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Now the forces and actions of bodies are circumscribed and measured, either by distances of space, or by moments of time, or by concentration of quantity, or by predominanisa of virtue; iind unless these iout things have been well and carefully welched, we shall have sciences, fau* perhaps in theory, but in practice inefficient. The four instances which are useful in this point of view I class under one head as Mathematical Instances and Instances of Hcasnremeati.'-^.d^U7» Orga/mmi^ lib. ii, Aph. xliv. NEW TORE D. APPLETON AND COMPANY 1, 8, AHD 6 BOND STEEET 1885 13s PEEFACE. . In Marcli 1879 Clifford died at Madeira j six years afterwards a posthumous work is for the first time placed before the public. Some explanation of this delay must be attempted in the present preface.' The original work as planned by Clifford was to have been entitled The First Principles of the Mathe- matical Sciences Explained to the Non-Mathematical, and to have contained six chapters, on Number, Space, Quantity, Position, Motion, and Mass respectively. Of the projected work Clifford in the year 1875 dictated the chapters on Number , and Space completely, the first portion of the chapter on Quantity, and somewhat later nearly the entire chapter on Motion. The first two chapters were afterwards seen by him in proof, but never finally revised. Shortly before his death he ex- pressed a wish that the book should only be published ' A still mbre serious delay seems likely to attend the puWieafion of the second and concluding part (Kinetic) of Clifford's Elements of Dynamic, the manuscript of which was left in a completed state. I venture td think the delay a calamity to the mathematical world. VI PEEPACE. after very careful revision, and that its title should be changed to The Common Sense of the Exact Sciences. Upon Clififbrd's death the labour of revision and completion was entrusted to Mr. E. C. Eowe, then Professor of Pure Mathematics at University College, London. That Professor Eowe fully appreciated the difficulty and at the same time the importance of the task he had undertaken is very amply evidenced by the time and care he devoted to the matter. Had he lived to complete the labour of editing, the work as a whole would have undoubtedly been better and more worthy of Clifford than it at present stands. On the sad death of Professor Eowe, in October 1884, 1 was requested by Messrs. Kegan Paul, Trench, & Co. to take up the task of editing, thus left incomplete. It was with no light heart, but with a grave sense of responsibility that I undertook to see through the press the labour of two men for whom I held the highest scientific admiration and personal respect. The reader will perhaps appre- ciate my difficulties better when I mention the exact state of the work when it came into my hands. Chapters I. and IL, /Space and Number ; half of Chapter III., Quantity (then erroneously termed Chapter IV.) ; and Chapter V., Motion, were in proof. With these proofs I had only some half-dozen pages of the corresponding manuscript, all the rest having un- PREFACE. VU fortunately been considered of no further use, and accordingly destroyed. How far the contents of the later proofs may have represented what Clifford dictated I" have had no means of judging except from the few pages of manuscript in my possession. In revising the proofs of the first two chapters, which ClifiFord himself had seen, I have made as little alteration as possible, only adding an occasional foot-note where it seemed necessary. From page 65 onwards, however, I am, with three exceptions in Chapter V., responsible for all the figures in the book. After examining the work as it was placed in my hands, and consulting Mrs. Clifford, I came to the conclusion that the chapter on Quantity had been misplaced, and that the real gaps in the work were from the middle of Chapter III. to Chapter V., and again at the end of Chapter V. As to the manner in which these gaps were to be filled I had no definite information whatever ; only after my work had been completed was an early plan of Clifford's for the book discovered. It came too late to be of use, but it at least confirmed our rearrangement of the chapters. Tor the latter half of Chapter III. and for the whole of Chapter IV. (pp. 116-226) I am alone responsible. Yet whatever there is in them of value I owe to Clifford ; whatever is feeble or obscure is my own. VIU PKEFACE. With Chapter V. my task has been by no means light. It was written at a time when Clifford was much occupied with his theory of ' Graphs,' and found it impossible to concentrate his mind on anything else : parts of it are clear and succinct, parts were such as the author would never have allowed to go to press. I felt it impossible to rewrite the whole without depriving the work of its right to be called Clifford's, and yet at the same time it was absolutely necessary to make considerable changes. Hence it is that my revision of this chapter has been much more extensive than in the case of the first two. With the result I fear many will be dissatisfied; they will, however, hardly be more conscious of its deficiencies than I am. I can but plead the conditions under which I have had to work. One word more as to this chapter. Without any notice of mass or force it seemed impossible to close a discussion on motion; something I felt must be added. I have accordingly introduced a few pages on the laws of motion. I have since found that Clifford intended, to write a concluding chapter on mass. How to express the laws of motion in a form of which Clifford would have approved was indeed an insoluble riddle to me, because I was unaware of his having written anytliino- on the subject. I have accordingly expressed, although with great hesitation, my own views on the subject; PREFACE. IX these may be concisely described as a strong desire to see the terms matter and force, together with the ideas associated with them, entirely removed from scientific terminology — to reduce, in, fact, all dynamic to kine- matic. I should hardly have ventured to put forward these views had I not recently discovered that they have (allowing for certain minor differences) the weighty authority of Professor Mach, of Prag.' But since writing these pages I have also been referred to a discourse delivered by Clifford at the Eoyal Institution in 1873, some account of which appeared in Nature, June 10, 1880. Therein it is stated that 'no mathematician can give any meaning to the language about matter, force, inertia used in cuiTent text-books of mechanics.' ^ This fragmentary account of the discourse undoubtedly proves that Clifford held on the categories of matter and force as clear and original ideas as on all subjects of which he has treated; only, alas! they have not been preserved. In conclusion I must thank those friends who have been ever ready with assistance and advice. Without their aid I could not have accomplished the little that • See his recent book, Bie Mechanik in ihrer EntwicTcelung. Leipzig, 1883. ^ Mr. E. Tucker, 'who has kindly searched ClifFord's note-hooks for anything on the subject, sends me a slip of paper Tvith the following words in Clifford's handwriting : ' Force is not a fact at all, but an idea embodying what is approximately the fact.' X PREFACE. has been done. My sole desire has been to give to the public as soon as possible another work of one whose memory will be revered by all who have felt the invigorating influence of his thought. Had this work been published as a fragment, even as many of us wished, it would never have reached those for whom Clifford had intended it. Completed by another hand, we can only hope that it will perform some, if but a small part, of the service which it would undoubtedly have fulfilled had the master lived to put it forth. K. r. Uniteesitt College, London: February 26. CONTENTS. CHAPTER I. NUMBER. SaonON PAGB 1. Number is Independent of the order of Counting , , . 1 2. A Sum is Independent of the order of Adding . . . . 2 3. A Product is Independent of the order of Multiplying . . 6 4. The Distributive Law 14 6. On Powers 16 6. Square of a + 1 17 7. On Powers of a + ft 19 8. On the Number of Arrangements of a Group of Letters . . 