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You may copy it, give it away or %% +%% re-use it under the terms of the Project Gutenberg License included %% +%% with this eBook or online at www.gutenberg.org %% +%% %% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Basic book class. Required. %% +%% amsmath: Basic AMS math. Required. %% +%% amssymb: Basic AMS symbols. Required. %% +%% inputenc: Basic Accept different input encodings. %% +%% Could be dispensed with by changing all %% +%% ISO-8859-1-specific characters. %% +%% graphicx: Basic graphics for images. Required. %% +%% yfonts: Support for old German fonts %% +%% Used for one word in \textgoth %% +%% %% +%% Producer's Comments: %% +%% %% +%% %% +%% %% +%% Things to Check: %% +%% %% +%% Spellcheck: OK %% +%% LaCheck: OK %% +%% Lprep/gutcheck: OK %% +%% PDF pages, excl. Gutenberg boilerplate: 47 %% +%% PDF pages, incl. Gutenberg boilerplate: 56 %% +%% ToC page numbers: OK %% +%% Images: One: cornell.png %% +%% Fonts: OK. Note \textgoth ygoth.pfb for one word %% +%% Advertisement \fboxes: OK %% +%% The first Advertisement page, "Logic Taught by Love", page 45 in %% +%% the PDF as compiled Dec 05, is uncomfortably long, with an overfull %% +%% \vbox. %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% Dec 05: jt. Compiled with %% +%% pdflatex 13447-t %% +%% pdflatex 13447-t %% +%% pdflatex 13447-t %% +%% %% +%% %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\documentclass[oneside]{book} +\listfiles +\usepackage[latin1]{inputenc} +\usepackage{amsmath} +\usepackage{graphicx} +\usepackage{yfonts} + +\begin{document} + +\thispagestyle{empty} +\small +\begin{verbatim} +Project Gutenberg's Philosophy and Fun of Algebra, by Mary Everest +Boole + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Philosophy and Fun of Algebra + +Author: Mary Everest Boole + +Release Date: September 12, 2004 [EBook #13447] +[Date last updated: December 3, 2005] + +Language: English + +Character set encoding: TeX + +*** START OF THIS PROJECT GUTENBERG EBOOK PHILOSOPHY AND FUN OF ALGEBRA *** + + + + +Produced by Joshua Hutchinson, John Hagerson, and the Project +Gutenberg On-line Distributed Proofreaders. This book was produced +from images provided by Cornell University. + + + + + + +\end{verbatim} +\normalsize +\newpage + +\frontmatter + +\bigskip\bigskip\bigskip +\begin{center} +\Huge PHILOSOPHY \& FUN \\ +OF ALGEBRA \\ + +\bigskip\bigskip +\normalsize BY \\ +\Large MARY EVEREST BOOLE \\ +\bigskip +\footnotesize AUTHOR OF \\ +``PREPARATION OF THE CHILD FOR SCIENCE,'' ETC. \\ + +\bigskip\bigskip\bigskip\bigskip +\normalsize LONDON: C.~W.~DANIEL, LTD. \\ +3 Tudor Street, E.C.~4. +\end{center} + +\newpage + +\begin{center} +\textbf{Production Note} +\end{center} + +Cornell University Library produced this volume to replace the +irreparably deteriorated original. It was scanned using Xerox +software and equipment at 600 dots per inch resolution and +compressed prior to storage using CCITT Group 4 compression. The +digital data were used to create Cornell's replacement volume on +paper that meets the ANSI Standard Z39.48-1984. The production of +this volume was supported in part by the Commission on Preservation +and Access and the Xerox Corporation. 1990. + +\bigskip\bigskip\bigskip\bigskip + +\begin{center} +\includegraphics[width=75mm]{images/cornell.png} \\ + +\bigskip \footnotesize BOUGHT WITH THE INCOME OF THE \\ +\smallskip \normalsize SAGE ENDOWMENT FUND \\ +\medskip \footnotesize THE GIFT OF \\ +\smallskip \normalsize HENRY W. SAGE \\ +\bigskip 1891 +\end{center} + +\newpage + +\begin{center} +\fbox{\parbox{11cm}{ +\begin{center} +\textsc{Works by} \\ +MARY EVEREST BOOLE \\ +\medskip \rule{4cm}{1pt} \\ +\smallskip \textsc{Logic Taught By Love.} 3s.\ 6d.\ net. \\ +\smallskip\textsc{Mathematical Psychology of Gratry and \\ +Boole for Medical Students.} 3s.\ 6d.\ net. \\ +\smallskip\textsc{Boole's Psychology as a Factor in Education.} + 6d.\ net. \\ +\smallskip\textsc{The Message of Psychic Science to the World.} + 3s.\ 6d.\ net. \\ +\smallskip\textsc{Mistletoe and Olive.} 1s.\ 6d.\ net. \\ +\smallskip\textsc{Miss Education and Her Garden.} 6d.\ net. \\ +\smallskip\textsc{Philosophy and Fun of Algebra.} 2s.\ net. \\ +\medskip C.W.\ DANIEL. \\ +\rule{4cm}{1pt} \\ +\smallskip\smallskip \textsc{The Preparation of the Child + for Science.} 2s. \\ +\smallskip\textsc{The Logic of Arithmetic.} 2s. \\ +\medskip CLARENDON PRESS. +\end{center}}} +\end{center} + +\newpage + +\begin{center} +\large \textgoth{To} \\ +\bigskip \textsc{BASIL and MARGARET} \\ +\rule{4cm}{1pt} +\end{center} + +\textsc{My Dear Children,} + +A young monkey named Genius picked a green walnut, and bit, through +a bitter rind, down into a hard shell. He then threw the walnut +away, saying: ``How stupid people are! They told me walnuts are good +to eat.'' + +His grandmother, whose name was Wisdom, picked up the +walnut---peeled off the rind with her fingers, cracked the shell, +and shared the kernel with her grandson, saying: ``Those get on best +in life who do not trust to first impressions.'' + +In some old books the story is told differently; the grandmother is +called Mrs Cunning-Greed, and she eats all the kernel herself. +Fables about the Cunning-Greed family are written to make children +laugh. It is good for you to laugh; it makes you grow strong, and +gives you the habit of understanding jokes and not being made +miserable by them. But take care not to believe such fables; +because, if you believe them, they give you bad dreams. + +\medskip \hfill MARY EVEREST BOOLE. + +\emph{January} 1909. + +\tableofcontents + +%% CONTENTS +%% +%% CHAP. +%% 1. FROM ARITHMETIC TO ALGEBRA +%% 2. THE MAKING OF ALGEBRAS +%% 3. SIMULTANEOUS PROBLEMS +%% 4. PARTIAL SOLUTIONS AND THE PROVISIONAL +%% ELIMINATION OF ELEMENTS OF COMPLEXITY +%% 5. MATHEMATICAL CERTAINTY AND REDUCTIO +%% AD ABSURDUM +%% 6. THE FIRST HEBREW ALGEBRA +%% 7. HOW TO CHOOSE OUR HYPOTHESES +%% 8. THE LIMITS OF THE TEACHER'S FUNCTION +%% 9. THE USE OF SEWING CARDS +%% 10. THE STORY OF A WORKING HYPOTHESIS +%% 11. MACBETH'S MISTAKE +%% 12. JACOB'S LADDER +%% 13. THE GREAT \emph{X} OF THE WORLD +%% 14. GO OUT OF MY CLASS-ROOM +%% 15. $\sqrt{-1}$ +%% 16. INFINITY +%% 17. FROM BONDAGE TO FREEDOM +%% APPENDIX + +\mainmatter + +\chapter{From Arithmetic To Algebra} + +Arithmetic means dealing logically with facts which we know (about +questions of number). + +``Logically''; that is to say, in accordance with the ``Logos'' or +hidden wisdom, \textit{i.e.} the laws of normal action of the human +mind. + +For instance, you are asked what will have to be paid for six pounds +of sugar at 3d.\ a pound. You multiply the six by the three. That is +not because of any property of sugar, or of the copper of which the +pennies are made. You would have done the same if the thing bought +had been starch or apples. You would have done just the same if the +material had been tea at 3s.\ a pound. Moreover, you would have done +just the same \textit{kind} of action if you had been asked the +price of seven pounds of tea at 2s.\ a pound. You do what you do +under direction of the Logos or hidden wisdom. And this law of the +Logos is made not by any King or Parliament, but by whoever or +whatever created the human mind. Suppose that any Parliament passed +an act that all the children in the kingdom were to divide the price +by the number of pounds; the Parliament could not make the answer +come right. The only result of a foolish Act of Parliament like that +would be that everybody's sums would come wrong, and everybody's +accounts be in confusion, and everybody would wonder why the trade +of the country was going to the bad. + +In former times there were kings and emperors quite stupid enough to +pass Acts like that, but governments have grown wiser by experience +and found out that, as far as arithmetic goes, there is no use in +ordering people to go contrary to the laws of the Logos, because the +Logos has the whip hand, and knows its own business, and is master +of the situation. Therefore children now are allowed to study the +laws of the Logos, whenever the business on hand is finding out how +much they are to pay in a shop. + +Sometimes your teachers set you more complicated problems +than:---What is the price of six pounds of sugar? For instance:---In +what proportion must one mix tea bought at 1s.\ 4d.\ a pound with +tea bought at 1s.\ 10d.\ a pound so as to make 5 per cent.\ profit +by selling the mixture at 1s.\ 9d.\ a pound? + +Arithmetic, then, means dealing logically with certain facts that we +know, about number, with a view to arriving at knowledge which as +yet we do not possess. + +When people had only arithmetic and not algebra, they found out a +surprising amount of things about numbers and quantities. But there +remained problems which they very much needed to solve and could +not. They had to guess the answer; and, of course, they usually +guessed wrong. And I am inclined to think they disagreed. Each +person, of course, thought his own guess was nearest to the truth. +Probably they quarrelled, and got nervous and overstrained and +miserable, and said things which hurt the feelings of their friends, +and which they saw afterwards they had better not have said---things +which threw no light on the problem, and only upset everybody's mind +more than ever. I was not there, so I cannot tell you exactly what +happened; but quarrelling and disagreeing and nerve-strain always do +go on in such cases. + +At last (at least I should suppose this is what happened) some man, +or perhaps some woman, suddenly said: ``How stupid we've all been! +We have been dealing logically with all the facts we knew about this +problem, except the most important fact of all, the fact of our own +ignorance. Let us include that among the facts we have to be logical +about, and see where we get to then. In this problem, besides the +numbers which we do know, there is one which we do not know, and +which we want to know. Instead of guessing whether we are to call it +nine, or seven, or a hundred and twenty, or a thousand and fifty, +let us agree to call it $x$, and let us always remember that $x$ +stands for the Unknown. Let us write $x$ in among all our other +numbers, and deal logically with it according to exactly the same +laws as we deal with six, or nine, or a hundred, or a thousand.'' + +As soon as this method was adopted, many difficulties which had been +puzzling everybody fell to pieces like a Rupert's drop when you nip +its tail, or disappeared like bats when the sun rises. Nobody knew +where they had gone to, and I should think that nobody cared. The +main fact was that they were no longer there to puzzle people. + +A little girl was once saying the Evening Hymn to me, ``May no ill +dreams disturb my rest, No powers of darkness me molest.'' I asked +if she knew what \textit{Powers of Darkness} meant. She said, ``The +wolves which I cannot help fancying are under my bed when all the +time I know they are not there. They must be the Powers of Darkness, +because they go away when the light comes.'' + +Now that is exactly what happened when people left off disputing +about what they did not know, and began to deal logically with the +fact of their own ignorance. This method of solving problems by +honest confession of one's ignorance is called +Algebra.