘法 | |
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93
dividend
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n.红利,股息;回报,效益 | |
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94
justify
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vt.证明…正当(或有理),为…辩护 | |
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95
auditors
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n.审计员,稽核员( auditor的名词复数 );(大学课程的)旁听生 | |
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96
axis
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n.轴,轴线,中心线;坐标轴,基准线 | |
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97
rotation
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n.旋转;循环,轮流 | |
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98
dread
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vt.担忧,忧虑;惧怕,不敢;n.担忧,畏惧 | |
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99
justifying
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证明…有理( justify的现在分词 ); 为…辩护; 对…作出解释; 为…辩解(或辩护) | |
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100
postulates
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v.假定,假设( postulate的第三人称单数 ) | |
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101
postulate
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n.假定,基本条件;vt.要求,假定 | |
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102
fixed
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adj.固定的,不变的,准备好的;(计算机)固定的 | |
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103
cylinder
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n.圆筒,柱(面),汽缸 | |
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104
cone
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n.圆锥体,圆锥形东西,球果 | |
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105
glide
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n./v.溜,滑行;(时间)消逝 | |
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106
incessant
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adj.不停的,连续的 | |
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107
artifice
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n.妙计,高明的手段;狡诈,诡计 | |
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108
philosophic
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adj.哲学的,贤明的 | |
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109
immediate
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adj.立即的;直接的,最接近的;紧靠的 | |
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110
datum
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n.资料;数据;已知数 | |
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111
trajectories
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n.弹道( trajectory的名词复数 );轨道;轨线;常角轨道 | |
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112
ascertainment
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n.探查,发现,确认 | |
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113
derive
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v.取得;导出;引申;来自;源自;出自 | |
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114
velocity
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n.速度,速率 | |
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115
acceleration
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n.加速,加速度 | |
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116
impervious
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adj.不能渗透的,不能穿过的,不易伤害的 | |
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117
analyze
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vt.分析,解析 (=analyse) | |
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118
crouching
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v.屈膝,蹲伏( crouch的现在分词 ) | |
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119
drawn
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v.拖,拉,拔出;adj.憔悴的,紧张的 | |
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120
elasticity
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n.弹性,伸缩力 | |
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121
molecules
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分子( molecule的名词复数 ) | |
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122
friction
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n.摩擦,摩擦力 | |
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123
mutual
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adj.相互的,彼此的;共同的,共有的 | |
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124
qualitative
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adj.性质上的,质的,定性的 | |
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125
equilibrium
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n.平衡,均衡,相称,均势,平静 | |
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126
attachment
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n.附属物,附件;依恋;依附 | |
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127
intensity
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n.强烈,剧烈;强度;烈度 | |
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128
disposition
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n.性情,性格;意向,倾向;排列,部署 | |
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129
frictionless
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adj.没有摩擦力的 | |
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130
concurrent
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adj.同时发生的,一致的 | |
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131
meanders
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曲径( meander的名词复数 ); 迂回曲折的旅程 | |
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132
illustrate
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v.举例说明,阐明;图解,加插图 | |
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133
apparatus
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n.装置,器械;器具,设备 | |
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134
strings
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n.弦 | |
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135
dynamics
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n.力学,动力学,动力,原动力;动态 | |
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136
latitude
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n.纬度,行动或言论的自由(范围),(pl.)地区 | |
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137
extol
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v.赞美,颂扬 | |
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138
fads
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n.一时的流行,一时的风尚( fad的名词复数 ) | |
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139
Founder
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n.创始者,缔造者 | |
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