There is no royal road to Mathematics. Mathematicians, like poets, cannot be made but they are born. Still I have firm conviction that the following guiding principles and cautions, if strictly observed, shall convert Mathematics from a cold, unsociable stranger with knit brows and frowning countenance into a warm-hearted, cheerful and loving friend.
1(a) Never approach Mathematics just after taking heavy meals. Let the food be well digested, and then apply yourself to this subject. Otherwise you will find it a very dry and rather repulsive study and most uninteresting.
(b) In days of hard Mathematical work you ought to take light simple food that you can digest very easily; and be temperate. Don’t take ghee in excess. High thinking and plain living should go side by side.
2(a) Don’t attack Mathematical problems or hard pieces of book work when you are sleepy or when about to go to bed. You will in that state find them quite invincible and impregnable. Not only will they offer passive resistance, but will then lay you flat down on your bed. Plainly speaking, you will in two or three minutes, after taking a difficult problem in hand, fall fast asleep. But you may, with advantage, at such a time, revise that part of Mathematics which you are already thoroughly conversant with, or work easy sums and simple riders that require very little mental exertion.
(b) In, order to excel in Mathematics you should always give to sleep what is its due. We cannot have a clear brain if we do not have enough of sleep. It is said of a great Mathematician, Des Cartes, that on account of his delicate health, he was permitted to lie in bed till late in the mornings; this was a custom which he always followed, and when he visited Pascal in 1647 he told him that the only way to do good work in Mathematics and to preserve his health was never to allow anyone to make him get up in the morning before he felt inclined to do so.
3 (a) If, however, circumstances oblige you to study difficult portions of Mathematics or solve hard problems just after taking meals, or just before retiring to bed, you ought to keep standing as you work, or be walking up and down while you think. Otherwise your efficiency of labour will be very small, and laziness will get the upper hand of you.
(b) Never neglect to take bodily exercise. This is a neglect which proves ruinous to most students. Irregular students waste the greater part of their time in idleness but overwork themselves just before the examination, taking no exercise and setting at nought the laws of health. Thus they succeed very easily in breaking their health though not in passing the examination. Then, is imputed to labour what is brought about in reality by laziness; the charge is laid at the door of hard work, whereas it was indolence that impaired their health. Remember it is not labour that kills a student, bat it is laziness or neglect of exercise that does so. Workers are sadly wanted in India, but no lazy workers.
4. When you begin a new book, it is advisable, first, to go through the book-work of the whole, at the same time doing the easy sums which come out on the first or at most at the second trial. After thus once passing through the book begin it anew, and omit no example. By adopting this system, you will, save a great deal of your time and labour and your work will, be most efficient.
5. As far as possible try to do everything with your own unaided efforts. Not only should you try to solve the examples by your own exertions, but try to do the book-work also without the aid of the author. Try, as it were, to re-discover everything. This will do you immense good. Read the heading in the case of each Article or the enunciation in the case of each Proposition and then shut your book, and try if you can give your own demonstration. Think over the subject for a time, if your exertions seem to be fruitless, read one or two sentences from the top in that Article or Proposition and then closing the book try to complete the proof; if even then your attempts avail nothing, read one or two sentences from the bottom of the same Article or Proposition, and do your best to supply the parts of the proof not seen by you. If, then also you fail, read a little more of the book, and try to fill up the gap yourself. Thus a part at least of each Article or Proposition must, by all means, be drawn out from your own brain, if you want to acquire a sound knowledge of Mathematics. You may, at first read very little by this method, but whatever is not learnt in this way forms but a very poor part of education. By and by your power will increase and this process will no longer be slow. Your progress will, after trying this method for a time, be both rapid and thorough, and you will find yourself quick to perceive and slow to forget. It is to such readers that the Roman proverb applies: “Beware of the man of few books.”
“The great danger” says a Mathematician, “which all mathematical students have to guard against is that of learning off book-work without fully mastering the essential points of the methods. Mathematics cannot be crammed; to be able to write out book-work faultlessly is not sufficient. The why and wherefore of each step must be fully grasped, and students must not rest content unless they fully understand in every case what is the property to be proved, what known results are assumed, and what methods are to be applied. Otherwise their memory will be unfairly taxed, the work will degenerate into mere drudgery, and all this will he, of little avail if the book-work so assiduously committed to memory should be set with some trifling alteration – a frequent artifice among examiners for finding out whether candidates really know their work.”
The solution of easy problems and riders, which is also practically indispensable, also depends almost entirely on a thorough knowledge of fundamental principles and those who do not clearly realize this are too often apt to rush on to results in their answers in the examination, and use the words “it is obvious” or “evident” to conceal their ignorance of the intermediate steps, which, however, deceives no one but the candidates themselves. On the other hand, those who will trouble to realise fully the methods of the book-work and the framework of facts on which each Proposition is built up, will possess sufficiency powerful machinery to solve any reasonable problems that may be set.
All that will then be required is readiness in applying their knowledge, and this can only be brought about by frequent practice in working examples.
6. Don’t disdain or pass over sums containing easy applications of the formulae, and never be satisfied with knowing merely the nay how to work out a rider; work it out actually, carry your theory into practice. Never forget the precious maxim “The way to more light is the faithful use of what we have.” By so doing you will acquire practice which alone makes us perfect. You know the greater part of your University Examination-papers will consist of such easy riders; and even those questions in which brainwork is most prominent, depend not a little for their full and leady solution on practical applications of the formulae. If you are already practised in that work you will finish in a very short time the whole of the paper, except those portions which require thinking, and out of the total amount of time allotted having got a great deal at your disposal for thinking only, you will most probably succeed in your efforts in this direction too, and thus do the whole of the paper. As it is not enough for a man to know the theory of swimming but he ought to have practice in that art if he wants to swim across a river; so is practice necessary for you if you want to swim across the troublous sea of University Examinations. Simple riders and easy sums are a great recreation to the student of Mathematics.