24 9. On a Theorem concerning any Power of a + ft . , .27 10. On Operations which appear to be without Meaning . . . 32 11. Steps 34 12. Extension of the Meaning of Symbols 38 13. Addition and Multiplication of Operations . . . .40 14. Division of Operations 42 16. General Kesult.s of our Extension of Terms . , . ,45 CHAPTER IL SPACE. 1. Boundaries take up no Eoom 47 2. Lengths can be Moved without Change 52 3. The Characteristics of Shape 55 4. The Characteristics of Surface Boundaries 63 6. The Plane and the Straight Line 66 6. Properties of Triangles 69 7. Properties of Circles j Belated Circles and Triangles . . 75 8. The Conic Sections 81 9. On Surfaces of the Second Order 87 10. How to form Curves of the Third and Higher Orders . • . 91 Xll CONTENTS. CHAPTEK III. QUANTITY. SECTIOK 1. The Measurement of Quantities . . . 2. The Addition and Subtraction of Quantities 3. The Multiplioation and Division of Quantities 4. The Arithmetical Expression of Eatios 6. The- Fourth Proportional 6. Of Areas ; Stretch and Squeeze . 7. Of Fractions 8. Of Areas ; Shear .... 9. Of Circles and their Areas 10. Of the Area of Sectors of Curves 11. Extension of the Conception of Area 12. On the Area of a Closed Tangle . 13. On the Volumes of Space-Figures . 14. On the Measurement of Angles . 15. On Fractional Powers PAOB . 95 . 99 . 100 . 102 . 105 . 113 . 116 . 120 . 123 . 130 . 131 . 135 . 138 , 141 , 144 CHAPTER rV. POSITION. 1. All Position is Relative . . ; i , , . . 147 2. Position may be Determined by Directed Steps . . . . 149 3. The Addition of Directed Steps or Vectors .... 153 4. The Addition of Vectors obeys the Commutative Law . , 158 5. On Methods of Determining Position in a Plane . . . 169 6. - -Polar Co-ordinates 164 7. The Trigonometrical Eatios 166 8. Spirals 167 9. The Equiangular Spiral 171 10. On the Nature of Logarithms 176 11. The Cartesian Method of Determining Position . . . 181 12. Of Complex Numbers 188 13. On the Operation which turns a Step through a given Angle . 192 14. Eelation of the Spin to the Logarithmic Growth of Unit Step 195 15. On the Multiplication of Vectors . . ' ] 98 16. Another Interpretation of the Product of Two Vectors . . 204 17. Position in Three-Dimensioned Space . .... 207 18. On Localised Vectors or Eotors ... . . 210 19. On the Bending of Space 214 CONTENTS. XUl CHAPTER V. MOTION. BECTIOX 1. On the Various Kinds of Motion . . . 2. Translation and the Curye of Positions 3. Uniform Motion 4. Variable Motion 5. On the Tangent to a Curve . . . 6. On the Determination of Variable Velocity 7. On the Method of Fluxions .... 8. Of the Eelationship of Quantities, or Functions 9. Of Acceleration and the Hodograph . 10. On the Laws of Motion . , . . 11. FA6B . 227 230 235 . 237 . 243 . 250 . 253 . 255 . 260 . 267 Of Mass and Force 269 THE COMMON SENSE OF THE EXACT SCIENCES. CHAPTEE I. NtTMBEK. § 1, Number is Independent of the order of Counting. The word which stands at the head of this chapter contains six letters. In order to find out that there are six, we count them ; n one, u two, w three, i four, e five, r six. In this process we have taken tlie letters one by one, and have put beside them six words which are the first six out of a series of words that we always carry about with us, the names of numbers. Afterputting these six words one to each of the letters of the word number, we found that the last of the words was six ; and accordingly we called that set of letters by the napae six. If we counted, the letters in the word ' chapter ' in the same way, we should find that the last of the numeral words thus used would be seven; and accor- dingly we say that there are seven letters. But now a question arises. Let us suppose that the letters of the word number are printed upon separate 2 THE COMMON SENSE OF THE EXACT SCIENCES. small pieces of wood belonging to a box of letters ; tbat we put these into a bag and shake them up and bring them out, putting them down in any other order, and then count them again ; we shall still find that there are six of tbem. For example, if they come out in the alphabetical order b e mnr u, and we put to each of these one of the names of numbers that we have before used, we shall still find that the last name will be six. In the assertion that any group of things con- sists of six things, it is implied that the word six will be the last of the ordinal words used, in whatever order we take up this group of things to count them. That is to say, the number of any set of things is the same in whatever order we count them. Upon this fact, which we have observed with regard to the particular number six, and which is true of all numbers whatever, the whole of the science of number is based. We shall now go on to examine some theorems about numbers which may be deduced from it. §2.4 Sum is Independent of the order of Adding. Suppose that we have two groups of things ; say the letters in the word ' number,' and the letters in the word * chapter,' We may count these groups separately, and find that they come respectively to the numbers six and seven. We may then put them all together, and we find in this case that the aggregate group which is so formed consists of thirteen letters. Now this operation of putting the things all together may be conceived as taking place in two different ways. We may first of all take the six things and put them in a heap, and then we may add the seven things to them one by one. The process of counting, if it is performed NUMBER. 6 in this order, amounts to counting seven more ordinal words after the word six. We may however take the seven things first and put them into a heap, and then add the six things one by one to them. In this case the process of counting amounts to counting six more ordinal words after 'the word seven. But from what we observed before, that if we count any set of things we cotoe to the same number in what- ever order we count them, it follows that the number we arrive at, as belonging to the whole group of things, must be the same whichever of these two processes we use. This number is called the sum of the two numbers 6 and 7 ; and, as we have seen, we may arrive at it either by the first process of adding 7 to 6, or by the second process of adding 6 to 7. The process of adding 7 to 6 is denoted by a short- hand symbol, which was first used by Leonardo da Vinci. A little Maltese cross ( + ) stands for the Latin plus, or the English increased by. Thus the words six increased hy seven are written in shorthand 6 + 7. Now we have arrived at the result that six increased hy seven is the same number as seven increased by six. To write this wholly in shorthand, we require a symbol for the words, is the same number as. The symbol for these is = ; it was first used by an Englishman, Robert Eecorde. Our result then may be finally written in this way : — 6 + 7 = 7 + 6. The proposition which • we have written in this symbolic form states that the sum of two numbers 6 and 7 is independent of the order in which they are added together. But this remark which we have made about two particular numbers is equally true of any two numbers whatever, in consequence of our funda- 4 THE COMMON SENSE OF THE EXACT SCIENCES. mental assumption that the number of things in any group is independent of the order in which we count them. For by the sum of any two numbers we mean a number which is arrived at by taking a group of things containing the first number of individuals, and adding to them one by one another group of things containing the second number of individuals ; or, if we like, by taking a group of things containing the second number of individuals, and adding to them one by one the group of things containing the first number of individuals. Now, in virtue of our fundamental assumption, the results of these two operations must be the same. Thus we have a right to say, not only that 6 + 7 = 7 + 6, but also that 6 + 13 = 13 + 5, and so on, whatever two numbers we like to take. This we may represent by a method which is due to Vieta, viz., by denoting each number by a letter of the alphabet. If we write a in place of the first number in either of