\footnote{\textit{See} Appendix.} + +The name Algebra is made up of two Arabic words. + +The science of Algebra came into Europe through Arabs, and therefore +is called by its Arabic name. But it is believed to have been known +in India before the Arabs got hold of it. + +Any fact which we know or have been told about our problem is called +a datum. The number of pounds of sugar we are to buy is one datum; +the price per pound is another. + +The plural of datum is data. It is a good plan to write all one's +data on one column or page of the paper and work one's sum on the +other. This leaves the first column clear for adding to one's data +if one finds out any fresh one. + +\chapter{The Making of Algebras} + +The Arabs had some cousins who lived not far off from Arabia and who +called themselves Hebrews. A taste for Algebra seems to have run in +the family. Three Algebras grew up among the Hebrews; I should think +they are the grandest and most useful that ever were heard of or +dreamed of on earth. + +One of them has been worked into the roots of all our science; the +second is much discussed among persons who have leisure to be very +learned. The third has hardly yet begun to be used or understood in +Europe; learned men are only just beginning to think about what it +really means. All children ought to know about at least the first of +these. + +But, before we begin to talk about the Hebrew Algebras, there are +two or three things that we must be quite clear about. + +Many people think that it is impossible to make Algebra about +anything except number. This is a complete mistake. We make an +Algebra whenever we arrange facts that we know round a centre which +is a statement of what it is that we want to know and do not know; +and then proceed to deal logically with all the statements, +including the statement of our own ignorance. + +Algebra can be made about anything which any human being wants to +know about. Everybody ought to be able to make Algebras; and the +sooner we begin the better. It is best to begin before we can talk; +because, until we can talk, no one can get us into illogical habits; +and it is advisable that good logic should get the start of bad. + +If you have a baby brother, it would be a nice amusement for you to +teach him to make Algebra when he is about ten months or a year old. +And now I will tell you how to do it. + +Sometimes a baby, when it sees a bright metal tea-pot, laughs and +crows and wants to play with the baby reflected in the metal. It has +learned, by what is called ``empirical experience,'' that tea-pots +are nice cool things to handle. Another baby, when it sees a bright +tea-pot, turns its head away and screams, and will not be pacified +while the tea-pot is near. It has learned, by empirical experience, +that tea-pots are nasty boiling hot things which burn one's fingers. + +Now you will observe that both these babies have learnt by +experience. Some people say that experience is the mother of Wisdom; +but you see that both babies cannot be right; and, as a matter of +fact, both are wrong. If they could talk, they might argue and +quarrel for years; and vote; and write in the newspapers; and waste +their own time and other people's money; each trying to prove he was +right. But there is no wisdom to be got in that way. What a wise +baby knows is that he \emph{cannot tell}, by the mere look of a +tea-pot, whether it is hot or cold. The fact that is most prominent +in his mind when he sees a tea-pot is the fact that \emph{he does +not know} whether it is hot or cold. He puts that fact along with +the other fact:---that he would very much like to play with the +picture in the tea-pot supposing it would not burn his fingers; and +he deals logically with both these facts; and comes to the wise +conclusion that it would be best to go very cautiously and find out +whether the tea-pot is hot, by putting his fingers near, but not too +near. That baby has begun his mathematical studies; and begun them +at the right end. He has made an Algebra for himself. And the best +wish one can make for his future is that he will go on doing the +same for the rest of his life. + +Perhaps the best way of teaching a baby Algebra would be to get him +thoroughly accustomed to playing with a bright vessel of some kind +when cold; then put it and another just like it on the table in +front of him, one being filled with hot water. Let him play with the +cold one; and show him that you do not wish him to play with the +other. When he persists, as he probably will, let him find out for +himself that the two things which look so alike have not exactly the +same properties. Of course, you must take care that he does not hurt +himself seriously. + +\chapter{Simultaneous Problems} + +It often happens that two or three problems are so entangled up +together that it seems impossible to solve any one of them until the +others have been solved. For instance, we might get out three +answers of this kind:--- +\begin{center}\begin{tabular}{c} + $x$ equals half of $y$; \\ + $y$ equals twice $x$; \\ + $z$ equals $x$ multiplied by $y$. +\end{tabular}\end{center} +The value of each depends on the value of the others. + +When we get into a predicament of this kind, three courses are open +to us. + +We can begin to make slap-dash guesses, and each argue to prove that +his guess is the right one; and go on quarrelling; and so on; as I +described people doing about arithmetic before Algebra was invented. + +Or we might write down something of this kind:--- + +The values cannot be known. There is no answer to our problem. + +We might write:--- +\begin{center}\begin{tabular}{c} + $x$ is the unknowable; \\ + $y$ is non-existent; \\ + $z$ is imaginary, +\end{tabular}\end{center} +and accept those as answers and give them forth to the world with +all the authority which is given by big print, wide margins, a +handsome binding, and a publisher in a large way of business; and so +make a great many foolish people believe we are very wise. + +Some people call this way of settling things Philosophy; others call +it arrogant conceit. Whatever it is, it is not Algebra. The Algebra +way of managing is this:--- + +We say: Suppose that $x$ were Unity (1); what would become of $y$ +and $z$? Then we write out our problem as before; only that, +wherever there was $x$, we now write 1. + +If the result of doing so is to bring out some such ridiculous +answer as ``2 and 3 make 7,'' we then know that $x$ cannot be 1. We +now add to our column of data, ``$x$ cannot be 1.'' + +But if we come to a truism, such as ``2 and 3 make 5,'' we add to +our column of data, ``$x$ may be 1.'' Some people add to their +column of data, ``$x$ is 1,'' but that again is not Algebra. Next we +try the experiment of supposing $x$ to be equal to zero (0), and go +over the ground again. + +Then we go over the same ground, trying $y$ as 1 and as 0. + +And then we try the same with $z$. Some people think that it is +waste of time to go over all this ground so carefully, when all you +get by it is either nonsense, such as ``2 and 3 are 7''; or truisms, +such as ``2 and 3 are 5.'' But it is not waste of time. For, even if +we never arrive at finding out the value of $x$, or $y$, or $z$, +every conscientious attempt such as I have described adds to our +knowledge of the structure of Algebra, and assists us in solving +other problems. + +Such suggestions as ``suppose $x$ were Unity'' are called ``working +hypotheses,'' or ``hypothetical data.'' In Algebra we are very +careful to distinguish clearly between actual data and hypothetical +data. + +This is only part of the essence of Algebra, which, as I told you, +consists in preserving a constant, reverent, and conscientious +awareness of our own ignorance. + +When we have exhausted all the possible hypotheses connected with +Unity and Zero, we next begin to experiment with other values of +$x$; \emph{e.g.}---suppose $x$ were 2, suppose $x$ were 3, suppose +it were 4. Then, suppose it were one half, or one and a half, and so +on, registering among our data, each time, either ``$x$ may be so +and so,'' or ``$x$ cannot be so and so.'' + +The method of finding out what $x$ cannot be, by showing that +certain suppositions or hypotheses lead to a ridiculous statement, +is called the method of \emph{reductio ad absurdum}. It is largely +used by Euclid. + +\chapter[Partial Solutions\ldots\/Elements of Complexity]{Partial Solutions +and the Provisional Elimination of Elements of Complexity} + +Suppose that we never find out for certain whether $x$ is unity or +zero or something else, we then begin to experiment in a different +direction. We try to find out which of the hypothetical values of +$x$ throw most light on other questions, and if we find that some +particular value of $x$---for instance, unity---makes it easier than +does any other value to understand things about $y$ and $z$, we have +to be very careful not to slip into asserting that $x$ \emph{is} +unity. But the teacher would be quite right in saying to the class, +``For the present we will leave alone thinking about what would +happen if $x$ were something different from unity, and attend only +to such questions as can be solved on the supposition that $x$ is +unity.'' This is what is called in Algebra ``provisional elimination +of some elements of complexity.'' + +It might happen that one of the older pupils, specially clever at +mathematics, but not very well disciplined, should start some point +connected with the supposition that $x$ is something different than +unity. It would be the teacher's business to remind her: ``At +present we are dealing with the supposition that $x$ \emph{is} +unity. When we have exhausted that subject we will investigate your +question. But, till then, please do not distract the attention of +the class by talking about what is not the business on hand at +present.'' + +If the girl forgot, the teacher might say: ``I should very much like +you to try your own suggestion in private, but please do not talk +about it in class till I give you leave.'' + +If she forgot again, the teacher might say,---I think I should be +inclined to say:---``If you cannot remember not to distract the +class by talking about what is irrelevant to the business on hand, I +shall have to request you to keep outside my class-room till you +can.'' + +In an orderly school the teachers have time to be polite, and it is +their business to set the example of being so. In history, +especially such history as that of half-civilised countries 3000 +years ago, teachers were under too much strain to cultivate either a +polite \emph{manner} of saying things, or, what is of far more +consequence, that genuine intellectual courtesy which is the +absolutely necessary condition for the development of any really +perfect mathematical system. The great Hebrew Algebra, therefore, +never became quite perfect. It was only rough hewn, so to speak; and +its manners and customs were rough too. The teachers had ways of +saying, ``Hold your tongue, or else go out of my class-room,'' which +perhaps we should now call bigoted and brutal. But what I want