Most students when asked to work cut a sum, sometimes after making a few feeble efforts but frequently before making any, give up in despair ejaculating the words “It is very difficult, it will not come out.” But the self-same students, after the problem has been explained to them, cannot help uttering “Oh, it was so easy!” I say, yes, it was so easy, but you could not get it cut because you did not enter into it. You had no courage, no strong will, no patience, or no Mathematical virtue.
7. Frequently revise the portions which you have already read; otherwise your further progress will be very very slow, and you will find yourself no match for the examiners. “Every Mathematical book that is worth anything” says Professor Chrystal, “must be read backwards and forwards. Go on but often return to strengthen your faith. When you come on a hard or dreary passage pass it over; and come back to it after you have seen its importance or found the need for it further on.”
8. In order to attain dexterity in analysis and calculation and become expert in giving ready solutions to problems, it is desirable to acquire the habit of performing mathematical investigations mentally. No other discipline is so effectual in strengthening the faculty of attention; it gives a facility of apprehension, an accuracy and steadiness to the conceptions; and what is a still more valuable acquisition, it habituates the mind to arrangements in its reasonings and reflections. To give an illustration of how much it improves the intellectual powers I may cite the case of Euler, who had always accustomed himself to that exercise; and having practised it with .assiduity he is an instance to what an astonishing degree it may be acquired.
“Two of Euler’s pupils had calculated a converging series as far as the seventeenth term, but found, on comparing the written results, that they differed one unit at the fiftieth figure; they communicated this difference to their master, who went over the whole calculation by head, and his decision was found to be the true one. For the purpose of exercising his little grandson in the extraction of roots, he has been known to form to himself the table of the first six powers of all numbers from 1 to 100, and to have preserved it actually in his memory.”
9. Mathematics requires of us a great deal of time and energy; we should be continually working at it. But though it requires our body to be always in motion ever working, and subject to the laws of Dynamics; it demands our mind to be always at rest, in equilibrium and in a state subject, as it were, to the laws of Statics. A man wanting to excel in Mathematics, should banish care and anxiety from his mind, think of nothing else bat his work, should have a serene and tranquil heart, should allow nothing to disturb his peace and calm of mind. His labour will bear little fruit unless he is able to keep his mind in perfect solitude; which in most cases, will require his body also to be in loneliness.
One lesson, Nature, lot me learn of thee,
One lesson which in every wind is blown,
One lesson of two duties kept at one
Though the loud world proclaim then enmity—
Of toil unsever’d from tranquillity!
Of labour, that in lasting fruit outgrows
Far noisier schemes, accomplished in repose,
Too great for haste, too high for rivalry!
Yes, while on earth a thousand discords ring,
Man’s senseless uproar mingling with his toil,
Still do thy quiet ministers move on,
Their glorious tasks in silence perfecting;
Still working, blaming still oar vain turmoil;
Labourers that shall not fail, when man is gone.
10. A student of Mathematics should always have a humble heart and a docile spirit.
Carefully store in every piece of knowledge, gather every bit of Mathematical truth; what, if you can make no immediate use of them, and what, if no pleasing result seems likely to spring from them.
“….because right is right, to follow right
Were wisdom in the scorn of consequence”
What a noble spirit of research was betrayed by the great Mathematician when he spoke of himself as having been all his life but “a child gathering pebbles on the sea-shore” – a similitude expressing not only his humility, but alluding likewise to “the spirit in which he had pursued his investigations, as having been that, not of selection and system -building, but of childlike alacrity in seizing upon whatever contributions of knowledge Nature threw at his feet.”
These directions may be summed up m a single one: Love the subject, (Love conquers all, ) and try, by every means possible, to keep yourself in a state in which you may be able to concentrate your mind and pay close and undivided attention to the subject. This is a faculty, which, if we consider the testimony of Newton sufficient evidence, is the great constituent of inventive power It is that complete retirement of the mind within itself, during which the senses are looked up; that intense’ meditation on which no idea can intrude; that firm, straightforward progress of thought, deviating into no irregular sally; that perfect yoga where the mind becomes one with the subject which can alone place Mathematical subjects in a light efficiently strong to illuminate them fully, and preserve the perceptions of the mind’s eye in the right order.
In the end I shall lay before you the secret of success in the study of Mathematics as well as in that of any other undertaking. It is seeking not our own aggrandisement, but the glory of God; it is like the Red Cross Knight to labour and struggle for the Faerie Queen Gloriana or the Glory of God. It is thus to make our whole life a continuous prayer by our acts. It is to carry into practice the noble advice of Lord Sri Krishna –
“………………………………in thy thoughts
Do all thou dost for me! Renounce for Me!
Sacrifice heart and* mind arid will to Me
Live in the faith, of Me!”
Let me close with the following strictly true Shakespeare:
“Heaven doth with us as we with torches do,
Not light them for ourselves; for if our virtues
Did not go forth of us, it were all alike
As if we had them not;
Spirits are not finely touch’d,
But to fine issues; nor Nature never lends
The smallest scruple of her excellence
But like a thrifty goddess she determines
Herself the glory of a creator,
Both thanks and use.”