For Global Peace with Social Justice in a Sustainable Environment

Geometry and Mahatma Gandhi

Prof. Dr. Yogendra Yadav

Gandhian Scholar

Gandhi Research Foundation, Jalgaon, Maharashtra, India

Contact No- 09404955338

E-mail- dr.yadav.yogendra@gandhifoundation.net

Geometry and Mahatma Gandhi

Geometry is one of the most popular branches of mathematics. It is very easy. All problems of it related to mind. It concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge. It is concerning length, area and volume with elements of a formal mathematical science. It is describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Mahatma Gandhi wrote about geometry; “Boys will be taught their own language that is, Gujarati or Hindi and, if possible, Tamil, as also English, arithmetic, history, geography, botany and zoology. Advanced pupils will also be taught algebra and geometry. It is expected that they can be brought up to the matriculation level.”1

Mahatma Gandhi wrote about geometry; “We are in the experimental stage and the first students have to be the victims. However, let him learn well what is given to him. I hope one of these days to examine him. He was sure of his geometry lessons, but he was found wanting. Let him cultivate regular and studious habits, and learn to rely on himself in his studies.”2 Mahatma Gandhi wrote about geometry; “Now let us take higher education. I have learned Geography, Astronomy, Algebra, Geometry, etc.”3 Mahatma Gandhi wrote about geometry; “Instruction in letters will be through the students’ own languages and will include History, Geography, Arithmetic, Algebra, Geometry, Economics, etc., the learning of Sanskrit, Hindi and at least one Dravidian language being obligatory.”4

Mahatma Gandhi wrote about geometry; “If we harbor even this thought, we depart from this doctrine of ahimsa. Those who join the Ashram have to literally accept that meaning. That does not mean that we practice that doctrine in its entirety. Far from it. It is an ideal which we have to reach, and it is an ideal to be reached even at this very moment, if we are capable of doing so. But it is not a proposition in geometry to be learnt by heart: it is not even like solving difficult problems in higher mathematics; it is infinitely more difficult than solving those problems.”5 Mahatma Gandhi wrote about geometry; “Mathematics will include Arithmetic, Algebra, Geometry and Trigonometry. In other words, the pupils will be brought up to the level of the present First Year of the College.”6 Mahatma Gandhi wrote about geometry; “The Hindus invented the decimal system. Geometry and Alge bra were first developed in India, and so too Trigonometry.”7

Mahatma Gandhi wrote about geometry; “It is almost like Euclid’s line which exists only in imagination, never capable of being physically drawn. It is nevertheless an important definition in geometry yielding great results. So may a perfect brahmachari exist only in imagination. But if we did not keep him constantly before our mind’s eye, we should be like a rudderless ship. The nearer the approach to the imaginary state, the greater the perfection.”8 Mahatma Gandhi wrote about geometry; “I was obliged to lay down my weapon. In this way, a particular idea strikes me only at a particular moment. When I was a student, I could not follow anything in geometry. Up to the time we had reached the thirteenth theorem, I simply could not understand what geometry was about. But then suddenly, as the teacher explained this theorem on the board, light dawned in my mind, and from that time onwards I followed the subject with interest. In the same manner, these last three or four days an idea has got hold of my mind. What should we do if we want non-co-operation to succeed, would like students to join it and wish to secure swaraj within a year? I shall place before you today what I have always believed. I have had unshakable faith in it from the very beginning, but one aspect of my reason for this faith I understand better today than I did before.”9

Mahatma Gandhi wrote about geometry; “Is it any wonder that a person who has learnt the theorems of geometry merely by rote without understanding them 1 At she Gujarat Political Conference held at Ahmadabad on August 27, 28 and 29, 1920. Should occasionally commit a howler? What would be his plight if, having memorized a step with “therefore”, he says “because”? Just as this person would betray his unintelligent cramming, so will anyone claiming to have fulfilled the All-India Congress Committee’s conditions without understanding them be able to proceed no further than the gate. On reaching it, he will find that he does not know the secret of opening it.”10 Mahatma Gandhi wrote about geometry; “I have been told that parents have tired of our educational programme. They are unhappy that the children are being educated through the mother tongue! I laughed on hearing this. The pain followed; when there is extreme suffering, one cannot cry, one laughs. What degeneration was this, I felt. The parents fear that their children will not be able to speak good English. They do not mind if they speak bad Gujarati. Has it ever occurred to them that, if they learn through Gujarati, they will bring a little of the education into their homes too? I myself do not know equivalents of technical terms in geometry, algebra and arithmetic.”11

