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( This tract was written and published by Rama while acting as Joint Professor of Mathematics at Forman Christian College , Lahore )

I am fully aware of the difficulties which I shall have to encounter in trying to enlist your interest in what is commonly called " a dry subject " . The usefulness of the study of Mathematics sounds like a paradox to the superficial observer . An ordinary person cannot help putting such questions as : —

( A ) Why should we bother our heads about the 47th Proposition of Euclid's First Book ?
( B ) Of what use in the world can the Binomial Theorem be ?
( C ) Why should we spend a considerable portion of our life at a , b , c , and longs ( f ) ?
( D ) What is the use of dealing with the Greek Mathematical signs ?

1 . The inability to answer questions like these , or the apparent uselessness of Mathematics makes this study very unpopular .

2 . Another reason why it is disliked is that it is a very hard subject which taxes both the memory and the intellect ; it is difficult to understand and more difficult to remember . To read it is not like walking on a smooth paved road ; but here the path is , so to speak , both slippery and rough , presents many stumbling blocks and rubs in the way .

3 . A third reason why Mathematics is felt so heavy and tedious is that generally it is not administered in proper doses or in an agreeable form ; in other words , teachers do not always try to make it attractive . Carbon dioxide , swallowed as in soda-water , is conducive to health but inhaled , it injures the system . Just so , Mathematics does us good only if taken or studied in the proper way .

Students , as a rule , complain against the University because Mathematics is made a compulsory subject in some examinations , they blame the Syndics and have all sorts of hard names to give to Mathematical writers .

To begin with , let us , for the sake of argument , assume that Mathematics has really no reward to offer , has nothing to pay . But , dear friends , let us not in whatever we undertake , be led and guided by a desire of reward . This mercenary spirit ought to be checked . The event or fruit of any action ought not to influence us ; let us do whatever we engage in , goaded by a sense of duty are not drawn by the bright future :—

If duty calls to brazen walls ,
How base the fool who flinches .

Let us work into life the following advice of the author of Bhagavad Gita :—

" . . . . . . . . . . . . . . . Find full reward
Of doing right in right ! Let light deeds be
Thy motive , not the fruit which comes from them .
And live in action ! Labour ! Make thine acts
Thy piety . . . . . . . . . "

Learn to acquire knowledge for its own sake ; hunger and thirst after knowledge . Learn a lesson from the life of Old Ruler Ulysses , who with one foot in the grave woos knowledge and asks the followers :

To follow knowledge like a sinking star ,
Beyond the utmost bound of human thought .

The assumption above made is far from being correct . The advantages of Mathematics are very many . They do not lie on the surface , but are hidden and concealed :—

( i ) Mathematics is like the ocean , rough , boisterous , and fearful on the surface ; but having precious pearls , and gems of the purest ray serene at the bottom ; or

( ii ) It may be compared to the statues of the old satyrs and sileni of Greece ; repulsive figures to look at , but enclosing within them the finished and fascinating statues of the most beloved gods of the Greeks .

( iii ) Like the solar light it appears quite colorless to the unthinking multitude , while it is in reality composed of the colours of the rainbow .

Mathematics ( Gr : Mathe-Matike ) in its original sense signifies " skill , knowledge , or science " . And in all its subsequent development it has had the idea of " skill , knowledge , or science " always underlying it . It has been feeding Art and Science . It is in no small measure to Mathematics that the world owes its Sciences of Astronomy , Optics , Acoustics , Statics , Dynamics , Hydrostatics , Hydrodynamics , Thermodynamics , Magnetism , etc ; and the Arts of Navigation , Engineering , Architecture , and the like .

Mathematics is well called an exact science and a sure and certain branch of knowledge ( cf the phrase " Mathematical certainty " ) .

" Geometry , " Pascal observes , " is almost the only subject in which we find truths wherein all people agree ; and one cause of this is that geometricians alone regard the true laws of demonstration . " So Geometry or Mathematics , we may say , has been like that solid and substantial food to science which goes for the most part to form bone or the supporting element .