you +to notice is that ``Hold your tongue, or get out of my class-room,'' +is not the same thing as ``My hypothesis is right, and yours ought +not to be tried anywhere.'' + +This latter is contrary to the essential basis of Algebra, viz., a +recognition of one's own ignorance. + +The other, a rough way of saying ``Get out of my class-room,'' is +only contrary to that fine intellectual courtesy which is essential +to the \emph{perfection} of mathematical method. + +\chapter[Mathematical Certainty\ldots]{Mathematical Certainty and +Reductio ad Absurdum} + +It is very often said that we cannot have mathematical certainty +about anything except a few special subjects, such as number, or +quantity, or dimensions. + +Mathematical certainty depends, not on the subject matter of our +investigation, but upon three conditions. The first is a constant +recognition of the limits of our own knowledge and the fact of our +own ignorance. The second is reverence for the As-Yet-Unknown. The +third is absolute fearlessness in meeting the \emph{reductio ad +absurdum}. In mathematics we are always delighted when we come to +any such conclusion as $2 + 3 = 7$. We feel that we have absolutely +cleared out of the way one among the several possible hypotheses, +and are ready to try another. + +We may be still groping in the dark, but we know that one +stumbling-block has been cleared out of our path, and that we are +one step ``forrader'' on the right road. We wish to arrive at truth +about the state of our balance sheet, the number of acres in our +farm, the time it will take us to get from London to Liverpool, the +height of Snowdon, the distance of the moon, and the weight of the +sun. We have no desire to deceive ourselves upon any of these +points, and therefore we have no superstitious shrinking from the +rigid \emph{reductio ad absurdum}. On some other subjects people do +wish to be deceived. They dislike the operation of correcting the +hypothetical data which they have taken as basis. Therefore, when +they begin to see looming ahead some such ridiculous result as $2 + +3 = 7$, they shrink into themselves and try to find some process of +twisting the logic, and tinkering the equation, which will make the +answer come out a truism instead of an absurdity; and then they say, +``Our hypothetical premiss is most likely true because the +conclusion to which it brings us is obviously and indisputably +true.'' + +If anyone points out that there seems to be a flaw in the argument, +they say, ``You cannot expect to get mathematical certainty in this +world,'' or ``You must not push logic too far,'' or ``Everything is +more or less compromise,'' and so on. + +Of course, there is no mathematical certainty to be had on those +terms. You could have no mathematical certainty about the amount you +owed your grocer if you tinkered the process of adding up his bill. +I wish to call your attention to the fact that \emph{even in this +world} there is a good deal of mathematical certainty to be had by +whosoever has endless patience, scrupulous accuracy in stating his +own ignorance, reverence for the As-Yet-Unknown, and perfect +fearlessness in meeting the \emph{reductio ad absurdum}. + +\chapter{The First Hebrew Algebra} + +The first Hebrew algebra is called Mosaism, from the name of Moses +the Liberator, who was its great Incarnation, or Singular Solution. +It ought hardly to be called an algebra: it is the master-key of all +algebras, the great central director for all who wish to learn how +to get into right relations to the unknown, so that they can make +algebras for themselves. Its great keynotes are these:--- + +When you do not know something, and wish to know it, state that you +do not know it, and keep that fact well in front of you. + +When you make a provisional hypothesis, state that it is so, and +keep that fact well in front of you. + +While you are trying out that provisional hypothesis, do not allow +yourself to think, or other people to talk to you, about any other +hypothesis. + +Always remember that the use of algebra is to \emph{free people from +bondage}. For instance, in the case of number: Children do their +numeration, their ``carrying,'' in tens, because primitive man had +nothing to do sums with but his ten fingers. + +Many children grow superstitious, and think that you cannot carry +except in tens; or that it is wrong to carry in anything but tens. +The use of algebra is to free them from bondage to all this +superstitious nonsense, and help them to see that the numbers would +come just as right if we carried in eights or twelves or twenties. +It is a little difficult to do this at first, because we are not +accustomed to it; but algebra helps to get over our stiffness and +set habits and to do numeration on any basis that suits the matter +we are dealing with. + +Of course, we have to be careful not to mix two numerations. If we +are working a sum in tens, we must go on working in tens to the end +of that sum. + +Never let yourself get fixed ideas that numbers (or anything else +that you are working at) will not come right unless your sum is set +or shaped in a particular way. Have a way in which you usually do a +particular kind of sum, but do not let it haunt you. + +You may some day become a teacher. If ever you are teaching a class +how to set down a sum or an equation, say ``This is my way,'' or +``This is the way which I think you will find most convenient,'' or +``This is the way in which the Government Inspector requires you to +do the sums at present, and therefore you must learn it.'' But do +not take in vain the names of great unseen powers to back up either +your own limitations, or your own authority, or the Inspector's +authority. Never say, or imply, ``Arithmetic requires you to do +this; your sum will come wrong if you do it differently.'' Remember +that arithmetic requires nothing from you except absolute honesty +and patient work. You get no blessing from the Unseen Powers of +Number by slipshod statements used to make your own path easy. + +Be very accurate and plodding during your hours of work, but take +care not to go on too long at a time doing mere drudgery. At certain +times give yourself a full stretch of body and mind by going to the +boundless fairyland of your subject. Think how the great +mathematicians can weigh the earth and measure the stars, and reveal +the laws of the universe; and tell yourself that it is all one +science, and that you are one of the servants of it, quite as much +as ever Pythagoras or Newton were. + +Never be satisfied with being up-to-date. Think, in your slack time, +of how people before you did things. While you are at school my +little book, \emph{Logic of Arithmetic}, will help you to find out +many things about your ancestors which may amuse and interest you; +but, as soon as you leave school and choose your own reading, take +care to read up the histories of the struggles and difficulties of +the people who formerly dealt with your own subject (whatever that +may be). + +If you find the whole of the data too complicated to deal with, and +judge that it is necessary to eliminate one or more of them, in +order to reduce your material within the compass of your own power +to manage, do it as a \emph{provisional} necessity. Take care to +register the fact that you have done so, and to arrange your mind, +from the first, on the understanding that the eliminated data will +have to come back. Forget them during the working out of your +experimental equation; but never give way to the feeling that they +are got rid of and done with. + +Be very careful not to disturb other people's relationships to each +other. For instance, if a teacher is explaining something to another +pupil, never speak till she has done. Beware of the sentimental +craving to be ``in it.'' Any studying-group profits by right working +relations being set up between any two members; and ultimately each +member profits. The whole group suffers from any distraction between +any two. Therefore listen and learn what you can; but never disturb +or distract.% +\footnote{D.\ Marks bases the Seventh Commandment on the +desirability of not distracting existing relations.} + +Take care not to become a parasite; do not lazily appropriate the +results of other people's labour, but learn and labour truly to get +your own living. Take care that everything you possess, whether +physical, mental, or spiritual, shall be the result of your own toil +as well as other people's; and remember that you are bound to pay, +in some shape or way, everyone who helps you. + +Do not make things easy for yourself by speaking or thinking of data +as if they were different from what they are; and do not go off from +facing data as they are, to amuse your imagination by wishing they +were different from what they are. Such wishing is pure waste of +nerve force, weakens your intellectual power, and gets you into +habits of mental confusion. + +When the time comes to stop grind-work, there is no better rest than +amusing your imagination by thinking of non-existent possibilities; +but do it on a free, generous scale. Give yourself a perfectly free +rein in the company of the Infinite. During such exercise of the +imagination, remember that you are in the company of the Infinite, +and are not dealing with, or tinkering at, the problem on your +paper. + +Keep always at hand, clearly written out, a good standard selection +of the most important formul\ae{}---Arithmetical, Algebraic, +Geometric, and Trigonometrical, and accustom yourself to test your +results by referring to it. + +These are the main laws of mathematical self-guidance. Once upon a +time ``Moses'' projected them on to the magic-lantern screen of +legislation. In that form they are known as the Ten Commandments; +or, to change the metaphors, we might call the Ten Commandments the +outer skin of the mathematical body. + +A great many people seem to suppose that, though everyone ought to +keep the Ten Commandments, it does not matter what happens to one's +mind. Just so, there are people who live unhealthy lives, and think +they can make all right by putting cosmetics on their skin. But I +hope you have learned in the hygiene class how stupid and futile all +that is. The way to have a healthy skin is to grow it, by leading a +hygienic life on a moderate allowance of pure wholesome food, and +taking a proper amount of exercise in pure fresh air. People who do +that with their minds grow the Ten Commandments naturally, just as +Moses grew them. The world has been trying the other plan---bad food +and air inside, and cosmetics outside---for at least 4000 years; and +not much seems to have come of it yet. The Ten Commandments have not +yet succeeded in getting themselves kept. Perhaps that is why some +schoolmasters and mistresses think they would like to try the other +plan now. Still, it is very good to have a normal model of what a +healthy human being ought to look like outside. It is good to have a +standard for reference. Therefore do not get too much immersed in +the mere details of your own problems. Learn the Ten Commandments +and a few other old standard formularies by heart, and repeat them +every now and then. And say to yourself, ``If I