these two cases, or in any other case, and 6 in place of the second number, then our formula will stand thus : — a + h = b + a. By means of this representation we have made a statement which is not about two numbers in particu- • lar, but about all numbers whatever. The letters a and h so used are something like the names which we give to things, for example, the name horse. When we say a horse has four legs, the statement will do for any particular horse whatever. It says of that particular horse that it has four legs. If we said ' a horse has as many legs as an ass,' we should not be speaking of any particular horse or of any particular ass, but of any horse whatever and of any ass whatever. Just in the same way, when we assert that a + b = h + a,yfe are NUMBER. O not speaking of any two particular numbers, but of all numbers whatever. We may extend this rule to more numbers than two. Suppose we add to the sum a + b a third number, c, then we shall have an aggregate group of things which is formed by putting together three groups, and the number of the aggregate group is got by adding together the numbers of the three separate groups. This number, in virtue of our fundamental assumption, is the same in whatever order we add the three groups together, because it is always the same set of things that is counted. Whether we take the group of a things first, and then add the group of h things to it one by one, and then to this compound group of a + 6 things add the group of c things one by one; or whether we take the group of c things, and add to it the group of b things, and then to the compound group of c + 6 things add the group of a things, the sum must in both cases be the same. We may write this result in the symbolic form a+b + c = c + b + a, or we may state in words that the sum of three numbers is independent of the order in which they are added together. This rule may be extended to the case of any number of numbers. However many groups of things we have, if we put them all together, the number of' things in the resulting aggregate group may be counted in various ways. We may start with counting any one of the original groups, then we may follow it with any one of the others, following these by any one of those left, and so on. In whatever order we have taken these groups, the ultimate process is that of counting the whole aggregate group of things ; and consequently the numbers that we arrive at in these different ways must all be the same. 6 THE COMMON SENSE OF THE EXACT SCIENCES. § 3. ^ Product is Independent of the order of Multiply mg. Now let us suppose that we take six groups of things vyhich all contain the same number, say 5, and that we want to count the aggregate group which is made by putting all these together. We may count the six groups of five things one after another, which amounts to the same thing as adding 5 five times over to 5. Or if we like we may simply mix up the whole of the six groups, and count them without reference to their previous grouping. But it is convenient in this case to consider the six groups of five things as arranged in a particular way. Let us suppose that all these things are dots which are made upon paper, that every group of five things is five dots arranged in a horizontal line, and that the six groups are placed vertically under one another as in the figure. o • e o • We then have the whole of the dots of these six groups arranged in. the form of an oblong which con- tains six rows of five dots each. Under each of the five dots belonging to the top group there are five other dots belonging to the remaining groups ; that is to say, we have not only six rows containing five dots each, but five columns containing six dots each. Thus the whole set NUMBBK. 7 of dots can be arranged in five groups of six each, just as well as in six groups of five each. The whole number of things contained in six groups of five each, is called six times five. We learn in this way therefore that six times five is the same number as five times six. As before, the remark that we have here made about two particular numbers may be extended to the case of any two numbers whatever. If we take any number of groups of dots, containing all of them the same number of dots, and arrange these as horizontal lines one under the other, then the dots will be arranged not only in lines but in columns ; and the number of dots in every column will obviously be the same as the number of groups, while the number of columns will be equal to the number of dots in each group. Consequently the number of things in a groups of 6 things each is equal to the number of things in b groups of a things each, no matter what the numbers a and 6 are. The number of things in a groups of h things each is called a times & ; and we learn in this way that a times h is equal to 6 times a. The number a times h is denoted by writing the two letters a and 6 together, a coming first ; ^lie^hat we may express our result in the symbolic form dh=-T[)a. Suppose now that we put together seven such com- pound groups arranged in the form of an oblong like that we constructed just now. They cannot now be repre- sented on one sheet of paper, but we may suppose that instead of dots we have little cubes which can be put into an oblong box. On the floor of the box we shall have six rows of five cubes each, or five columns of six cubes each ; and there will be seven such layers, one on •the top of another. Upon every cube therefore which is in the bottom of the box there will be a pile of six 2 8 THE COMMON SENSE OP THE EXACT SCIENCES. cubes, and we shall have altogether five times six such piles. That is to say, vire have five times six groups of seven cubes each, as well as seven groups of five times six cubes each. The whole number of cubes is indepen- dent of the order in which they are counted, and con- sequently we may say that seven times five times six is the same thing as five times six times seven. But it is here very important to notice that when we say seven times five times six, what we mean is that seven layers have been formed, each of which contains five times six things; but when we say five_ times six times seven, we mean that five times six columns have been formed, each of which contains seven things. Here it is clear that in the one case we have first multi- plied the last two numbers, and then multiplied the result by the first mentioned (seven times five times six = seven times thirty), while in the other case it is the first two numbers mentioned that are multiplied together, and then the third multiplied by the result (five times six times seven = thirty times seven). Now it is quite evident that when the box is full of these cubes it may be set upon any side or upon any end ; and in all cases there will be a number of layers of cubes, either 5 or 6 or 7. And whatever is the number of layers of cubes, that wiU also be the number of cubes in each pile. Whether- therefore we take seven layers containing five times six cubes each, or six layers containing seven times five cubes each, or five layers containing six times seven cubes each, it comes to exactly the same thing. We may denote five times six by the symbol 5x6, and then we may write five times six times seven, 5x6x7. But now this form does not tell us whether we are to multiply together 6 and 7 first, and then take 5 NUMBER. 