According to Roger Bacon , Mathematics is the " gateway and the key to other sciences " . Professor Ball says , " It is interesting to note that the advance in our knowledge of Physics is largely due to the application to it of Mathematics , and every year it becomes more difficult for an experimenter to make any mark in the subject unless one is also a Mathematician " .

What generally happens is that the Mathematician takes the results of some every day observations and raises on them splendid superstructures which attract the attention of the Experimentalist , who steps forward and verifies by experiment the results thought out by the Mathematician . Then the labours of the two combined enrich the world with inventions and discoveries ; give to it its railways , telegraphs balloons and what not .

Happy the person who is a Mathematician and Experimentalist in one .

" The most general division of Mathematics " , says Herbert Spencer , " dealing with number guides all industrial activities , be they those by which processes are adjusted or estimates framed or commodities bought and sold or accounts kept . No one needs to have the value of this division of Abstract Science insisted upon . "

" For the higher arts of construction " , the same writer continues to say , " some acquaintance with the more special division of Mathematics is indispensable . The village carpenter who lays out work by empirical rules , equally with the builder of Brittanica-Bridge , makes hourly reference to the laws of space-relations . The surveyor who measures the land purchased ; the architect in designing a mansion to be build on it ; the builder when laying out the foundation ; the masons in cutting the stones ; and the various artisans who put up the fittings ; are all guided by geometrical truths . Railway-making is regulated from beginning to end by geometry ; alike in the preparation of plans and sections ; in staking cut the line ; in the mensuration of cuttings and embankments ; in the designing and building of bridges , culverts , viaducts , tunnels , stations . Similarly with the harbours , docks , piers and various engineering and architectural works that fringe the coasts and overspread the country as well as the mines that run underneath it . And now-a-days even the fanner , for the correct laying out of the drains , has recourse to the level —  that is , to geometrical principles .

On the application of Mechanics ( a branch of Applied Mathematics ) depends the success of modern manufactures . The properties of the lever , the wheel-and-axle , etc , are recognised in every machine , and to machinery in these times we owe all production . "

The following is the case in England and will in no long time be the case here too :

" Trace the history of the breakfast roll . The soil out of which it came was drained with machine-made tiles ; the surface was turned over by a machine ; the wheat was reaped thrashed and winnowed by machines ; by machinery it was ground and bolted ; and had the flour been sent to Gosport , it might have been made into biscuits by a machine . Look round the room in which you sit . If modern , probably the bricks in its walls are machine-made and by machinery — the flooring was sawn and planed , the mantelshelf sawn and polished , the paper-hangings made and painted . The veneer on the table , the turned legs of the chairs , the carpet , the curtains are all products of machinery .

Your clothing — plain , figured or printed — is it not wholly woven , nay , perhaps ever sewn by machinery ? And the volume you are reading are not its leaves fabricated by one machine and covered with those words by another ? Add to this that for the means of distribution over land and sea , we are similarly indebted . And then observe that according as knowledge of mechanics is well or ill applied to these ends , comes success or failure . The engineer , who miscalculates the strength of materials , builds a bridge that breaks down . The manufacturer who uses a bad machine cannot compete with another whose machine wastes less in friction and inertia . The ship-builder adhering to the old model is outsailed by one who builds on the mechanically justified wave-line principle . And as the ability of a nation to hold its own against other nations depends on the skilled activity of its units , we see that on mechanical knowledge may turn the national fate . "

Let us now see to whom most of the modern inventions and discoveries of which the world is so proud owe their origin ?

By whom was the first steam-engine made ?
James Watt , a Mathematical Instrument-maker .
By whom was the dock invented ?
Galileo , a Mathematician .
By whom was the first telescope made ?
Galileo , a Mathematician .
By whom , the Barometer ?
Pascal , a Mathematician .
Who found out the amount of alloy in the golden Crown of Ruler Hiero of Syracuse ?
Archimedes , a Mathematician .
Who was it that discovered the Law of Gravitation ?
Newton , the great of Mathematicians .

In a word , directly or indirectly , almost all our conveniences and articles of comfort are due to this branch of Philosophy or Science which we call Mathematics .

Professor Adams , the Mathematician , foretold the existence in the heavens of a satellite not known to the world before , and then the practical astronomer actually discovered the same .