Mahatma Gandhi wrote about geometry; “In geometry there are certain postulates. So also there are some in the political science. When the Princes are guaranteed their status it is presumed that they will also guarantee progressive and enlightened government to their subjects. I am shortly to preside over the Kathiawar States Conference at Bhavnagar and I have reserved all my detailed say about the Indian States for this occasion. I was invited to preside over this Conference long ago. But before my incarceration I had unfurled the banner of satyagraha and I thought that my acceptance of the presidentship of any of these conferences might jeopardize the position of the Chiefs. I did not wish to put them in an awkward position and thus to mar the harmonious relations which existed between them and me. You know that I am an out-and-out believer in non-violence. To embitter the sweet relations between Princes and their subjects would be contrary to my principles. But I need not assure you that I can never forget the people of these States and their legitimate claims. My desire is to maintain sweet reasonableness in my handling the question of Indian States and I do not wish to prejudice any party. My sole desire is that the status of Princes and the rights of their subjects should be respected. I earnestly wish to enlist the sympathies of the Princes in this propaganda of khadi and the Charkha. I have pinned my faith to the spinning-wheel. On it I believe the salvation of this country depends.”12

Mahatma Gandhi wrote about geometry; “Some propositions in geometry, for instance, were very difficult of comprehension for me. I took them for granted, and today I not only can understand them, but can lose myself in a study of geometry as easily as I can do in my present work. If you have faith and ply the wheel, you take it from me that someday you will admit that what an old man once told you about it Was literally true. No wonder that one learned in the lore applied the following text from the Gita to the spinning-wheel.”13 Mahatma Gandhi wrote about geometry; “One person putting in labour worth a rupee makes no difference, but a town with a population of seven thousand earning seven thousand rupees in this manner does make a difference. By working the spinning-wheel, some other virtues are automatically acquired. It is accompanied by simplicity, sincerity and regularity; and regularity in one matter leads to regularity in one's entire life, just as, if one angle of a crooked square is set right, the other angles automatically resume their shape. This is a law of geometry.”14

Mahatma Gandhi wrote about geometry; “You can ply it even while you are talking with friends, or relaxing in their company, or thinking about problems in geometry. Shepherds on our side ply it as Gandhiji held up his takli. They walk. They watch their sheep and ply the takli at the same time. If you do not understand what I am saying, go and ask Satish Babu. He has joined this movement having given up his chemical works and his income.”15 Mahatma Gandhi wrote about geometry; “My definition of swaraj is the will for the time being of the people of India, as expressed through their representatives. There cannot be any hard and fast definition of swaraj, as you have, for instance, of a straight line in geometry. It has a varying value according to the variation in the temperament of the people dominated by various circumstances.”16

Mahatma Gandhi wrote about geometry; “The head master put the visitors’ book before me for my remarks. On turning over the pages, I read a long report from the Inspector of Schools. Whilst he had no prejudice against spinning, his experience of the experiment, wherever made, went to show that they were a failure as at this school. The Inspector thought that the experiment to be called a success should be self-supporting. I do not know why a spinning class should be self-supporting any more than a geometry class. The success of the latter would be measured by the progress made by the boys in geometry.”17 Mahatma Gandhi wrote about geometry; “But I can conceive the impossibility of people assimilating higher or subtler truths, unless they have undergone preliminary training, even as those who have not made preliminary preparations are quite unfit to breathe the rarefied atmosphere in high altitudes, or those who have no preliminary training in simple mathematics are unfit to understand or assimilate higher geometry or algebra.”18