9 times the result, or whetlicr we are to multiply 5 and 6 first, and take that numher of sevens. The distinction between these two operations may" be pointed out by means of parentheses or brackets ; thus, 5 x (6 x 7) means that the 6 and 7 must be first multiplied to- gether and 5 times the result taken, while (5 x 6) x 7 means that we are to multiply 5 and 6 and then take the resulting number of sevens. We may now state two facts that we have learned about multiplication. First, that the brackets make no difference in the result, although they, do make a difference in the pro- cess by which the result is attained; that is to say, 5x(6x7) = (5x6)x7. Secondly, that the product of these three numbers is independent of the order in which they are multi- plied together. The first of these statements is called the associa- tive law of multiplication, and the second the commuta- tive law. Now these remarks that we have made about the result of multiplying together the particular three numbers, 5, 6, and 7, are equally applicable to any three numbers whatever. We may always suppose a box to be made whose height, length, and breadth will hold any three num- bers of cubes. In that case the whole number of cubes will clearly be independent of the position of the box ; but however the box is set down it will contain a certain number of layers, each layer containing a cer- tain number of rows, and each row containing a certain number of cubes. The whole number of cubes in the box will then be the product of these three numbers ; and it will be got at by taking any two of the three 10 THE COMMON SENSE OF THE EXACT SCIENCES. numbers, multiplying them together, and then multi- plying the result by the third number. This property of any three numbers whatever may now be stated symbolically. In the first place it is true that a{hc) = {ab)c; that is, it comes to the same thing whether we multiply the product of the second and third numbers by the first, or the third number by the product of the first and second. In the next place it is true that abc = acb = hca, &c., and we may say that the product of any three numbers is independent of the order and of the mode of group- ing in which the multiplications are performed. We have thus made some similar statements about two numbers and three numbers respectively. This naturally suggests to us that we should inquii-e if cor- responding statements can be made about four or five numbers, and so on. We have arrived at these two statements by con- sidering the whole group of things to be counted as arranged in a layer and in a box respectively. Can we go any further, and so arrange a number of boxes as to exhibit in this way the product of four numbers? It is pretty clear that we cannot. Let us therefore now see if we can find any other sort of reason for believing that what we have seen to be true in the case of three numbers — viz., that the re- sult of multiplying them together is independent of the order of multiplying— is also true of four or more numbers. In the first place we wiU show that it is possible to interchange the order of a pair of these numbers which are next to one another in the process of multiplying, ■without altering the product. NUMBER. 11 Consider, for example, the product of four numbers, abed. We will endeavour to show that this is the same thing as the product acbd. The sjonbol abed means that we are to take c groups of d things and then b groups like the aggregate so formed, and then finally a groups of bed things. Now, by what we have already proved, b groups of cd things come to the same number as c groups of bd things. Consequently, a groups of bed things are the same as a groups of cbd things ; that is to say, abcd= acbd. It will be quite clear that this reasoning will hold no matter how many letters come after d. Suppose, for example, that we have a product of six numbers abcdef. This means that we are to multiply / by e, the result by d, then def by c, and so on. How in this case the product def simply takes the place which the number d had before. And b groups of c times def things come to the same number as c groups of b times de/" things, for this is only the product of three numbers, b, c, and def. Since then this result is the same in whatever order b and c are written, there can be no alteration made by multiplications coming after, that is to say if we have to multiply by ever so many more numbers after multiplying by a. It follows therefore that no matter how many numbers are multi- plied together, we may change the places of any two of them which are close together without altering the product. In the next place let us prove that we may change the places of any two which are not close together. For example, that abedef is the same thing as aeedbf, where b and e have been interchanged. We may do this by first making the e march backwards, changing 12 THE COMMON SEJ^SE OF THE EXACT SCIENCES. places successively with d and c and b, when the product is changed into aebcdf; and then making b march forwards so as to change places successively with c and d, whereby we have now got e into the place of b. Lastly, I say that by such interchanges as these we can produce any alteration in the order that we like. Suppose for example that I want to change abcdef into dcfbea. Here I will first get d to the beginning; I therefore interchange it with a, producing dbcaef. Next, I must get c second ; I do this by interchanging it with b, this gives dcbaef. I must now put / third by interchanging it with b, giving dcfaeb, next put b fourth by interchanging it with a, producing dcfbea. This is the form required. By five such interchanges at most, I can alter the order of six letters in any way I please. It has now been proved that this alter- ation in the order may be produced by successive in- terchanges of two letters which are close together. But these interchanges, as we have before shown, do not alter the product ; consequently the product of six numbers in any order is equal to the product of the same six numbers in any other order ; and it is easy to see how the same process wUl apply to any number of numbers. But is not all this a great deal of trouble for the sake of proving what we might have guessed before- hand ? It is true we might have guessed beforehand that a product was independent of the order and group- ing of its factors ; and we might have done good work by developing the consequences of this guess before we were quite sure that it was true. Many beautiful theorems have been guessed and widely used be- fore they were conclusively proved; there are some even now in that state. But at some time or other the NUMBER. 13 inquiry has to be undertaken, and it always clears up our ideas about tbe nature of the theorem, besides giving us the right to say that it is true. And this is not all ; for in most cases the same mode of proof or of in- vestigation can be applied to other subjects in such a way as to increase our knowledge. This happens with the proof we have just gone through ; but at present, as we have only numbers to deal with, we can only go backwards and not forwards in its application. We have been reasoning about multiplication ; let us see if the same reasoning can be applied to addition. What we have proved amounts to this. A certain result has been got out of certain things by taking them in a definite order ; and it has been shown that if we can interchange any two consecutive things without altering the result, then we may make any change whatever in the order without altering the result. Let us apply this to counting. The process of counting consists in taking certain things in a definite order, and applying them to our fingers one by one ; the result depends on the last finger, and its name is called the number of the things so counted. We learn then that this result will be independent of the order of counting, provided only that it remains unaltered when we interchange any two consecutive things ; that is, provided that two adjacent fingers can be crossed, so that each rests on the object previously under the other, without employing any new fingers or setting free any that are already employed. With this assumption we can prove that the number of any set of things is independent of the order of counting ; a statement which,- as we have seen, is the foundation of the science of number. 14 THE COMMON SENSE OF THE EXACT SCIENCES. § 4, The Distributive Law. There is another law of multiplication which is, if possible, still more important than the two we have already considered. Here is a particular case of it: the number 5 is the sum of 2 and 3, and 4 times 5 is the sum of 4 times 2 and 4 times 3. We can make this visible by an arrangement of dots as follows : — Here we have four rows of five dots each, and each row is divided into two parts, containing respectively two dots and three dots. It is clear that the whole number of dots may be counted in either of two ways ; as four rows of five dots, or as four rows of two dots together with four rows of three dots. By our general principle the result is independent of the order of counting, and therefore 4 X 5 = (4 X 2) + (4 X 3) ; or, if we put in evidence that 5 = 2 + 3, 4 (2 + 3) = (4 X 2) + (4 X 3). The process is clearly applicable to any three num- bers whatever, and not only to the particular numbers 4, 2, 3. We may construct an oblong containing a rows of b + c dots ; and this may be divided by a vertical line into a rows of 6 dots and a rows of c dots. Counted in one way, the whole number of dots is a{b-^c); NDMBER. 