Mathematics enables us to calculate accurately distances , billions of miles in length , as the distances of stars , etc ; and it also enables us to measure magnitudes about one billionth part of a cubic inch in volume , like the size of a molecule or atom . From finite quantities it leads us on to the region of the infinite .

By Mathematics we discover some of the Universal Laws of Nature written with inerasible ink on the faces of substances by the unerring finger of the Almighty . In the lines and figures of Geometry we learn " those characters " to use an expression of Galileo " in which the great book of the universe is written " .

In Statics and Dynamics the Mathematician deals with forces varying according to different laws ; and in case a new kind of energy should come to light and give rise to forces obeying laws different from those which the forces of ordinary Nature obey , the Mathematician will be found fully equipped to reserve it ; whereas the mere experimentalist if not calling Mathematics to aid , will be at a loss how to deal with it at the first sight . Let a new fluid be discovered and its fundamental property known ; it will find itself already registered in the works on Higher Hydrostatics as an old servant with specified duties to discharge .

There is a variety among individuals of all species ; again the different species of the same genus are in no instance exactly alike ; and the different , genera differ widely . So , I presume that different planets of the same Solar System have no monotony and the different Solar Systems are not alike in every respect . They are , in all probability , governed by new law ; and are blessed with new materials , new liquids and new kinds of Energy . Mathematics embraces the properties of these new things as well as those of the old familiar ones . This is knowledge of intrinsic worth .

Its rules and laws govern the phenomena and facts that can even take place on the background of Eternity . " The old order ceaseth , yielding place to new " but the mathematical dogmas remain still controlling all these vicissitude and undergoing no change in themselves .

Says Herbert Spencer , " Of course as those facts which concern all humanity throughout all times , must be held of greater moment than those which concern only a portion of them during the continuance of a fashion , it follows that in a rational estimate , knowledge of such facts , being knowledge of intrinsic worth , must , other things being equal , take precedence of knowledge that is of quasi-intrinsic or conventional worth " .

Hence you can judge of the importance of Mathematics which , beyond doubt , imparts knowledge of the kind of facts here alluded to .

If most people pride themselves on possessing a knowledge of Law , ( Law dealing with matters of this transient world ) , why should a knowledge of the eternal laws dealing with all worlds and possibly with the world to come be disdained ?

" That very law which moulds a tear ,
And bids it trickle from its source ,
That law preserves the earth a sphere ,
And guides the planets in their course . "

This law and many similar laws are treated in the works on Mathematics .

To show that the sphere of Mathematics is not confined to the physical objects alone , but extends over the mental and psychic phenomena as well , I may refer to the distinguished writers on the Calculus of Probabilities , who have applied it to Belief , and also to Edgeworth and Jevons , who have shown it to be capable of application to Feelings .

Milton holds that a part of the happiness of the pious will consist in the consciousness of the knowledge which they acquired in this world . If this be true , Mathematics is sure to make you happier in the world to come , as it embodies knowledge of the widest application .

I have been discussing so far the value of Mathematics as knowledge . Now , let us discuss its value in the way of discipline . And here , without question , it holds a supreme place .

The Vernacular word for Mathematics is " Riyazi " and this very name signifies " pertaining to Riyazat " or " discipline " . The study of Mathematics involves a mental exercise best fitted for strengthening the faculties .

The advantages of Physical exercise are not apparent to an ordinary Indian youth ; and Physical exercise is not as pleasant to one as eatables — being ignorant of the fact that in proportion as one takes more physical exercise , one will enjoy and digest the eatables better .

Similarly the advantages of mental exercise involved in the study of Mathematics are not apparent to an ordinary Indian student , and so , one reads Mathematics with great reluctance , not knowing that in proportion as one studies more of Mathematics , one will relish and master other subjects better .

" I have mentioned Mathematics " , says Locke , " as a way to settle in the mind a habit of reasoning closely and in train ; not that I think it necessary that all people should be deep Mathematicians , but that having got the reasoning , which that study necessarily brings the mind to , they might be able to transfer it to other parts of knowledge as they shall have occasion . "

There are people who are already physically strong , yet physical exercise will make them still stronger . Similarly there are people already intellectually very strong , yet a study of Mathematics will most certainly add to their intellectual powers .