really am doing my +algebra quite rightly, \emph{this} (the standard formularies) is how +I shall think and feel and wish. I shall wish to behave thus, not +because anybody ordered me to do so, but from sheer liking and sense +of the general fitness of things.'' + +\chapter{How to Choose Our Hypotheses} + +The faculties by means of which we get our positive data are called +the senses (sight, hearing, etc.). + +The faculty by means of which we get our hypothetical data is called +the Imagination. + +Some persons are prone to warn young people against what they call +an excessive exercise of the imagination. Of course, to say that +``excessive'' anything is too much is a mere truism, but nobody +knows yet what is the proper amount of use for the imagination. What +we do know is that there is a good deal of excessive mis-use of the +imagination, by which I mean that there is a frightful amount of +using it contrary to the laws of its normal action. A kind of use of +it, such as, when we find a child doing it with its eyes, we say, +``Do not learn the habit of squinting''; or if it does the analogous +thing with its legs, we say, ``Go and run about, or do some +gymnastics; do not stand there lolloping crooked against the wall.'' + +Squinting and lolloping crooked are things that it is best to avoid +doing much of with any part of one's self. + +Moreover, it is bad to spend too many hours over either a microscope +or a telescope, or in gazing fixedly at some one-distance range. The +eyes need change of focus. So does the imagination. + +There has been in modern Europe a shocking riot in mis-use of the +imagination. The remedy is to learn to use it. But the same kind of +people who would like to bandage a child's eyes lest it should learn +to squint, like to bandage the imagination lest it should wear +itself out by squinting. + +In a school which professes to be conducted on hygienic principles, +we have nothing to do with that sort of pessimistic quackery. We use +the imagination as freely as the hands and eyes. + +But when we come to the end of our arithmetic we do not content +ourselves with guesses; we proceed to algebra--that is to say, to +dealing logically with the fact of our own ignorance. One of the +data that we do know is that all great nerve-centres affect each +other. Mis-use of any one tends more or less to produce distorted +action in the others. And, quite apart from that consideration, any +energetic and continued action of one tends more or less to suppress +the action of the others, for the time being, by drawing the blood +from the organs which are the seat of them; and then, when normal +circulation is restored, to produce for a time an unusual +sensitiveness in the others. There is nothing abnormal or wrong in +this, provided that we recognise the fact, and, as I said, are +careful to deal logically with the fact of our own ignorance +whenever anything happens either to our eyes or to our imagination +which we do not at the moment quite understand. + +If you ever arrive at using your imagination strongly and rightly in +the construction of any sort of algebra, you may find that it +affects to some extent your sense-organs. It certainly will affect +them more or less whether you know it or not. What I mean is that it +may affect them in a way that forces you to be aware of the fact. If +ever this should happen, take it quite naturally; and as long as you +are too young to understand how it happens, just say to yourself, +``This is $x$, one of the things that I do not know, and perhaps +shall know some day if I go on quietly acting in accordance with +strict logic, and remembering my own ignorance.'' + +The ancient Hebrews used their imaginations very freely, and +sometimes really very logically. And sometimes the free use of the +imagination produced sensations in the eyes and ears as if of seeing +and hearing. They considered this quite natural, as it really was. +Many great mathematicians in modern Europe have had these +sensations. + +The Hebrews called these sensations by a Hebrew word which is +translated by the English word ``angel,'' from the Greek +``angelos,'' a messenger. The Hebrews were quite right. The +sensations are messengers from the Great Unknown. They bring no +information about outside facts. No angel tells you how many petals +there are in a buttercup; if you want to know that, you are supposed +to ask the buttercup itself. No angel tells you the price of sugar; +you ought to ask your grocer. No angel tells you how to invest your +money; you ought to ask your banker or your lawyer. There are people +foolish enough to ask angels about investments, or about which horse +will win a race; which is just as foolish as asking your banker in +town how many blossoms there are on the rose tree in your country +garden. It is not his business, and if he made a guess it would most +likely turn out a wrong one. All that sort of thing is quackery and +superstition. + +But the angels do bring us very reliable information from a vast +region of valuable truth about which most of us know very little as +yet. They guide us how to frame our \emph{next provisional working +hypothesis}, how to choose the particular hypothesis which at our +present stage of knowledge and development will be most illuminating +for us. Some of the angels come during sleep; we call them dreams. +Dreams sometimes suggest the best working hypothesis to experiment +on next. More often they warn us against thinking upon some +hypothetical basis which for the present will not suit us. + +And here comes in the value of such formul\ae{} as the Ten +Commandments. They are the laws of the \emph{normal} working of the +brain machinery. + +The angel (or imaginary messenger) suggests to you the one among +possible working hypotheses on which your brain will most readily +work. Now the formularies of which I spoke give you the laws of +healthy brain action. Therefore, if the angel suggests something +contrary to the registered formulas, he is suggesting the hypothesis +which you ought carefully to avoid thinking out or using at that +time. It is of all paths towards disease the one which will lead +you, in your present condition, most rapidly towards disease. But if +the imaginary angel suggests nothing contrary to the formularies, +then the image or idea which he suggests is likely to be one on +which your mind for the time being can work safely, and \emph{the} +one along which it can work most easily and profitably. + +When your imagination is acting strongly in providing you with +working hypotheses, there are a few little precautions which you +ought to observe. + +Do not at such times take either very rapid or very much prolonged +physical exercise. + +Be rather particular not to eat anything either indigestible or +highly flavoured. + +Even if you were in the habit of taking any kind of alcoholic +stimulant (which, while you are young, I hope you will not do), +avoid it during the process of framing hypotheses. Be extra careful, +at such times, to keep up any routine exercises of slack muscles and +slow breathing which you find suit you. + +Take a little extra care, at such times, not to catch cold. You are +rather less liable than usual to take cold at such times; but, on +the other hand, you are less conscious than usual of ordinary +physical sensations, and may be very cold without knowing it. A +chill may settle locally, and produce permanent mischief. + +Above all, be very careful, while the imaginative fit is on, to +avoid letting the subject as to which your imagination is stirred +become the object of either fun, vanity, or gossip. The vision which +you see may quite harmlessly and legitimately become a source of fun +to yourself and your friends at some future time, but take care +never to gossip or joke about it until it has passed from the +condition of imaginative vision to that of working hypothesis. But +the most important precaution of all is incessant reverence for the +Great Unknown, the sacred $x$: or, in other words, a constant +awareness of your own ignorance. + +Remember always that Genius means conscientious, careful work on +suggestions of the imagination taken as provisional hypotheses. + +To take suggestions of the Imagination as fact is Insanity. When you +hear of a man that he has unquestionable genius but is a little mad, +that means that he sometimes takes the products of his imagination +as working hypotheses, but sometimes mistakes them for facts. + +All the above precautions may be summed up in one sentence: Remember +that the more active the imagination is, the less the physical and +moral instincts are on the alert; therefore, conscious precaution +should supplement instinct at such times, until self-protection has +become so fixed by habit as to become in its turn automatic and +instinctive. + +If you observe these precautions you need not fear using your +imagination freely. When you hear of some brilliant imaginative +writer who has come to grief physically, mentally, or morally, after +a short and brilliant career, you will find it advantageous to try +to find out which of the precautions he has been neglecting. + +In future letters I hope to point out to your notice some famous +cases of disaster due to such neglect. + +\chapter{The Limits of the Teacher's Function} + +One of the greatest causes of mental and moral confusion, as well as +of absolute insanity, in modern Europe, is the fact that numbers of +people plunge into the second and third great Hebrew algebras before +they rightly understand the first. Even if they are silent about +their results, this distracts their own minds, and sows the seeds of +bad habits and mental confusion in their own constitutions. Many of +these people give to the world their own wild guesses about the +second and third algebras, and that puts the rest of the world into +confusion. We are, therefore, not going to enter on the question of +the second algebra till I have provided you with the possibility of +understanding and practising the first. In the next few chapters I +hope to give you a series of stories of people who used, and +sometimes mis-used, the algebra of Moses, in order that you may see +how to work the rules strictly and how mistakes might creep in. + +But, before we begin our stories, there is one principle to which I +must call your attention: it is the business of your teachers at +school to see that you acquire skill in using certain implements or +tools; it is not their business nor mine to decide what use you +shall make, when you are grown up, of the skill which you have +acquired. It is their business to see that you learn to read and to +speak properly; it is not their business to decide beforehand +whether you shall recite in public or only read to your own family +and your sick friends. It is their business to see that you know how +to sew; but not to settle whether you shall, in future, make your +own clothes or work for the poor. So it is with the tools of the +mind, such as algebra and logic. It is our business to see that you +know how to use algebraic and logical method accurately and +skilfully; it is not our business to decide whether, in the future, +you shall use your skill to deceive other