Mahatma Gandhi wrote about geometry; “We see from the report of this school that all the subjects are taught in it through the mother tongue. The teachers claim to teach history and geography according to new methods. It is no small advantage to the pupils to be taught subjects like geometry through Gujarati.”19 Mahatma Gandhi wrote about geometry; “Aparigraha is an ideal condition. It can be said that an ideal is never realized perfectly. But we should not lower our ideal on this account. No one has ever been able to draw the ideal straight line of geometry but we may not, for that reason, change its definition. If we draw a straight line, keeping the ideal one in our view, we shall succeed in drawing a line which will serve our purpose. But if we modify the definition, we will be like a boat without the rudder. There is nothing wrong with money as a piece of metal; evil comes through its use.”20

Mahatma Gandhi wrote about geometry; “We cannot give any arbitrary meaning to the term yajna. We can adopt only a meaning which is consistent with the use of the word in the Gita. We may draw all possible conclusions from the principles of geometry, but they should be such as Euclid would not question or oppose.”21 Mahatma Gandhi wrote about geometry; “If read superficially, this verse is likely to mislead the reader. We shall not find anywhere in the world a perfect example of such a person; as in geometry we require imaginary, ideal figures, so in practical affairs, too, we require ideal instances when discussing ethical issues.”22 Mahatma Gandhi wrote about geometry; “I remember my own school days when geometry was extremely unpopular amongst the boys. The reason of its unpopularity was not in the boys but in the teacher himself. Not having a full grasp of the subject, he rattled away for all he was worth at the propositions which he drew up on the board before the boys who never followed him. Now, personally I consider that geometry is a most fascinating study and when I understood its fascination, I really could never appreciate objections that boys very often raised to that subject. But you will find if you were to go deep into such things that wherever a particular subject is uninteresting or could not be popular among the boys and girls, it is not the fault of the subject or of the boys and girls, but essentially of the teachers.”23

Mahatma Gandhi wrote about geometry; “You will apply your reason, we do not want you to deaden your reason, but you yourselves, even as we, will come to the conclusion that reason which God has given is after all a limited thing, and that which is a limited thing will not be able to reach the limitless. Therefore, go through these preliminary conditions, even as when you want to study geometry or algebra, you have to go through preliminary processes, however trying and tiresome. Observe them and then you will find that what we tell you with our own experience will be also yours.”24 Mahatma Gandhi wrote about geometry; “Some people are of the view that such subjects as grammar, compound interest, higher geometry, etc., which the learners are apt to forget in after years, should not be included in the courses to be framed for purposes of national education. Do you agree to this? If you do, why should not Urdu also I be put in the same category? When Hindus and Muslims feel the urge to come into close contact with each other and to understand each other’s culture, then only will the knowledge of Sanskrit and Urdu prove useful and lasting. Knowledge of Urdu will be put to active use and hence increase only when there is respect for and a desire to learn the culture of which Urdu is the vehicle. Until then it is bound to remain no more than a religious rite like the worship of Ganesh a formal affair without any practical value.”25

Mahatma Gandhi wrote about geometry; “I do not understand why grammar, compound interest and higher geometry have all been classed together. I have always believed that grammar is absolutely necessary for the mastery of a language, and that grammar and higher geometry are highly interesting subjects. Both provide innocent, intellectual entertainment. I will, therefore, accord a place to both these subjects in national education for those who go in for higher education or wish to study the science of language. In the same way, he who wants to be good at accounts cannot do so without learning compound interest. Therefore, all the three things mentioned by the correspondent in the question will have their due place in the syllabus for national education. The point is that there are things which are common to all schemes of education. Today, we have to differentiate between Government education and National education because the former is detrimental to national development.”26 Mahatma Gandhi wrote about geometry; “Geometry was a new subject in which I was not particularly strong, and the English medium made it still more difficult for me. The teacher taught the subject very well, but I could not follow him. Often I would lose heart and think of going back to the third standard, feeling that the packing of two years’ studies into a single year was too ambitious. But this would discredit not only me, but also the teacher; because, counting on my industry, he had recommended my promotion. So the fear of the double discredit kept me at my post. When, however, with much effort I reached the thirteenth proposition of Euclid, the utter simplicity of the subject was suddenly revealed to me. A subject who only required a pure and simple use of one’s reasoning powers could not be difficult. Ever since that time geometry has been both easy and interesting for me.”27