Rev Dr Chalmers has stated : " I am not aware that as an expounder to the people of the lessons of the Gospel , I am much the better for knowing that the , three angles of a triangle are together equal to two right angles ; or that the square on the hypotenuse is equal to the square of the two containing sides in a right-angled triangle . But I have a strong persuasion that both the power to apprehend and the power to convince maybe mightily strengthened — that the habit of clear and consecutive reasoning may be firmly established by the successive journeys which the mind is called on to perform along the pathway of Geometrical Demonstration . The truth is that as a preparative whether for the bar or for the pulpit , I have more value in Mathematics for the exercise which the mind takes as it travels along the road , than for all the spoil which it gathers at the landing-place . "

The author of The History And Philosophy Of The Inductive Sciences has shown in Thoughts On The Study of Mathematics , that Mathematical studies judiciously pursued form one of the most effective means of developing and cultivating the reason ; and that :—

" the object of a liberal education is to develop the whole mental system of a person to make one's speculative inferences coincide with one's practical convictions ; to enable one to give a reason for the belief that is in one , and not to leave one in the condition of Solomon's sluggard , who is wiser in conceit than seven people who can render a reason . "

To this may be subjoined the judgment of John Stuart Mill , which has been recorded in Mill's invaluable system of Logic ( Vol II ) in the following terms :—

" The value of Mathematical instruction as a preparation for the more difficult investigations ( physiology , society , government , etc ) consists in the application of its method . Mathematics will ever remain the most perfect type of the Deductive Method in general ; and the applications of Mathematics to the branches of Physics furnish the only school in which philosophers can learn the most difficult and important portion of their art , the employment of the laws of the simpler phenomena for explaining and predicting those of the more complex . These grounds are quite sufficient for deeming Mathematical training an indispensable basis of scientific education and regarding , With Plato , one who is ayewuerpnros , as wanting in one of the most essential qualifications for the cultivation of the higher branches of philosophy " .

The study of Mathematics strengthens both the intellect and memory and tends to impart to us an assimilative memory rather than a sensuous one inasmuch as it teaches us to remember things by the aid of the intellect or thinking faculties , and discourages us from memorising a demonstration and the like by endless repetition . It gives us a Memory which has brought immense wealth to Professor Loisette . The nature of the subject admits of no such thing as cramming . We cannot cram Mathematics ; whatever we learn of it must be got up intelligently .

It is true that Mathematics at first appears to be a very dry subject and most distasteful ; but for that very reason we ought to study it with zest and zeal . In so doing , we shall be the stronger in will-power .  " Perhaps " , says Huxley , " the most valuable result of all education is the ability to make yourself do the thing you have to do when it ought to be done , whether you like it or not . It is the first lesson that ought to be learned , and however early one's training begins , it is probably the last lesson that one learns thoroughly " .

The abstruse nature of the subject compels a student to concentrate the attention Mathematics is the best cure for mind-wandering . Bacon says :—

" If one's wits wander , let one study Mathematics , for in demonstration if one's wits be called away ever so little , one must begin again . "

Now , if on no other account , on account of this grand virtue which it inculcates , viz , concentration of attention , we ought to value Mathematics . No one who is stricken with absent-mindedness can make one's mark in any department of human activity .

" The path to proficiency in Mathematics is so rough , and so hard an application is necessary that on the way we lose all our roughness and become perfectly smooth and frictionless , as it were , just as the wooden harrow used in this country becomes smooth by passing over the rough and uneven ground turned into clods by the plough .

Now a smooth ball or the like if put in rolling or sliding motion on the College floor will come to rest very long after a rough ball that was put in motion simultaneously with it . So , brains that have lost a considerable amount of their friction by working in the rugged field of Mathematics and have now been smoothed down , so to speak , when once put in motion or set to some hard task will , other things being equal , stop or be tired out long after those brains that have not been similarly trained .

Not only does the study of Mathematics thus habituate us to steadfastness and perseverance , but it engenders in us a strong inclination to work .