people or to show them the +truth. It \emph{is} our business to see that you do not deceive +yourself, because deceiving \emph{yourself} distorts your brain and +ruins the possibility of using logical methods skilfully to arrive +at the knowledge of truths. + +When you have found out a truth, then the question whether you shall +or shall not tell it to other people is a matter of conscience. You +will have to settle it alone with the Great Power which no man +knows. Self-deception, slipshod logic, and bad algebra are things +which it is the business of your elders to protect you from while +you are young, in order that you may not \emph{lose the power} of +being honest in case you wish to be so. My business is not to judge +what is good or bad conduct, but to see that you learn how to be +perfectly honest with yourself. I wish you to notice this, because +in the books of the Hebrew algebra you will sometimes find good kind +people spoken of very harshly; and some of the most dishonest and +selfish people in the world praised and spoken of as blessed. This +puzzles many good people, because they choose to fancy that the +Hebrew books are sermons about right and wrong feelings; and do not +like to recognise that they are really about the algebra of logic. + +As I said before, people who really conduct their minds strictly +according to the algebra of logic are very prone to grow kindness +and honesty towards other people, without thinking about it, as a +matter of taste, of choice. They \emph{like} being kind and honest +better than being selfish and dishonest, and they become kind and +honest without thinking much about it. But honesty to other people +and honesty to yourself \emph{are} two different things, and must be +kept apart in your mind, just as, in physiology class, you keep +apart the flesh of an animal and its skin. You believe that if the +flesh is thoroughly healthy it will grow a good skin; but, while you +are studying, you do not mix up statements about the one with +guesses about the other. If we find that a man's logic was good, and +his conduct what we should call bad, we must do what a doctor would +do if he found a spot on a patient's skin which he could not account +for by anything wrong in his circulation or digestion. He ought not +to say either, ``That spot is not there,'' or, ``I suppose it is +right that spot should be there,'' nor, on the other hand, to jump +to the conclusion that that patient had been eating some +particularly unwholesome thing. He ought to register in his mind, as +one of his data, the fact of his own ignorance of how that spot came +there. I shall have to tell you in another chapter the story of one +of the most selfish and deceitful persons that ever lived, as to his +conduct towards other people, but who was said to be blessed, +apparently for no reason except that he was absolutely straight with +his logic and honest with himself. + +Besides, no one who is consciously and deliberately dishonest to +serve his own selfish purposes can ever do as much harm to other +people as is done every day by men and women who have muddled their +own brains with crooked logic. + +\chapter{The Use of Sewing Cards} + +When you go for holidays perhaps your friends will ask you what is +the use of sewing curves on cards. I should like you to know exactly +what to say. + +The use of the single sewing cards is to provide children in the +kindergarten with the means of finding out the exact nature of the +relation between one dimension and two. + +There is another set of sewing cards which is made by laying two +cards side by side on the table and pasting a tape over the crack +between them. This tape forms a hinge. You can lay one card flat and +stand the other edgeways upright, and lace patterns between them +from one to the other. + +The use of this part of the method is to provide girls in the higher +forms with a means of learning the relation between two dimensions +and three. + +There is another set of models, the use of which is to provide +people who have left school with a means of learning the relation +between three dimensions and four. + +The use of the books which are signed George Boole or Mary Everest +Boole is to provide reasonable people, who have learned the logic of +algebra conscientiously, with a means of teaching themselves the +relations between $n$ dimensions and $n + 1$ dimensions, whatever +number $n$ may be. + +The above is a quite accurate account of the real Boole Method; as +much as there is any need for you to know while you are at school. + +I should feel grateful to you if you will each copy it out in a +clear handwriting, and keep it by you, and take it home whenever you +go away from school for the holidays. It would be all the better if +you learned it by heart. + +And now I will tell you why I am so anxious about this. + +The Boole method is a conveyance which will take you safely to +wherever the Great Unknown directs you to go. Some people mistake it +for the carpet in the \emph{Arabian Nights}, which took whoever +stepped on it wherever he or she \emph{wished} to go--which is a +quite different thing. The true Boole method depends essentially on +making a right use of imaginary hypotheses. The magic carpet depends +for its efficacy on making a wrong use of imaginary hypotheses. + +People get to very queer places on that carpet. I have been for +several excursions on it, so I know. + +One of the places it can take you to is a town where all the front +doors open on to a street very like Regent Street; with the most +gorgeous millinery, jewellery, and fruits in shop windows; and all +the back doors open to wild country where blue roses, black tulips, +and the fattest double carnations of all colours (including green +ones) grow wild in the hedges and fields; and where all the pigs +have wings. + +Another place that it can take you to is one where pigs can wallow +in all the filth they like without soiling their wings; and moths +fly into candles without singeing theirs. + +The carpet will take you straight \emph{to} whatever place you wish +to go to. It is by no means warranted to take you safely back. + +The advantage of Boole's method is that it \emph{is} warranted to +bring you safe down somewhere on solid earth,---not always the exact +place you started from, but a safe and clean place of some +kind---and to deposit you steady on your feet, with a compass in +your pocket which will show you a straight way home. + +\chapter{The Story of a Working Hypothesis} + +In an old Hebrew book there is a story of a person named Jacob, +which means the Supplanter. If you want to know why, you had better +read the story for yourself some day. It is not entirely a pretty +story, but it is very instructive. Jacob had a dream in which he saw +``angels'' coming down a ladder. It would be a very profitable +exercise of your imagination to ask yourselves why this particular +patriarch saw angels on a ladder, whereas so many other Hebrews saw +them in clouds, or flying down on wings, or mixed up with flames and +other romantic, pretty, moving things. + +Jacob had another dream, and saw an angel who wrestled with him, and +apparently left him with sciatica for life; which is not surprising, +for he had been sleeping out of doors on bare ground, just when +\emph{he} had been wrestling with very serious difficulties caused +by his own dishonest tricks. At such times, as I told you before, +people had better be a little extra careful not to catch cold; +because colds caught under such conditions are rather prone to leave +unpleasant traces, which last a long time, and sometimes all one's +life. + +Well, the angel who gave Jacob sciatica gave him something else: a +new name. Why did he give him a new name? Taking a new name was an +ancient ceremony which meant entering a new service. Sixty years ago +servants in Devonshire were called by their employer's name. A +gardener would have two names---his own, which he got from his +father, and his master's. I have even heard dogs called by their +master's names, for instance, Toby Smith, or Ponto Jones. + +You will often notice in old books that when people were converted, +that is to say, when they either took up a new religion or turned +from bad ways to good ones, the people who persuaded them to be +converted gave them a new name, very often the teacher's own name. +Well, the angel who wrestled with Jacob appears to have converted +him. He seems to have persuaded Jacob that there are other ways of +getting on in the world and promoting the fortunes of one's children +and grandchildren besides cheating everybody, including one's own +nearest relations. + +Therefore Jacob was not to be called ``the Supplanter'' any more: +his new name was to be Israël. Jacob's descendants are called +Hebrews, and also ``the people of Israël.'' Israël was the new name +which Jacob got when he turned from cheating to a better way of +getting on in life. + +What was that better way? That is our $x$, our first unknown. What +does the word Israël mean? That is our $y$, our second unknown. I +may as well tell you at once that, so far as I am concerned, $y$ +remains unknown. I want you to take notice that \emph{I} do not know +what the word Israël means. But some twenty years ago my imagination +supplied me with a working hypothesis:--Suppose Israël meant rhythm. + +Now if I had gone telling people that \emph{Israël means rhythm}, I +should have been contradicted and laughed at and told that I had no +proof of what I said and was talking of what I knew nothing about; +and whoever said so would have been perfectly right. I should have +been cheating myself and getting into bad slipshod habits. What I +did was to post up inside my brain as a working hypothesis: +``\emph{Suppose} Israël means rhythm, what would be the consequence +of that hypothesis?'' Then I read through old books of the Hebrews, +putting in my mind the word ``rhythm'' wherever I found the word +``Israël,'' and ``the people of rhythm'' instead of ``the people of +Israël.'' + +In the stories that are told about Jacob and his grandfather Abraham +the angels are represented as telling the two men that if they would +obey the angels, not only they themselves would be blessed, but all +their descendants would be blessed too, and be made, at last, the +means of conferring a great blessing on all the world; Moses warned +them that, if they did not obey their own special angels, some +special trouble would come to them. + +My imagination suggested to me that perhaps getting into the swing +of rhythmic beats is good for all people, but more good for the +people of Israël than for anybody else; and that wandering off into +irregular un-rhythmic freaks is more bad for the people of Israël +than for anybody else. + +This, again, you will observe, is purely imaginary hypothesis. I had +not the faintest warrant for saying anything of the kind; therefore +I did not say it; but I experimented at treating my Hebrew friends +and acquaintance \emph{as if} they were natural born ministers, or +servants, of the principle of rhythmic beat; as if it was their +business to introduce respect for rhythm and an orderly arrangement +of time into the general morals of the world; and as