Mahatma Gandhi wrote about geometry; “This shloka though seemingly somewhat baffling is not really so. The Gita on many occasions presents the ideal to attain which the aspirant has to strive but which may not be possible completely to realize in the world. It is like definitions in geometry. A perfect straight line does not exist, but it is necessary to imagine it in order to prove the various propostions.”28

Mahatma Gandhi wrote about geometry; “However, we do not seek solutions to such problems by regarding them as matters of absolute dharma. Relative dharma does not proceed on a straight path like a railway track. It has, on the contrary, to make its way through a dense forest where there is not even a sense of direction. Hence in this case, even one step is sufficient. Many circumstances have to be considered before the second step is taken and, if the first step is towards the north, the second may have to be taken towards the east. In this manner, although the path may appear crooked, since it is the only one which is correct, it can also be regarded as the straight one. Nature does not imitate geometry.”29 Mahatma Gandhi wrote about geometry; “May God give you inner strength. This is not a question of utilizing your intellect. If something has to be made convincing through the use of the intellect, it can be set out in the same manner as a proposition of geometry with a Q.E.D. at the end of it. However, here the intellect becomes helpless if there is no strength of heart. The intellect is a handmaid of the heart.”30

Mahatma Gandhi wrote about geometry; “Let me point out a flaw in your calculation. If you “return the compliment”, you should find out the love notes I send every week. So if love may be measured arithmetically your notes must be as many times long as all my notes put together. But thank God! Love ignores and falsifies both arithmetic and geometry.”31 Mahatma Gandhi wrote about geometry; “One never goes to join any work unless the wish is there. But children who are not properly trained never feel the desire to do a good thing. The best rule, therefore, is to join in good work. A child who does this regularly comes to like the work in course of time, that is, he feels a desire to join it. If we always follow only our desires, we would become self-indulgent. We should try our best to save ourselves from that fate. You may say of me that my favourite subjects were languages and geometry.”32

Mahatma Gandhi wrote about geometry; “To keep a watch1 over Narandas means that, when he seems to have taken an excessive burden upon himself you should caution him and also inform me. I have discovered no ambiguity in any of my statements. If they are ambiguous, they are so unintentionally and because of my imperfect command over language. My statements are short, and hence they leave many things unsaid. But in this respect they are like propositions in geometry.”33 Mahatma Gandhi wrote about geometry; “I will certainly not draft trivial rules. I neither have the inclination nor it is good to draft such trivial rules. With the change of place and time, new sub-rules will emerge from great principles. At the same time, some rules will become obsolete while new ones will come in force. If you have a thorough understanding of the basic principles, then you would know the source of the sub-principles and be able to draft the rules easily. Just as a person who knows the derivation of the principles of geometry is able to find the correct time.”34

Mahatma Gandhi wrote about geometry; “And now concerning hypothetical questions. I had understood your purpose in asking the questions to be exactly what you explain it to be, but I would describe such questions as hypothetical. In some cases questions of this nature may be asked, but it would be better not to ask any. In any case, you should not make it a habit of asking such questions. Anybody who does that commits the same error which a student of geometry who asks his professor to solve riders does. Such a student will never learn geometry well.”35 Mahatma Gandhi wrote about geometry; “During this stage the child should acquire a general knowledge of world history and geography, botany, astronomy, arithmetic, geometry, and algebra. What you say is logically correct. But if one does not have a clear idea as to what the inner voice means, the principle cannot be followed in practice. I believe that the inner voice is perfect knowledge or realization of the Truth. And because we do not see perfect Truth, because the truth that we see is imperfect, we look upon the seers of the world as our guides and follow them. Definite rules have been laid down to help us realize truth, and we can know Truth only by following them. Hence, just as we cannot know geometry without studying it , so also it is not possible for anybody to hear the inner voice without the necessary effort and training. Hence, accor-ding to my definition, a murderer cannot cite the inner voice in defence of his act.”36