It tends to make us bitter opponents to inaction , it stores in us immense energy . The student of Mathematics being compelled to work very hard and long for the sake of success in the subject , goes on working hard even after this impressed force is withdrawn , being then impelled by the energy accumulated in one , just as a railway train continues travelling for a long time even after the steam is shut off . But alas ! for the rash youth , who no sooner are set free from the great motive power — Mathematics or some other branch of learning and enter life , then they come to a dead stop on account of the brake of sensual indulgence ; or at best get their motion considerably retarded by that brake .

Mathematics , startling as it may sound , aids Religion in a most remarkable manner and strengthens the foundation of moral character . Every now and then it puts us in a most humiliating mood , it makes us realize our own incapability , it repeatedly brings us face to face with something which we think we cannot surmount . It makes us humble and meek . It tends to do away with our vanity and self-conceit . It breaks us down and consequently exercises the will of GOD on us .

" Do you " , says Theore Monod , a French Divine , " know what is GOD's chief difficulty with us ? It is not the making us , it is the breaking us . It is not the edifying us , it is the putting us down . And therefore it is that GOD's chief instrument for edification is the pick axe . GOD must break us down , down , down , and whatever is given us to do for GOD's service , GOD will first of all show us that we are not able to do it . O GOD , take me , break me and make me . "

The value of Mathematics in this respect is well pointed out in the following remark by Locke :—

" A person in the study of Mathematics will see , that however good one may think one's understanding yet in many things , and those very visible , it may fail one . This would take off that presumption that most people have of themselves in this part , and they would not be apt to think their minds wanted no helps to enlarge them , that there could be nothing added to the acuteness and penetration of their understandings . "

All this shows that the sharp discipline to which it subjects a person has a wonderful influence in smoothing down one's asperities , in accustoming one , as a rule , to the habits of patience , perseverance , self denial and humility .

" True Science " ,  says Huxley , " ( including Mathematics undoubtedly ) , and true religion are twin siblings , and the separation of either from the other is sure to procure the death of both . Science prospers exactly in proportion as it is religious ; and religion flourishes in exact proportion to the scientific depth and firmness of its basis . The great deeds of philosophers have been less the fruit of their intellect than of the direction of that intellect by an eminently religious tone of mind . Truth has yielded rather to their patience , their love , their single heartedness and their self-denial than to their logical acumen . "

Isaac Todhunter in the Essays on Education says that of all the subjects required for passing University Examinations , Mathematics furnishes the most reliable test of a one's working powers . A student may do remarkably well in the Examination in a language ; and yet this may have been owing to keeping constant company with one who always speaks that language and is a thorough master of it . A student may distinguish oneself in History in passively hearing other students while they were preparing that subject for their Examination . One may obtain very high marks in a Practical Science Examination ; and yet this may be on account of having familiarized oneself with the Science — Apparatus and its use for amusement's sake . And so with the other subjects . But one , who excels in Mathematics , could not have done so ? Expect by dint of hard labour . One proves oneself capable of facing difficulties and doing one's duty well , however disagreeable that duty may be .

Nothing particular has as yet been said about " problems " as against " book-work " in Mathematics . They are hard nuts to crack for the student . But once cracked they yield an ambrosial kernel ; and the student thus drives an exquisite pleasure from the sweets of intellectual conquest . No other branch of knowledge can present a like phenomenon . After a hard problem has been solved , you will often observe the Mathematician's eye brighten , and at length , with a pleasure ( of which the ecstasy of Archimedes was but a simple expansion ) , hear one exclaim " I have got it ! I have got it ! "

It may not be out of place to say something as to how charming and fascinating this subject has been to some persons , or to what extent people of yore have been impressed by its importance . Plato loved it to such a degree that the inscription over the entrance to Plato's school ran :—

" Let none ignorant of Geometry enter my door " , and on one occasion an applicant who knew no Geometry is said to have been refused admission .

It is related of a Mathematician that while absorbed in solving some problem , the besieged city in which the house lay was taken by the enemy , and to the spot where the Mathematician sat musing , came up with a drawn sword in hand , a soldier who was about to break the slate of the life . The Mathematician , who had been quite ignorant of the capture of the city , did not , even now , lift up the head and look at the soldier . The astonished soldier shouted at the top of the voice to make the poor victim prepare for death . At this the Mathematician raised the eyes and said : " Wait a moment : I am about to solve it " ( the problem ) . The city was captured by the enemy , but the heart had been captivated by Mathematics .