if they would, +of course, become degraded more than other people, if they allowed +themselves to drift into being irregular and disorderly. Now you +will observe that, though all this was purely imaginary hypothesis, +it was of a harmless kind; there is nothing contrary to the ten +commandments, or to any other register of safe rules, in treating +one's Hebrew acquaintance as if one expected them to be more orderly +as to time than other people. + +The registered rules allowed me to consider this a safe road; and my +imagination showed me that it was one along which I could travel +quickly; therefore I started to go along it and waited to see where +I got to. One consequence which came was that some of the people of +Israël began telling me that I seemed to know things about their old +books (even some old books that I had never read), which they +themselves had never observed before; I had enabled them to get at +real values for the $x$'s and $y$'s; of some of their problems. + +Please notice that all this is pure imaginary hypothesis. Ancient +peoples made a hypothesis, for which they had no authority, about +angels; and I made one, for which I had no authority, about some of +those supposed angels. And, by dealing logically with these +imaginations, we got to some very real knowledge. + +\chapter{Macbeth's Mistake} + +The whole question of choosing one's next working hypothesis has +been fogged, owing to people's neglect of a very simple principle. +Suppose you are out bicycling in a strange place. You come to a bit +of smooth, good road, which is either flat or goes very gently down +hill; and presently curves in a nice, big, easy sweep round a bit of +wood or a cliff, so that you cannot see far along it. What you know +at once is that you can, \emph{if you choose}, get up great speed +without overmuch exertion. That is obvious, and needs no discussion. +The question you have to settle is: Shall you choose to do it? + +If you have heard the whole road spoken of, in general terms, as a +nice safe one to go on, you probably do choose to make use of the +specially easy bit of the road to get up a lively spin. + +But supposing that, at the beginning of the gentle slope down, you +come upon a notice board with an inscription ``Go slowly,'' or +``Dangerous to cyclists,'' I hope you would have sense enough not to +think---``What do those old fogies know about the needs of the young +generation? I have a right to go fast if I choose, and I shall have +my jolly spin in spite of them.'' Nor would you say: ``I can take +care of myself, and if I run into somebody else that is his look +out.'' If you are an experienced cyclist you would keep on your +seat, and go cautiously; if you are still a very inexperienced one, +it would be wise to get off your cycle, and not mount again till you +had come to the curve, and gone round it, and seen what is beyond. + +The notice board is not an actual prohibition to go along the +``King's highway'' if you choose. The people who put up the board +have no authority over you. But your own instincts of +self-preservation, and I hope also your instinct of loyalty and good +comradeship with the possible other cyclist who may be at the bottom +of the hill, would suggest to you not to throw away the guardianship +of a caution from those who know more than you do about the road. + +Having given you this general indication of the principle which I am +trying to explain, we will go back to the question of an imaginary +working hypothesis. + +My imagination, as I told you, showed me that my mind would travel +quickly and easily along the road opened up by supposing that Israël +means Rhythm. Looking back in my memory, I could not find the +smallest indication that anybody had either come to grief himself or +offended any Hebrew person by behaving as if the people of Israël +were the People of Rhythm; and there is nothing in the Ten +Commandments to suggest that there is any harm in doing so. So I +started off on a glorious, easy, rapid spin; and arrived, without +any mishap, at several very interesting bits of scenery. + +Now let us take the case of the old Scotch legend of Macbeth, as +told by Shakespeare. + +Macbeth and his wife appear to have been, at first, very +well-intentioned, good people, as human beings go; better than most +people; and enormously better than Jacob, or his mother, or his +uncle, or most of the people belonging to him. Macbeth was a +brilliant and successful soldier; his imagination suggested to him +that he had it in him to rise rapidly to fortune and power. He might +become Thane of Cawdor, and some day even King of Scotland. His +imagination was so vivid that he pictured three old women going +through some heathen incantation and predicting to him that he would +be Thane of Cawdor and King. Here was a road open, along which it +was quite sure that his mind would travel easily if he would let it +do so. The question was: Should he let it go along that road? Now +there were living at the time a Thane of Cawdor and a King of +Scotland. While they lived, he could not be either. The commandments +say, ``Thou shalt not covet thy neighbour's goods.'' Here was a +danger signal. If Macheth had known as much as Shakespeare knew +about the art of sound thinking, he would immediately have said to +himself, ``\,`Cawdor' and `King' are the roads that I had better not +travel along just now, for fear the wheels of my mind should get too +much way on, and carry me into danger.'' But Macbeth had either not +learnt algebra at school, or, if he had, he had only crammed it up +for examination out of a textbook, and not learned it as the Science +of the \emph{Laws of Thought}. + +Another day his imagination showed him a dagger. A dagger is a thing +to kill people with. As a soldier, he had probably used a real one +in war. But, if he had had any proper nerve training, he would have +known that when his imagination was so vivid that he did not, for +the moment, know an imaginary dagger from a real one, he ought +immediately to ``go slack''; to lie down and think about the moors +or the sky, or about anything or anybody that was not connected with +doing anything in particular, with planning anything, with taking +any resolution, and especially with breaking any of the Ten +Commandments. He had already told his wife about the three old +women. If she had been a sensible woman, she would have told him +that she wanted to go away from home; and got him to take her right +away for a few weeks; and kept him busy and amused in thinking of +other things; till he left off seeing things that were not there. +But neither Macbeth nor his wife knew as much as Shakespeare did +about the value of danger signals and the conditions for making a +safe working hypothesis. + +You had better read the story of Macbeth and see for yourselves what +they did do. + +Next to the old Hebrew books, Shakespeare is the best road map that +I know of for people who wish to travel safely about the country of +the imagination. + +\chapter{Jacob's Ladder} + +In Chapter X.\ I set you children a question:---Why did Jacob's +angels come down a ladder, whereas other Hebrews saw angels mixed up +with romantic pretty things such as wings and clouds? + +I hope some of you have made a guess before now; but some are not +good at guessing. I will tell you what may help you to find out. + +If a bird wants to go up and down from the roof to the garden, it +trusts to its wings. A man has to use a ladder: +step,---step,---step. + +If a bird is not fully fledged or has a broken wing, it has to find +something more or less like a ladder; and go up and down bit by bit: +hop,---hop,---hop. + +If an artist wishes to draw a parabola, he does it freehand, that is +to say, he just draws the curve He does not take all the trouble +which Mrs Somervell's book makes little children take, of getting +the curve step by step by the method of Finite Differences. + +Jacob wished to be rich. Some angel, but a very bad one, inspired +him with an idea of getting rich in one big sweep, by cheating his +father and brother. By wanting to do things in that sort of quick, +easy way, when he did not yet know how to do things both quickly and +rightly, he got into terrible trouble and had to leave his country. + +Now I suppose that the angels who converted him meant to say +something like this: ``It is all very well for good, holy, +God-fearing men like your father and grandfather to go where they +are taken by angels who can move about on wings; but you are at +present a stupid, clumsy person; your wings have not grown yet, or +you have broken them by being covetous. We are going to show you how +\emph{you} should go about: step,---step,---step. Have patience, and +take pains; and don't go about on magic carpets.'' + +\chapter{The Great $x$ of the World} + +A great question which people like to quarrel about is:---Who or +What made things be as they are? As soon as people grew clever +enough to think about anything except scrambling for food and taking +care of their own babies, they began quarrelling about Who or What +made things be. Nobody knew anything about it; and most people had a +great deal to say about it. Moses saw that there was no hope of +getting a country orderly while all this confusion was going on; so +he said to the Hebrews, ``I must not allow all this confusion to go +on among a people that I am made responsible for. None of us have +ever seen the Maker of things. We can see the things growing, but +not the force that makes them. \emph{That} is our X; our Unknown. We +are going to begin by stating that we don't know. We are going to +call the Maker of things `I Am,' or `That which is, whatever it is'; +and we are going to make two hypotheses to start with. We are going +to try thinking of `I Am' as Unity; one, and not several or a +fraction. We will also try thinking of `I Am' as No-Thing,---we are +not going to suppose at present that any particular kind of thing +made the rest; we will suppose that `I Am' is not a thing. When we +find that any particular proceeding or behaviour destroys men, or +makes them too sickly or weak or stupid or quarrelsome to manage +other creatures and keep the upper hand of the world, we will say, +for short, that `I Am' does not like or does not intend the people +of Israël to go on with that kind of proceeding or behaviour. + +``Now these two hypotheses are as much as we can deal with for the +present. Anybody who wants to think out other hypotheses than those +will have to think to himself, or go out of the country that I am to +manage. + +``Now we will arrange all the facts that we know round the statement +of our own ignorance; and then try our hypotheses on them. + +``We know that eating the flesh of certain uncleanly animals gives +people certain diseases; we will say, for short, `I Am' does not +intend the Hebrews to eat the flesh of those animals. We know that +if people are dirty in their habits and careless in preparing their +food and in washing their hands before they touch food, they get +fevers; we will suppose that `I Am' does not intend the people of +Israël to be dirty in their habits. We know that if people burn +things the smoke of which makes them drunk and silly, they