15 counted in another way, it is ab + ae. Hence we must always haye a (6 + c) = a6 + ac. This is the first form of the distributive law. Now the result of multiplication is independent of the order of the factors, and therefore a {b + c) = (b + c) a, ab = ba, ac = ca; so that our equation may be written in the form (b + c) a =^ ha + ca. This is called the second form of the distributive law. Using the numbers of our previous example, we say that since 5 is the sum of 2 and 3, 5 times 4 is the sum of 2 times 4 and 3 times 4. This form may be arrived at in- dependently and very simply as follows. We know that 2 things and 3 things make 5 things, whatever the things are ; let each of these things be a group of 4 things ; then 2 fours and 3 fours make 5 fours, or (2x4) + (3x4)=5x4. The rule may now be extended. It is clear that our oblong may be divided by vertical lines into more parts than two, and that the same reasoning will apply. This figure, for example, makes visible the fact that just as 2 and 8 and 4 make 9, so 4 times 2, and 4 times 3, and 4 times 4 make 4 times 9. Or generally — 16 THE COMMON SENSK OP THE EXACT SCIENCES. a (b + c + d) = ah + ac + ad, (b + c + d) a = ba + ca + da; and tlie same reasoning applies to the addition of any number of numbers and tbeir subsequent multiplication. § 5. Ow Powers. When a number is multiplied by itself it is said to be squared. The reason of this is that if we arrange a number of lines of equally distant dots in an oblong, the number of lines being equal to the number of dots in each line, the oblong wUl become a square. If the square of a number is multiplied by the number itself, the number is said to be cubed ; because if we can fill a box with cubes whose height, length, and breadth are all equal to one another, the shape of the box will be itself a cube. If we multiply together four numbers which are all equal, we get what is called the fourth power of any one of them ; thus if we multiply 4 3's we get 81, if we multiply 4 2's we get 16. If we multiply together any number of equal num- bers, we get in the same way a power of one of them which is called its fifth, or sixth, or seventh power, and so on, according to the number of numbers multiplied together. Here is a table of the powers of 2 and 3 : — Index 1 2 3 4 6 6 7 8 Powers of 2 . . . 2 4 8 16 32 64 128 256 » 3 . . . 3 9 27 81 243 729 2187 6561 The number of equal factors multiplied together is called the index, and it is written as a small figure above the line on the right-hand side of the number whose power is thus expressed. To write in shorthand NUMBEE. 17 the statement that if you multiply seven threes together you get 2187, it ia only needful to put down : — 3^ = 2187. It is to be observed that every number is its own first power ; thus 2' = 2, 3' = 3, and in general a* = a. § 6. Square ofa+1. We may illustrate the properties of square numbers by means of a common arithmetical puzzle, in which one person teUs the number another has thought of by means of the result of a round of calculations per- formed with it. Think of a number .... say 3 Square it 9 Add 1 to the original number ... 4 Square that 16 Take the difference of the two squares . 7 This last is always an odd number, and the number thought of is what we may call the less half of it ; viz., it is the half of the even number next below it. Thus, the result being given as 7, we know that the number thought of was the half of 6, or 3. We will now proceed to prove this rule. Suppose that the square of 5 is given us, in the form of twenty- five dots arranged in a square, how are we to form the square of 6 from it? We may add five dots on the right, and then five dots along the bottom, and then one dot extra in the corner. That is, to get the square of 6 from the square of 5, we must add one more than twice 5 to it. Accordingly — 36 = 25 + 10 + 1. 18 THE COMMON SENSE OF THE EXACT SCIENCES. And, conversely, the number 5 is the less half of the difference between its square and the square of 6. e e o o o o e • » e e e • • • • e • • • • • e • • • • • o o e e e o e o The form of this reasoning shows that it holds good for any number -whatever. Having given a square of dots, we can make it into a square having one more dot ia each side by adding a column of dots on the right, a row of dots at the bottom, and one more dot in the corner. That is, we must add one more than twice the number of dots in a side of the original square. If, therefore, this number is given to us, we have only to take one from it and divide by 2, to have the num- ber of dots in the side of the original square. We will now write down this result in shorthand. Let a be the original number ; then a + 1 is the number next above it; and what we want to say is that the square of a + 1, that is {a+l)^, is got from the square of a, which is a% by adding to it one more than twice a, that is 2a + 1. Thus the shorthand expression is (a + 1) = = a2 + 2a + 1. This theorem is a particular case of a more general one. which enables us to find the square of the sum of NUMBER. 19 any two numbers in terms of the squares of the two numbers and their product. We will first illustrate this by means of the square of 5, which is the sum of 2 and 3. • o • mo • e • • • • • • o • • • • • • • e e • • The square of twenty-five dots is here divided into two squares asd two oblongs. The squares are respec- tively the squares of 3 and 2, and each oblong is the product of 3 and 2. In order to make the square of 3 into the square of 3 + 2, we must add two columns on the right, two rows at the bottom, and then the square of 2 in the comer. And in fact, 25 = 9 + 2x6 + 4. § 7. On Powers of a + b. To generalise this, suppose that we have a square with a dots in each side, and we want to increase it to a square with a + h dots in each side. We must add 6 columns on the right, b rows at the bottom, and then the square of b in the corner. But each column and each row contains a dots. Hence what we have to add is twice ab together with b^, or in shorthand : — (a + by = a" + 2ah + bK The theorem we previously arrived at may be got from this by making 6 = 1. 20 THE COMMON SENSE OF THE EXACT SCIENCES. Now this is quite completely and satisfactorily proved ; nevertheless we are going to prove it again in another way. The reason is that we want to extend the proposition still further; we want to find an ex- pression not only for the square of (a + h), but for any other power of it, in terms of the powers and products of powers of a and h. And for this purpose the mode of proof we have hitherto adopted is unsuitable. We might, it is true, find the cube of a + 6 by adding the proper pieces to the cube of a ; but this would be some- what cumbrous, while for higher powers no such repre- sentation can be used. The proof to which we now pro- ceed depends on the distributive law of multiplication. According to this law, in fact, we have (a + b)^ = (a + b) (a + h) = a {a + I) + b(a + b) = aa + ab + ba + bb = a^ + 2ab + 6^ It will be instructive to write out this shorthand at length. The square of the sum of two numbers means that sum multiplied by itself. But this product is the first number multiplied by the sum together with the second number multiplied by the sum. Now the first number multiplied by the sum is the same as the first number multiplied by itself together with the first number multiplied by the second number. And the second number multiplied by the sum is the same as the second number multiplied by the first number to- gether with the second number multiplied by itself. Putting all these together, we find that the square of the sum is equal to the sum of the squares of the two numbers together with twice their product. Two things may be observed on this comparison. First, how very much the shorthand expression gains NUMBER. 