Isaac Newton , oftentimes when busy at some Mathematical theorem , used to forget taking meals . I may add two amusing anecdotes :—

( 1 ) Newton invited a friend to dinner and forgot it . The friend arrived and found the philosopher in a fit of abstraction . Dinner was brought up for one . The friend , without disturbing Newton , sat down and despatched it . Newton recovering from reverie , looked at the empty dishes , and said : " Really , if it was not for the proof to the contrary before my eyes , I could have sworn I have not yet dined . "

( 2 ) Once when riding home from Grantham , Newton dismounted to lead the horse up a steep hill ; when turning at the top to remount , Newton found that the bridle was in hand , while the horse had slipped it and gone away .

Galileo had very long been purposely kept in ignorance of Mathematics , but one day , by chance , hearing a lecture on Geometry , was so fascinated by the Science that , thenceforward devoted all spare time to this study , and finally got leave to discontinue former studies . Galileo preserved enthusiasm for the subject in spite of poverty , public ridicule , and prosecution .

And so did Kepler , notwithstanding domestic troubles , poverty and oilier inconveniences .

Archimedes could not disengage from Mathematical dreams even when walking or when bathing as is evidenced by the well-known story which says that Archimedes one day while taking a bath was so much elated at the discovery then made , that unable to be contained , immediately ran almost naked into the street crying " Eureka ! Eureka ! I have found it ! I have found it ! "

It is related of Euler that even in the perusal of Virgil's poetry , Euler met with images that would recall the associations of Euler's more familiar studies , and lead Euler back from the fairy scenes of fiction to the element , more congenial to the nature , of Mathematical abstraction .

Amongst the ancient Hindus , Mathematics was extensively so loved , so very many were well versed in the subject .

Amongst the rich , Mathematics has exercised its sway over Boyle , Cavendish , Napier , Lord Kelvin , and others . Amongst people of letters Milton , Bacon , Locke , Carlyle Helps , Froude . . . and many others may be counted among its fervent admirers , if not votaries .

Perhaps some of you can still see no connection between abstract and practical science , and hold the former in little esteem , despising mental discipline unless you perceive its direct reference to the actual business of life , and so reject Mathematics as of little practical interest , calling it with Alexander Pope as :—

" Trick to show the stretch of human brain ,
Mere curious pleasure or ingenious pain . "

Remember , immediate usefulness alone is a fallacious recommendation for a branch of learning . Dont shun Pure Mathematics on the ground of its " purely speculative character " .

Says Professor De Morgan in the remarkable introduction to the London Edition of Ram Chandra's Maxima and Minima :—

" That sound judgment which gives people well to know what is best for them , as well as that faculty of invention which leads to development of resources and to the increase of wealth and comfort , are both materially advanced , perhaps cannot rapidly be advanced , without a great taste for pure speculation among the general mass of the people down to the lowest of those who can read and write . . . "

After giving a most satisfactory proof of the above statement , the above mentioned writer puts the conclusion in the following words :—

" The History of England as well as of other countries has impressed me with a strong conviction that pure speculation is a powerful instrument in the progress of a nation . Plato advised the Athenians to betake themselves to the study of Mathematics , in order to evade the pestilence incident to the international war which was raging in Greece . "

Mathematics is knowledge and consequently it is power . It is a weapon , though a very heavy one . If we cannot wield that weapon , the fault is all our own ; because we could wield it if we would , by dint of patience and perseverance ; and once wielded , that weapon is something awful in our hands . Knowledge of Mathematics is like an estate which should be watered and cultivated laboriously before it yields abundant crops . Many people have reaped rich harvests out of this apparent barren land .

The processes of the Differential Calculus seem far remote from the Propositions of Physical Science , yet Newton was led by their aid to found a system of Mechanics equally suited to determine the motion of the stone falling to the ground , or the revolutions of the Planetary bodies .