manage +their affairs badly, and make mistakes, and do not grow their crops +properly, and are not ready to fight when enemies attack them. The +people in neighbouring countries say that the Maker of things likes +or dislikes to smell the smoke of these drugs; they know no more +than we do what \emph{He} likes to smell, but we are going to +suppose that `I Am' does not like \emph{us} to smell them.'' + +The Hebrews never found out what ``I Am'' is; but those who stuck +loyally by the hypotheses of Moses, and refused to be distracted +from the matter in hand, or to talk about anything except the +experiment which they were trying, found out several things that +were very useful to them. For instance, about weather and the +electricity of the atmosphere, and how to take care of their health, +and how to use their imagination to supply them with working +hypotheses for a variety of sciences, and how to use their dreams to +show them where they had been making mistakes and spoiling their +brains. Whereas the people who would insist on shouting and arguing +and quarrelling about things which were only wild guesses got on +very slowly with learning Science. + +\chapter{Go Out of My Class-Room} + +A story is told of one of the orderly pupils of Mosaism who got to +know a good deal about weather and electricity; and at last he got +out of patience with the people who wanted to shout and argue. And +he said to them: ``What is the good of all this arguing backwards +and forwards about things that we do not know and cannot settle? Let +us try a fair experiment. You go on shouting and doing whatever +\emph{you} think the Unseen Powers like; and I will do what \emph{I} +think will get them to do what \emph{I} like. And let us agree that +whichever of us can draw a spark out of a thundercloud shall be +considered to know most about how to come to an understanding with +`I Am.'\,'' + +So the other people shouted and jumped about, and cut themselves +with knives; because they had taken it into their heads to imagine +that the Maker of things liked to see that kind of behaviour. + +Why they thought so I cannot conceive. But there's no end to the +rubbish that people get to think when they argue about what X is, +instead of trying hypotheses in an orderly manner. + +The Unknown Powers let them shout all day long; and then Elijah got +a spark out of a thundercloud. + +The same sort of thing happened again about a hundred and fifty +years ago. Various sorts of priests were shouting and arguing about +what ``I Am'' wished people to believe and to think; and then +Benjamin Franklin and his friends, who had not been mixing up with +the argument or making wild guesses, but quietly experimenting and +dealing logically with the fact of their own ignorance, sent up a +kite into a thundercloud, and got a spark down; and the consequence +of that is that all kinds of people say, ``What a wonderful man +Benjamin Franklin was!'' and all sorts of people are able to ride +about in electric trams. + +But the curious part of the matter is that many people use electric +trams to go to meetings, on purpose to shout and argue and make wild +guesses about things they know nothing about! + +However, what they choose to do is not our business. You are living +in an orderly school; and of course you do not argue about things +you know nothing about. Let us go back to our Hebrew electrician. + +He had shown the people of Israël what comes of sticking peaceably +to one's working hypothesis. If he had been thoroughly logical he +would have gone on sticking to it. He would have said to the people +of Israël, ``Now you see that I can teach you electricity; this land +is going to be my class-room; make those shouting people hold their +tongues, or else go away; so that we can go on with our lessons in +peace. When they want to learn electricity properly, they can come +back.'' But he was in too great a hurry to make a complete and final +settlement. A good teacher sends a noisy, troublesome pupil out of +his class-room for the time, but does not expel her from the school +merely for being troublesome. The shouting people were among the +facts which ``I Am'' put before Elijah to deal with. He found it +necessary to eliminate them in order to reduce his data within the +compass of his power to manage, but he should have done it as a +provisional necessity. He should have arranged his mind on the +understanding that the eliminated data would have to come back. + +Instead of that he used his power and science to kill them; and gave +way to the feeling that they were got rid of and done with. + +And then his mind began to go wrong. He lost his nerve. He began to +talk nonsense about things \emph{he} knew nothing about, and led a +great many people into mistakes. + +\chapter{$\sqrt{-1}$} + +When you come to quadratic equations you will be confronted with an +entity (or non-entity) whose name is written this way---$\sqrt{-1}$, +and pronounced "square root of minus one." Many people let this +nonentity persuade them to foolish courses. A story is told of a man +at Cambridge who was expected to be Senior Wrangler; but he got +thinking \emph{about} the square root of minus one as if it were a +reality, till he lost his sleep and dreamed that \emph{he} was the +square root of minus one and could not extract himself; and he +became so ill that he could not go to his examination at all. +Angels, and square roots of negative quantities, and the other +things that have no existence in three dimensions, do not come to us +to gossip about themselves; or the place they came from; or where +they are going to; or where we are going to in the far future. They +are messengers from the As-Yet-Unknown; and come to tell us where we +are to go next; and the shortest road to get there; and where we +ought not to go just at present. When square root of minus one comes +to you, behave reasonably about him. Treat him logically, exactly as +if he were six or nine; only always remember to keep well in front +of you \emph{the fact of your own ignorance}. You may never find out +any more about him than you know now; but if you treat him sensibly +he will tell you plenty of truths about your $x$'s and $y$'s, and +other unknown things. + +Please don't suppose that I have always behaved sensibly to angels. +I have often made serious mistakes in dealing with them. I have +acted in haste and have had plenty of reason to repent at leisure. +But one thing they have taught me is that we need never be +\emph{afraid} of angels, whether white or black, as long as we keep +the laws of logic. Another thing they have shown me is that angels +never really gossip. They have often pretended to gossip to me; but +I have found out afterwards that they have been talking clever +nonsense in order to test me and prove me; so that I might see in +what an illogical state of mind I have met them. Angels leave real +gossip to old women who have done their life's work and have time to +sit in the chimney corner and tell tales about their past +experiences to their child friends. + +\chapter{Infinity} + +You remember the angel who looks like this, $\sqrt{-1}$. Now I am +going to introduce you to another angel. It is called ``Infinity.'' +When you come to it, remember what I told you before---Angels are +messengers from the great world of the ``As-Yet-Unknown.'' They +never gossip about their private affairs, or those of other angels. +They come to tell you either about what you are to do next, or about +something you had better not do next; and if you ask them +impertinent questions about things that do not concern you for the +time being, they will give you headaches and make your head spin: +just to teach you to mind your own business. This particular angel +always comes with a message about a broken link or a loosened chain. +It comes, when an hypothesis has been fully worked out, to tell you +that you are now free from the bonds of that hypothesis and at +liberty to start experimenting on a fresh one. But its message is +never: ``You have got out of that particular house of bondage and +therefore you may, for all the rest of your life, run riot, and eat, +and drink, and do, whatever you please.'' Its message always is: +``You have outgrown that master; now you may take a holiday and have +a fling before you go into a higher class; but, just because you are +set free, look out for danger traps; and mind your Ten +Commandments.'' + +You will understand Infinity's messages better if you will read +carefully what is written about it in Chapter XV. of ``The Logic of +Arithmetic.'' It brought the answer to the question: ``How many +children could pass through a school-room without the apples all +being eaten up, supposing that none of the children ate any?'' + +Let us go over that ground again. Suppose there is a cake on the +table. How many children can go through the room without the cake +being all eaten up? + +Well, that depends on two things: the size of the cake, and the +share which each child eats. If the cake weighs two pounds, and each +child eats two ounces, it will be all eaten up when sixteen children +have gone through the room. If the cake weighs only one pound, it +will be eaten up when eight children have gone through the room. But +if each child eats only one ounce, then again sixteen children will +have to go through the room before the cake is eaten up, and so on. +Many questions could be asked, all depending on the size of the cake +and the size of each child's share. + +All this time you are tied to an hypothesis that the children eat +cake (more or less). + +But now suppose we are freed from that hypothesis. Suppose no cake +is given to the children. How many can pass through the room before +it is all eaten up? + +The answer to that is: ``An infinite number.'' Infinity does not +mean any particular number, or a very large number. It means a +loosened chain, a discarded hypothesis, escape from the rule we were +working under. Something else, not the size of the cake, determines +the number of children. Infinity does not mean that there are enough +children in the world now to go on passing through the room +\emph{for ever}, but that the number of children who pass through +the room, now that the share of each child is 0 (zero), will have to +be determined by the number of children that there are in the +school, or the parish, or wherever it is that the children are +supposed to come from; \emph{and not by the size of the cake}. The +size of the cake has no longer anything to do with answering the +question: ``How many children can pass through the room before the +cake is all eaten?'' + +\chapter{From Bondage to Freedom} + +Moses had said, from the first, that the people of Israël would have +to think of ``I Am'' as the deliverer from bondage; but they were +not, at the time when he said it, advanced enough in their algebra +to understand that idea properly. So he gave them, as an hypothesis +to work on for the time being, that ``I Am'' did not like the +\emph{people of Israël} to eat and drink and smell unwholesome +things. He wished to make them attend to their own affairs, and +think as little as possible about what was done and thought outside +of their own