21 in clearness from its brevity. Secondly, tliat it is only shorthand for something which is just straightforward common sense and nothing else. We may always depend upon it that algebra, which cannot be translated into good English and sound common sense, is bad algebra. But now let us put this process into a graphical shape which will enable us to extend it. We start with two numbers, a and h, and we are to multiply each of them by a and also by h, and to add all the results. a . + 6 /\ /\ aa ha ab hi Let us put in each case the result of multiplying by a to the left, and the result of multiplying by h to the right, under the number multiplied. The process is then shown in the figure. If we now want to multiply this by a + & again, so as to make (a + 6)^, we must multiply each part of the lower line by a, and also by b, and add all the results, thus : — ab bb y\ y\ y\ aba bba, aab hah abb bib Here we have eight terms in the result. The first and last are a* and b^ respectively. Of the remaining six, three are baa, aba, aab, containing two a's and one b, and therefore each equal to a% ; and three are bba, bob, abb, containing one a and two b's, and therefore each equal to ab^. Thus we have : — (a + 6)» = a^ + da;'b + Sab' + b\ 22 THE COMSION SENSE OF THE EXACT SCIENCES, For exampHtll' = 1331. Here a = 10,1 = 1. and (10 + l)»=10' + 3xl0'' + 3xl0 + l, for it is clear that any power of 1 is 1. We shall carry this process one step further, before making remarks which will enable ns to dispense with it. In this case there are sixteen terms, the first and last being a* and h* respectively. Of the rest, some have three a's and one b, some two a's and two 6's, and some one a and three &'s. There are four of the first kind, since the i may come first, second, thii-d, or fourth; so also there are four of the third kind, for the a occurs in each of the same four places ; the remaining six are of the second kind. Thus we find that, {a + by = a* + 4a3& + 6a^&« + 4a¥ + 1*. We might go on with this process as long as we liked, and we should get continually larger and larger trees. But it is easy to see that the process of classifying and counting the terms in the last line would become very troublesome. Let us then try to save that trouble by making some remarks upon the process. If we go down the tree last figured, from a to abaa, we shall find that the term NUMBEE. 23 aiaa is built up from right to left as we descend. The a that we begin with is the last letter of abda ; then in descending we move to the right, and put another a before it ; then we move to the left and put b before that; lastly we move to the right and put in the first a. From this there are two conclusions to be drawn. First, the terms at the end are all different ; for any divergence in the path by which we descend the tree makes a difference in some letter of the result. Secondly, euery possible arrangement of four letters which are either a's or b's is 'produced. For if any such aiTangement be written down, say abah, we have only to read it backwards, making a mean * turn to the left ' and b ' turn to the right,' and it will indicate the path by which we must descend the tree to find that arrangement at the end. We may put these two remarks into one by saying that every such possible arrangement is produced once and once only. Now the problem before us was to count the number of terms which have a certain number of 6's in them. By the remark just made we have shown that this is the same thing as to count the number of possible arrangements having that number of &'s. Consider for example the terms containing one 6. When there are three letters to each term, the number of possible arrangements is 3, for the b may be first, second, or third, baa, aha, aab. So when there are four letters the number is 4, for the b may be first, second, third, or fourth ; baaa, abaa, aaba, aaab. And generally it is clear that whatever be the number of letters in each term, that is also the number of places in which the b can stand. Or, to state the same thing in shorthand. 24 THE COMMON SEKSE OP THE EXACT SCIENCES. if n be the number of letters, there are n terms con- taining one h. And then, of course, there are n terms containing one a and all the rest Va. And these are the terms which come at the beginning and end of the wth power of a+&; viz. we must have {a + hf = a^+ na"-^b + other terms + waS*^"' + 6", The meaning of this shorthand is that we have n {a + iys multiplied together, and that the result of that multiplying is the sum of several numbers, four of which we have written down. The first is the product of n a's multiplied together, or a" ; the next is n times the product of & by (w— 1) a's, namely, na''~^b. The last but one is n times the product of a by («— 1) 6's, namely, waft"" ' ; and the last is the product of n 5's multiplied together, which is written fe". The problem that remains is to fill up this state- ment by finding what the ' other terms ' are, containing each more than one a and more than one b. § 8. On the Number of Arrangements of a Group of Letters. This problem belongs to a very useful branch of applied arithmetic called the theory of * permutations and combinations,' or of arrangement and selection. The theory tells us how many arrangements may be made with a given set of things, and how many selec- tions can be made from them. One of these questions is made to depend on the other, so that there is an advantage in counting the number of arrangements first. With two letters there are clearly two arrangements, ab and ba. With three letters there are these six : abc, acb, bca, bac, cah, cba, NUMBER. 25 namely, two with a at the beginning, two with 6 at the beginning, and two with c at the beginning ; three times two. It would not be much trouble to write down all the arrangements that can be made with four letters ahcd. But we may count the number of them without taking that trouble ; for if we write d before each of the six arrangements of ahc, we shall have six arrangements of the four letters beginning with d, and these are clearly all the arrangements which can begin with d.- Similarly, there must be six beginning with a, six beginning with h, and six beginning with c ; in all, four times six, or twenty-four. Let us put these results together : With two letters, number of arrangements is two = 2 „ three „ three times two . . = 6 „ four „ four times three times two = 24 Here we have at once a rule suggested. To find the number of arrangements which can he made with a given group of letters, multiply together the numbers two, three, four, &c., up to the number of letters in the group. We have found this rule to be right for two, three, and four letters ; is it right for any number whatever of letters ? We will consider the next case of five letters, and deal with it by a method which is applicable to all cases. Any one of the five letters may be placed first ; there are then five ways of disposing of the first place. For each of these ways there are four ways of disposing of the second place ; namely, any one of the remaining four letters may be put second. This mates five times four ways of disposing of the first two places. For each of these there are three ways of disposing of the third place, for any one of the remaining three letters may 26 THE COMMON SENSE OF THE EXACT SCIENCES, be put third. This makes five times four times three ways of disposing of the first three places. For each of these there are two ways of disposing of the last two places; in all, five times four times three times two, or 120 ways of arranging the five letters. Now this method of counting the arrangements wiU clearly do for any number whatever of letters ; so that our rule must be right for all numbers. We may state it in shorthand thus : the number of arrangements of n letters is Ix2x3x... xm;or putting dots instead of the sign of multiplication, it is 1.2.3 ... w. The 1 which begins is of course not wanted for the multiplication, but it is put in to in- clude the extreme case of there being only one letter, in which case, of course, there is only one arrange- ment. The product 1.2.3...