" Conics " is a branch of pure Mathematics dealing with the sections of a cone . It could hardly be imagined as susceptible of any useful or interesting application whatever . But Kepler came and applied it to the motions of heavenly bodies , thus clearing up most intricate difficulties in Astronomy ; Moreover , the same Conic Section was found to apply to the motion of anything whatever projected here on our own planet ; be it a cricket ball , an arrow or a bullet , or even our own bodies in the act of jumping The process of finding the HCF of any two numbers in Algebra has been made use of by Sturm in solving with great case Equations of any degree whatever .

The Theory of Quadratic Equations was made use of by our India's own , Master Ram Chandra of Delhi , in working out problems of great practical interest in Maxima and Minima . In Trigonometry and Algebra we meet with what are called Exponential Functions and Imaginary or Impossible Quantities .

When you first study them , I suppose , you will be inclined to say " Of what use in the real world are Imaginary quantities , why should we waste our time on Impossibilities ? "

My friends , let me inform you that what you will thus cast off with disdain , has lately been made the corner stone of a new mansion in the world of Science , being developed into Hyperbolic Functions . The symbols e and pi ( meaningless to the unthinking student ) represent numbers which enter into analysis from whatever file Science and Art are approached .

An anecdote might be quoted for illustration . De Morgan was explaining to an actuary what was the chance that at the end of a given time a certain proportion of some group of people would be alive ; and quoted the actuarial formula involving pi , which was explained stood for the ratio of the circumference of a circle to its diameter .

De Morgan's acquaintance , who had so far listened with interest , interrupted and exclaimed , " My dear friend , that must be a delusion ; what can a circle have to do with , the number of people alive at the end of a given time ? " Dont be surprised to know that Ball writes of a distinguished Professor remarking that " it is impossible to conceive of a universe in which e and pi should not exist . "

I sympathise with those of you to whom the abstract principles involved in Mathematics appear to have scarcely any use or aim ; but if you continue your inquiries , your matured judgment will rectify your first opinion and at length you will find yourself possessed of , to me the words of Professor Hall , " an instrument of matchless power and of universal application ; a language which Nature must hear , and to which It shall always reply " .

Even if the study of Mathematics bear no fruit at all , do not regard your labour spent on it as wasted . Nothing is wasted or lost in nature , matter is indestructible and cannot be lost , energy is indestructible and cannot be lost , and so I maintain labour is indestructible and cannot be lost . Rivers take away with them a great deal of earth and other substances from the plains , and so far as we can see the earth carried away is lost , but the same earth collects in the sea , and in course of time forms islands there . The Sun dries up m the summer tanks , pools and lakes , and we think the water is lost ; but before autumn is ushered in , the same water comes down again in the form of rain . Similarly kinetic energy is converted into potential energy , thermal energy , electric or any other form of energy , but it is never lost , although it may so appear to us .  

Just in the same way , rest assured , labour is never lost : it is sometimes changed into experience , at other times it becomes , as it were , stored up for future use ; but it is never lost . The labour of Columbus , although it did not bring forth the desired result , was far from being lost ; the attempts of Englishmen at finding the North-West passage to India , although apparently fruitless , caused the Arctic Ocean to be explored . Similarly attempts at finding the philosopher's stone led to the discovery of the Science of Chemistry . Again attempts at unreal Astrology led to real Astronomy .

So , the apparently bootless endeavours of geometricians at the duplication of a cube , the trisection of an angle , and the squaring of a circle , were the cause of Conic Sections being discovered . The vain struggles and efforts to construct a perpetual-motion machine advanced most considerably the Science of Dynamics . The celebrated John Hunter occupied a great deal of time in studying most carefully the growth of a deer's horn ( a sheer waste of time and energy in the opinion of most of us ) ; but this apparently useless knowledge well applied in the case of a dying patient was one of the causes which rendered John Hunter's name immortal . The labour was not lost and so will not your labour be lost which you devote to the study of Mathematics , but will reproduce itself in other forms of fruitful energy .