land. + +But, after the time of Elijah, there came a change. A higher kind of +algebra came into use. Its incarnation was called Isaiah. + +The imagination of the Hebrews broke loose from the hypothesis that +``I Am'' had wishes and likes about the people of Israël different +from what was right for all the rest of the world. + +When that hypothesis was taken away, the imagination of such people +as Isaiah took wings and flew to---well---we do not know where, but +we call it Infinity. We know nothing about Infinity; except that it +comes when a chain is loosened. + +If you want to understand what it was that happened to Isaiah, and +what Infinity means in algebra, this is how you can find out. Get a +bowl and dip up some of the water out of a barrel in which a gnat +has laid her eggs. Little wigglers are born from those eggs. If you +watch them you will see that they swim in different positions, some +with their tails uppermost, some with their heads uppermost. There +may also be some worms, who do not swim much, but wriggle about at +the bottom of the bowl. Perhaps if we could hear them talking we +should hear them quarrelling about which was the right position. +Some of them might be disputing about what would happen to them in +the future. They might quarrel till the end of the world, and know +no more about it at the end than at the beginning. They are all tied +by the same hypothesis:---that everybody lives under water. It is a +very good working hypothesis for them; for if one of them got out of +the water it would die. If they knew algebra properly, they would +understand that water is only their present working hypothesis, and +that it is quite possible there may be people who live out of it. +But it is not sure that they do know enough algebra to be +\emph{aware of their own ignorance}. + +If you watch them carefully, you will some day see a wiggler come +out of the water. He has got wings. The water-hypothesis no longer +concerns him. Some link in the chain that bound him down to water +has opened; he is set free; Infinity has come to him. + +That is what happened to Isaiah when he got out of the kind of +Mosaism by which such people as Joshua and Samuel were tied down. +That is what will happen to you (if you learn your algebra properly) +when you are no longer tied down to $a$, $b$, $c$, and $\sqrt{-1}$, +as the values of $x$; and learn to see that the answer to a problem +may sometimes be +\begin{equation*} +X = Infinity. +\end{equation*} + +Please notice that if a winged gnat fell back into the water he +would die. You will find this a good working rule:---Whenever +anything comes near your imagination which calls itself either +``Infinity'' or ``The Liberation from Bondage,'' go slack for a few +minutes; say over the Ten Commandments; and make a mind-picture of +the gnat-grub in the water. Tell yourself that his best chance of +growing strong wings and being able to fly, when Infinity comes and +calls him to go up higher, is to stay in the water till the wings +have grown strong and work out the water-hypothesis to its logical +conclusion. + +Then make another mind-picture:---The gnat who has got wings, and +\emph{therefore must not try to amuse himself in the water}. + +Please observe:---There is nothing in this rule contrary to any +commandment. Moreover, there is nothing slavish or degrading in it; +nothing in the least like giving up your own liberty, or hampering +your own initiative, or being a slave to past ages; nothing which +prevents your being up to date and fit for the generation to which +you belong. + +You are not asked to have any opinions about it; or to think that it +is a duty in itself; or to think that you are better than other +people because you do it, or that every one is wrong who does not do +it. If you do it, it will be for no reason that you know of, except +that an old woman who has been trying to amuse you asks you to do it +as a token of friendly feeling towards her. + +\chapter{Appendix} + +The essential element of Algebra:---the habitual registration of the +exact limits of one's knowledge, the incessant calling into +consciousness of the fact of one's own ignorance, is the element +which Boole's would-be interpreters have left out of his method. It +is also the element which modern Theosophy omits in its +interpretation of ancient Oriental Mind Science. + +Men who wish to exploit other men fear nothing in logic or science +except this element. They fear nothing in earth, heaven, or hell, so +much as a public accustomed to realise exactly \emph{how much has +been proved, and where its own ignorance begins}. Exploiters fear +this about equally, whether they call themselves priests, +schoolmasters, college dons, political leaders, or organisers of +syndicates and trusts. + +As long as general readers can be kept from the habit of registering +at every step the fact of their own ignorance and the limits of +their own knowledge, a clever charlatan can deceive them about +anything he pleases:---``from pitch-and-toss to manslaughter''; from +Zero to Infinity; from the contents of a meat tin to the contents of +an engineer's report; from the interpretation of a bill before +Parliament to the interpretation of Isaiah. + +Once get any fair proportion of the public into the steady habit of +algebraising ignorance, and you will have done much towards reducing +all kinds of parasitic creatures to the alternative of starvation, +suicide, or earning their own living by rendering some kind of real +service to the organism which supports them. + +\markright{ADVERTISEMENT.} +\begin{center} +\fbox{\parbox{11cm}{ +\begin{center} +\Huge\textbf{Logic Taught} \\ +\textbf{by Love} \\ +\bigskip\Large\textbf{RHYTHM IN NATURE AND} \\ +\textbf{IN EDUCATION} \\ +\smallskip \normalsize A set of articles chiefly on the light thrown +on each other \\ +by Jewish Ritual and Modern Science \\ +\smallskip \textbf{By} \\ +\Large\textbf{Mary Everest Boole} \\ +\smallskip \normalsize Crown 8vo, Cloth, 3s.\ 6d.\ net. \\ +\smallskip \textbf{LIST OF CHAPTERS} \\ +\end{center} +\footnotesize\begin{itemize} + \item[1.] In the Beginning was the Logos. + \item[2.] The Natural Symbols of Pulsation. + \item[3.] Geometric Symbols of Progress by Pulsation. + \item[4.] The Sabbath of Renewal. + \item[5.] The Recovery of a Lost Instrument. + \item[6.] Babbage on Miracle. + \item[7.] Gratry on Logic. + \item[8.] Gratry on Study. + \item[9.] Boole and the Laws of Thought. + \item[10.] Singular Solutions. + \item[11.] Algebraizers. + \item[12.] Degenerations towards Lunacy and Crime. + \item[13.] The Redemption of Evil. + \item[14.] The Science of Prophecy. + \item[15.] Why the Prophet should be Lonely. + \item[16.] Reform, False and True. + \item[17.] Critique and Criticasters. + \item[18.] The Sabbath of Freedom. + \item[19.] The Art of Education. + \item[20.] Trinity Myths. + \item[21.] Study of Antagonistic Thinkers. + \item[22.] Our Relation to the Sacred Tribe. + \item[23.] Progress, False and True. + \item[24.] The Messianic Kingdom. + \item[25.] An Aryan Seeress to a Hebrew Prophet. + \item[ ] Appendix I. + \item[ ] Appendix II. +\end{itemize} \normalsize +\begin{center} +\rule{10cm}{1pt} +\textbf{London: C.W.\ DANIEL, 11 Cursitor Street, E.C.} +\end{center}}} +\end{center} + +\newpage +\begin{center} +\fbox{\parbox{11cm}{ +\begin{center} +\Huge\textbf{The Message of} \\ +\textbf{Psychic Science to} \\ +\textbf{The World} \\ +\smallskip \normalsize \textbf{By} \\ +\Large\textbf{Mary Everest Boole} \\ +\smallskip \normalsize Crown 8vo, Cloth, \textbf{3s.\ 6d.\ } net. \\ +\smallskip \textbf{LIST OF CHAPTERS} \\ +\end{center} +\footnotesize\begin{itemize} + \item[1.] The Forces of Nature. + \item[2.] On Development, and on Infantile Fever as a Crisis of Development. + \item[3.] On Mental Hygiene in Sickness. + \item[4.] On the Respective Claims of Science and Theology. + \item[5.] Thought Transference. + \item[6.] On Hom\oe{}opathy. + \item[7.] Conclusion. + \item[ ] \quad APPENDIX:--- + \item[ ] On Phrenology. + \item[ ] Notes. +\end{itemize} +\normalsize +\begin{center} +\rule{10cm}{1pt} +\textbf{London: C.W.\ DANIEL, 11 Cursitor Street, E.C.} +\end{center}}} +\end{center} + +\newpage +\begin{center} +\fbox{\parbox{11cm}{ +\begin{center} +\Huge\textbf{Mistletoe and Olive} \\ +\normalsize An Introduction for Children to the Life of + Revelation \\ +\Large\textbf{By Mary Everest Boole} \\ +\smallskip \normalsize Royal 16 mo. Cloth, \textbf{1s.\ 6d.\ } net. \\ +\smallskip \textbf{LIST OF CHAPTERS} \\ +\end{center} +\footnotesize\begin{itemize} + \item[1.] Greeting the Rainbow. + \item[2.] God hath not left Himself without a Witness. + \item[3.] Out of Egypt have I called my Son. + \item[4.] Holding up the Leader's Hands. + \item[5.] Greeting the Darkness. + \item[6.] Blind Guides. + \item[7.] Hard Lessons made Easy. + \item[8.] The Cutting of the Mistletoe. + \item[9.] Genius comes by a Minus. + \item[10.] The Rainbow at Sea, or the Magician's Confession. +\end{itemize} +\begin{center} +\rule{10cm}{1pt} +\smallskip \Huge\textbf{Miss Eduction} \\ +\textbf{and Her Garden} \\ +\smallskip \normalsize A Short Summary of the Educational \\ +Blunders of half a century \\ +\Large\textbf{By Mary Everest Boole} \\ +\smallskip \normalsize Royal 16 mo. Cloth, \textbf{6d.\ } net. \\ +\rule{10cm}{1pt} +\textbf{London: C.W.\ DANIEL, 11 Cursitor Street, E.C.} +\end{center}}} +\end{center} + +\newpage +\begin{center} +\fbox{\parbox{11cm}{ +\begin{center} +\Huge\textbf{Mathematical} \\ +\textbf{\quad Psychology of} \\ +\textbf{Gratry and Boole} \\ +\smallskip \normalsize \textbf{for MEDICAL STUDENTS} \\ +\smallskip Dedicated, by permission to Dr. H. \textsc{Maudesley} \\ +as a contribution to the science of brain, showing \\ +the light thrown on the natur of the human \\ +brain by the evolution of mathematical process. \\ +\smallskip \textbf{By} +\Large\textbf{Mary Everest Boole} \\ +\smallskip \normalsize \textbf{Crown 8vo. Cloth, 3s.\ 6d.\ net.} \\ +\smallskip LIST OF CHAPTERS \\ +\end{center} +\footnotesize \begin{itemize} + \item[1.] Introductory. + \item[2.] Geometric Co-ordinates + \item[3.] The Doctrine of Limits. + \item[4.] Newton and Some of his Successors. + \item[5.] The Law of Sacrifice. + \item[6.] Inspiration \emph{versus} Habit. + \item[7.] Examples of Practical Application of the Mathematical Laws of Thought. + \item[8.] The Sanity of True Genius. + \item[ ] Appendix. +\end{itemize}\normalsize +\begin{center} +\rule{10cm}{1pt} +\textbf{London: C.W.\ DANIEL, 11 Cursitor Street, E.C.} +\end{center}}} +\end{center} + +\newpage +\pagestyle{empty} +\small \pagenumbering{gobble} +\begin{verbatim} + +End of the Project Gutenberg EBook of Philosophy and Fun of Algebra +by Mary Everest Boole + +*** END OF THIS PROJECT GUTENBERG EBOOK PHILOSOPHY AND FUN OF ALGEBRA *** + +***** This file should be named 13447-t.tex or 13447-t.zip ***** +***** or 13447-pdf.pdf or 13447-pdf.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/1/3/4/4/13447/ + +Produced by Joshua Hutchinson, John Hagerson, and the Project +Gutenberg On-line Distributed Proofreaders. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..44c3203 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #13447 (https://www.gutenberg.org/ebooks/13447) |