%, or, as we may say, the product of the first n natural numbers, occurs very often in the exact sciences. It has therefore been found convenient to have a special short sign for it, just as a parliamentary reporter has a special sign for ' the remarks which the Honourable Member has thought fit to make.' Different mathematicians, however, have used different symbols for it. The symbol ]n is very much used in England, but it is difficult to print. Some continental writers have used a note of admira- tion, thus,'w !'i Of this it has been truly remarked that it has the air of pretending that you never saw it before. I myself prefer a symbol which has the weighty authority of Gauss, namely a Greek 11 (Pi), which may be taken as short for product if we like, thus, IIw. We may now state that — ni=i, n2=2, n3=6, n4=24, n5=i20, n6=720, and generally that NUMBER. 27 n (« + 1) = (w + 1) Un, for the product of tlie first n+l numbers is equal to the product of the first n numbers multiplied hj n + l. § 9. 0» a Theorem concerning any Power of a + h. We -will now apply this rule to the problem of counting the terms in (a + 6)"; and for clearness' sake, as usual, we will begin with a particular case, namely the case in which n = 5. We know that here there is one term whose factors are all a's, and one whose factors are all 6's ; fiye terms whiclfare the product of four a's by one h, and five which are the product of one a and four 6's. It remains only to count the number of terms made by multiplying three a's by two 6's, which is naturally equal to the number made by multiplying two a's by three b'a. The question is, therefore, how many different arrangements can he made with three a's and two Vs ? Here the three a's are aU alike, and the two 6's are alike. To solve the problem we shall have to think of them as different ; let us therefore replace them for the present by capital letters and small ones. How many different arrangements can be made with three capital letters AB C and two small ones de ? In this question the capital letters are to be con- sidered as equivalent to each other, and the small letters as equivalent to each other; so that the arrange- ment ABGde counts for the ^ame arrangement as GA-Bed. Every arrangement of capitals and smalls is one of a group of 6 x 2 = 12 equivalent arrangements; for the 3 capitals may be arranged among one another in US, = 6 ways, and the 2 smalls may be arranged in 112, = 2 ways. Now it is clear that by 28 THE COMMON SENSE OF THE EXACT SCIENCES. taking all the arrangements in respect of capital and small letters, and then permuting the capitals among themselves and the small letters among themselves, we shall get the whole number of arrangements of the five letters ABGde; namely 115 or 120. But since each arrangement in respect of capitals and smalls is here repeated twelve times, and since 12 goes into 120 ten times exactly, it appears that the number we require is ten. Or the number of arrangements of three a's and two 6's is 115 divided by nS and 112. The arrangements are in fact — bbaaa, babaa, baaba, baaab abbaa, ababa, abaab aabba, aabab aaabb The first line has a 6 at the beginning, and there are four positions for the second b ; the next line has a 6 in the second place, and there are three new positions for the other b, and so on. We might of course have ar- rived at the number of arrangements in this particular case by the far simpler process of direct counting, which we have used as a verification ; but the advantage of our longer process is that it will give us a general formula applicable to all cases whatever. Let us stop to put on record the result just obtained ; viz. we have found that (a + b)« = a^ + Sa'b + 10a^¥ + 10a^¥ + Bab* + &'. Observe that 1 + 5 + 10 + 10 + 5 + 1 = 32, that is, we have accounted for the whole of the 32 terms which would be in the last line of the tree appropriate to this case. We may now go on to the solution of our general problem. Suppose that p is the number of a's and q is NUMBER. 29 the number of h's whicli are multiplied together in a certain term ; we want to find the number of possible arrangements of these p a's and q b's. Let us replace them for the moment hjp capital letters and g small, ones, making p + q letters altogether. Then any ar- rangement of these in respect of capital letters and small ones is one of a group of equivalent arrangements got by permuting the capitals among themselves and the small letters among themselves. Now by per- muting the capital letters we can make lip arrange- ments, and by permuting the small letters Ilg ar- rangements. Hence every arrangement in respect of capitals and smalls is one of a group of Up x Uq equivalent arrangements. Now the whole number of arrangements of the p + q letters is U.{p + q); and, as we have seen, every arrangement in respect of capitals and smalls is here repeated Up x n^ times. Conse- quently the number we are in search of is got by di- viding Il{p + q) by Tip x Tlq. This is written in the form of a fraction, thus : — n (ff + g) Hp.nq' although it is not a fraction, for the denominator always divides the numerator exactly. In fact, it would be absurd to talk about half a quarter of a way of arranging letters. We have arrived then at this result, that the number of ways of arranging p a's and q h's is n (p + g) np . Uq ' This is also (otherwise expressed) the number of ways of dividing p + q places into p of one sort and q of 30 THE COMMON SENSE OF THE EXACT SCIENCES. another ; or again, it is the number of ways of selecting 2» things out ofp + g things. Applying this now to the expression of (a + 6)", we find that each of our other terms is of the form Up.nq where p + q = n; and that we shall get them all by giving to q successively the values 1, 2, 3, &c., and to p the values got by subtracting these from n. Tor example, we shall find that (a + ir = .« + ea^l + ^a^.^ + ^^^ aW ■ ^^ aW + 6ab' + ¥. n2-n4 The calculation of the numbers may be considerably shortened. Thus we have to divide 1.2. 3. 4. 5. 6 by 1.2.3.4; the result is of course 5 . 6. This has to be further divided by 2, so that we finally get 5 . 3 or 15. Similarly, to calculate ns . 03' we have only to divide 4.5.6 by 1 . 2 . 3 or 6, and we get simply 4 . 5 or 20. To write down our expression for (a + h)" we re- quire another piece of shorthand. We have seen that it consists of a number of terms which are all of the form ^"^ aPh\ Up.Uq but which differ from one another in having forp and q different pairs of numbers whose sum is n. Now just NUMBER. 31 as we used the Greek letter n for a product so we use the Greek letter S (Sigma) for a sum. Namely, the sum of all such terms will be written down thus : — Now we may very reasonably include the two extreme terms a" and b" in the general shape of these terms. For suppose we made p = n and 2=0, the corresponding term would be : — Ilw . IIo ' and this wiU be simply cH^ii nO = l and h'>=l. Of course there is no sense in ' the product of the first no numbers ' ; but if we consider the rule n (ra + 1) = (w + 1) Un, which holds good when n is any number, to be also true when n stands for nothing, and consequently w + 1 = 1, it then becomes ni = no, and we have already seen reason to make III mean 1. Next if we say that V means the result of multiplying 1 hjh q times, then &" must mean the result of multi- plying 1 by 6 no times, that is, of not multiplying it at all; and this result is 1. Making then these conventional interpretations, we may say that [a + &)» = 2jjiI?L.^a''&', [p + q= ti], it being understood that p is to take all values from n down to 0, and q all values from up to n. This result is called the Binomial Theorem, and was originally givejj by Sir Isaac Newton. An expression 32 THE COMMON SENSE OF THE EXACT SCIENCES. eontaiaing two terms, like a + b, is sometimes called Unomial ; and the name Binomial Theorem is an abbre- viation for theorem concerning any power of a binomial expression. § 10. 