Mahatma Gandhi wrote about geometry; “You were quite right in writing to me at length on nudism. I have no hesitation in agreeing with you in theory. But theories are not always capable of being reduced to practice. Not even in exact mathematics, like geometry, are theories capable of always being reduced to practice. The imaginary right angle of geometry will not build houses but the nearly perfect right angle which masons and carpenters use is responsible for many marvellous things.”37 Mahatma Gandhi wrote about geometry; “While teaching me the economics of carpentry, he will have taught me arithmetic and geometry. All this will make up a course of some seven years.”38 Mahatma Gandhi wrote about geometry; “I want you to drive away illiteracy from these villages, find out means whereby villagers can obtain food and clothing, and take the message of winning swaraj through truth and nonviolence to the villages. This responsibility rests on you. It is your dharma to work with this ideal in view. I have presented my scheme after mature deliberation. If it fails, the teachers would be to blame. It is through handicrafts that instruction in geometry, history, geography and arithmetic will be given and an attempt will be made to meet the expenses of the school.”39

Mahatma Gandhi wrote about geometry; “The pillory began with the fourth year. Everything had to be learnt through English geometry, algebra, chemistry, astronomy, history, geography. The tyranny of English was so great that even Sanskrit or Persian had to be learnt through English, not through the mother tongue. If any boy spoke in the class in Gujarati which he understood, he was punished. It did not matter to the teacher if a boy spoke bad English which he could neither pronounce correctly nor understand fully. Why should the teacher worry? His own English was by no means without blemish. It could not be otherwise. English was as much a foreign language to him as to his pupils. The result was chaos. We the boys had to learn many things by heart, though we could not understand them fully and often not at all. My head used to reel as the teacher was struggling to make his exposition on geometry understood by us. I could make neither head nor tail of geometry till we reached the 13th theorem of the first book of Euclid. And let me confess to the reader that in spite of all my love for the mother tongue I do not to this day know the Gujarati equivalents of the technical terms of geometry, algebra and the like. I know now that what I took four years to learn of arithmetic, geometry, algebra, chemistry and astronomy I should have learnt easily in one year if I had not to learn them through English but Gujarati. My grasp of the subjects would have been easier and clearer.”40

Mahatma Gandhi wrote about geometry; “Take geometry next. What can be a better demonstration of a circle than the disc of the takli? I can teach all about circles in this way, without even mentioning the name of Euclid.”41 Mahatma Gandhi wrote about geometry; “My interest in Geometry never gets stale and I shall be your first ‘school-boy’. Whether I shall succeed in cutting off the two triangles or not I do not know.”42 Mahatma Gandhi wrote about geometry; “I have explained the ideal to you. You yourself can, by keeping it before you, answer all the questions in the same way as every student of geometry can draw a line by keeping Euclid’s ideal line in his mind. Now try to understand.”43

Mahatma Gandhi wrote about geometry; “So we need not worry about the production of commercial khadi. Whatever difficulties may arise from the above have to be overcome by the workers. To ask whether this or that comes within this framework is a sign of mental laziness and ignorance. He who cannot draw deductions cannot be said to know geometry. The same is true of all sciences.”44 Mahatma Gandhi wrote about geometry; “You should give me the solution to your triangles. I admit my failure. I thought I knew geometry fairly well. You have demolished my pride. Having performed the act of destruction, you should now instruct by teaching me in small doses. Nothing of your correspondence is destroyed. Indeed I preserve it for days in the hope of giving time to the study of the problems set by you. And then it is destroyed.”45 Mahatma Gandhi wrote about geometry; “Would there be State power in an ideal society or would such a society be Stateless? I think the question is futile. If we continue to work towards the building of such a society, to some extent it is bound to be realized and to that extent people will benefit by it. Euclid has defined a straight line as having no breadth, but no one has yet succeeded in drawing such a line and no one ever will. Still we can progress in geometry only by postulating such a line. This is true of every ideal.”46

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