It is rather sacrilegious to think of lost labour in connection with a subject of which , in the words of no less an authority than Helmholtz , we may say , " Of all branches of human knowledge , there is none which , Kite it , has sprung as a completely armed Minerva from the head of Jupiter ; none before whose death-dealing Aegis doubt and inconsistency have so little dared to raise their eyes ! "

The vibrations of a lamp suspended from the ceiling taught Galileo how to construct the first pendulum-clock ; a falling apple gave Newton a lesson on the mysteries of the solar system ; a boiling kettle instructed George Stephenson how to make the steam-engine ; a frog's leg twitching when placed in contact with different metals directed Galvani to come to the important results wherein lay the germ of the Electric Telegraph . If apparently insignificant objects could teach such important lessons , will not Mathematics ( which means Knowledge and Science itself ) be able to teach you a great deal ?

Only a third eye is wanted ( an eye in the head or brain , Mahadeva's third eye ) to discover the Parvati of joy and glory on the mountains of Mathematics . Oh ! For the keen penetrating eye to which —

" There are tongues in trees , books in the running brooks .
Sermons in stones and good in everything " .

We are reaping abundantly the fruits of the labours of others . We travel by rail , the most desirable kind of conveyance ; we get our errands run by electricity harnessed for our sake , we live in comfortable houses , wear the clothes cut and sewn to suit our convenience , get our food cooked and prepared in such a way as to keep us in good health and many other things we enjoy which have been thought out and worked out for us by others .

Let us not forget that we also ought to do something for others in return . We owe a heavy debt to humanity . Let us try to leave the world better than we found it . Let us try to leave some footprints on the sands of time . Let us try to dive deep into the Ocean of Science and Mathematics and bring out , if possible , some pearls which may adorn the world .

Then work , work ; work with all your heart , with all your might , remembering that work is worship and remembering also that work is life —

" We live in deeds , not days ,
In thoughts , not breaths , in feelings ,
Not in figures on a dial .
One lives most who thinks most ,
Feels the noblest , acts the best . "

Genuine work will be found to be its own reward , Work is the normal state of the human .


|| < : 10 PRINCIPLES & CAUTIONS : > ||

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 .

|| 01 ||

( 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 . Dont take ghee in excess . High thinking and plain living should go side by side .

|| 02 ||

( A ) Dont 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 , Descartes , that on account of delicate health , Descartes was permitted to lie in bed till late in the mornings ; this was a custom which was always followed , and when visiting Pascal in 1647 , Descartes said that the only way to do good work in Mathematics and to preserve health was never to allow anyone to cause rising in the morning before inclined to do so .

|| 03 ||

( 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 , but it is laziness or neglect of exercise that does so . Workers are sadly wanted in India , but no lazy workers .

|| 04 ||

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 .

|| 05 ||

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 rediscover 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 person 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 be 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 .

|| 06 ||

Dont 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 ready 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 one to know the theory of swimming but one ought to have practice in that art if one 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 out 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 out because you did not enter into it . You had no courage , no strong will , no patience , or no Mathematical virtue .

|| 07 ||

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 . "

|| 08 ||

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 been accustomed to that exercise ; and having practised it with assiduity Euler 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 the master's decision was found to be the true one . For the purpose of exercising Euler's little grand-offspring in the extraction of roots , Euler has been known to form the table of the first six powers of all numbers from 1 to 100 , and to have preserved it actually in memory . "

|| 09 ||

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 .

One wanting to excel in Mathematics , should banish care and anxiety from the mind , think of nothing else but the work , should have a serene and tranquil heart , should allow nothing to disturb the peace and calm of mind . The labour will bear little fruit unless one is able to keep the mind in perfect solitude ; which in most cases , will require the 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 ,
The human's senseless uproar mingling with 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 the human is gone . ( Matthew Arnold )

|| 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 speaking of having been all the life but " a child gathering pebbles on the seashore " — a similitude expressing not only humility , but alluding likewise to " the spirit in which one had pursued one's 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 one's feet " .

|| < : SUMMARY : > ||

These directions may be summed up in 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 locked 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 aggrandizement , but the glory of GOD ; it is like the Red Cross Knight to labour and struggle for the Faerie 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 and 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 Its excellence
But like a thrifty god It determines
Itself the glory of a creator ,
Both thanks and use . "