0» Operations which appear to be without Meaning. We have so far considered the operations by which, when two numbers are given, two others can be deter- mined from them./ First, we can add the two numbers together and get their sum. Secondly, we can multiply the two numbers together and get their product. To the questions what is the sum of these two numbers, and wbat is the product of these two numbers, there is always an answer. But we shall now consider questions to which there is not always an answer. Suppose that I ask what number added to 3 will" produce 7. I know, of course, that the answer to this is 4, and the operation of getting 4 is called subtracting 3 from 7, and we denote it by a sign and write it 7-3=4. Bu^ if I ask, what number added to 7 will make 3, althoiigh this question seems good English when ex- pressed in words, yet there is no answer to it ; and if I write down in symbols the expression 3 — 7, 1 am asking a question to which there is no answer. There is then an essential difference between adding and subtracting, for two numbers always have a sum. If I write down the expression 3 + 4, I can use it as meaning something, because I know that there is a number which is denoted by that expression. But if I write down the expression 3 — 7, and then speak of it NUMBEB. 33 as meaning sometliing, I shall be talking nonsense, because I shall have put together symbols the realities corresponding to which will not go together. To the question, what is the result when one number is taken from another, there is only an answer in the case where the second number is greater than the first. In the same way, when I multiply together two numbers I know that there is always a product, and I am therefore free to use such a symbol as 4x5, because I know that there is some number that is denoted by it. But I may now ask a question; I may say, What number is it which, being multiplied by 4, produces 20 ? The answer I know in this case is 5, and the operation by which I get it is called dividing 20 by 4. This is denoted again by a symbol, 20-T-4 = 5. But suppose I say divide 21 by 4. To this there is no answer. There is no number in the sense in which we are at present using the word — that is to say, there is no whole number — which being multiplied by 4 wiU produce 21 : and if I take the expression 21-T-4, and speak of it as meaning something, I shall be talking nonsense, because I shall have put together symbols whose realities will not go txjgether. U^' The things that we have observed here wiU occur again and again in mathematics : for every operation that we can invent amounts' to asking a question, and this question may or may not have an answer according to circumstances. If we write down the symbols for the answer to the question in any of those cases where there is no answer and then speak of them as if they meant something, we shall talk nonsense. But this nonsense is not to be thrown away as useless rubbish. We have learned by 34 THE COMMON SENSE OF THE EXACT SCIENCES. very long and varied experience ttat nothing is more valuable than the nonsense which we get in this way ; only it is to be recognised as nonsense, and by means of that recognition made into sense. We turn the nonsense into sense by giving a new meaning to the words or symbols which shall enable the question to have an answer that previously had no answer. Let us now consider in particular what meaning we can give to our symbols so as to make sense out of the at present nonsensical expression, 3 — 7. § 11. Steps. The operation of adding 3 to 5 is written 5 + 8, and the result is 8. We may here regard the + 3 as a way of stepping from 6 to 8, and the symbol +3 may be read in words, step forward three. In the same way, if we subtract 3 from 5 and get 2, we write the process symbolically 6 — 3 = 2, and the symbol —3 may be regarded as a step from 5 to 2. If the former step was forward this is backward, and w6 may accordingly read — 3 in words, step backwards three. A step is always supposed to be taken from a number which is large enough to make sense of the result. This restriction does not affect steps forward, because from any number we can step forward as far as we like ; but backward a step can only be taken from numbers which are larger than the step itself. . The next thing we have to observe about steps is that when two steps are taken in succession from any number, it does not matter which of them comes first. If the two steps are. taken in the same direction this is clear enough. +3 + 4, meaning step forward 3 and NUMBER. 35 then step forward 4, directs us to step forward by tlxe number which is the sum of the numbers in the two steps ; and in the same way —3— 4 directs us to step backward the sum of 3 and 4, that is 7. If the steps are in opposite directions, as, for example, +3 — 7, we have to step forward 3 and then backward 7, and the result is that we must step backwards 4. But the same result wOTld have been attained if we first stepped backward 7 and then forward 3. The result, in fact, is always a step which is in the direction of the greater of the two steps, and is in magnitude equal to their difference. We thus see that when two steps are taken in suc- cession *they are equivalent to one step, which is inde- pendent of the order in which they are taken. We have now supplied a new meaning for our symbols, which makes sense and not nonsense out of the symbol 3 — 7. The 3 must be taken to mean + 3, that is, step forward 3 ; the — 7 must be taken to mean step backward 7, and the whole expression no longer means take 7 from 3, but add 3 to and then subtract 7 from any number which is large enough to make sense of the result. And accordingly we find that the result of this operation is —4, or, as we may write it, + 3-7 =-4. From this it follows by a mode of proof precisely analogous to that which we used in the case of multi- plication, that any number of steps taken in succession have a resultant which is independent of the order in which they are taken, and we may regard this rule as an extension of the rule already proved for the addition of numbers. A step may be multiplied or taken a given number of times, for example, 2 (—3) =—6; that is to say. 36 THE COMMON SENSE OP THE EXACT SCIENCES. if two backward steps of 3 be possible, they are equiva- lent to a step backwards of 6. In this operation of multiplying a step it is «lear that what we do is to multiply the number whi^Jh is stepped, and to retain the character of the step. On multiplying a step forwards we still hare a step for- wards, and on multiplying a step backwards we still have a step backwards. This multiplying may be regarded as an operation by which we change one step into another. Thus in the example we have just considered the multiplier 2 changes the step backwards 3 into the step backwards 6. But this opei-ation, as we have observed, will only change a step into another of the same kind, and the question naturally presents itself. Is it possible to find an operation which shall change a step into one of a different kind ? Such an operation we should naturally call reversal. , We should say that a step forwards is reversed, when it is made into a step backwards ; and a step backwards is reversed when it is made into a step forwards. If we denote the operation of reversal by the letter r, we can, by combining this with a multiplication, change —3 into +6, a step backwards 3 into a step forwards 6 ; viz. we should have the expression r2( — 3)= + 6. Now the operation, which is performed on one step to change it into another, may be of two kinds : either it keeps a step in the direction which it originally had, or it reverses it. If to make things symmetrical we insert the letter Tt when a step is kept in its original direction, we may write the equation A;2( — 3) = — 6 to express the operation of simply multiplying. Of course it is possible to perform on any given step a succession of- these operations. If I take the KUMBEK. 37 step +4, treble it, and reverse it, I get —12. If I double this and keep it, I get —24, and this may be written, fe2(r3)( + 4) = — 24. But this is equal to rC( + 4), which tells us that the two successive opera- tions which we have performed on this step, trebling and reversing it, doubling and keeping it, are equiva- lent to the single operation of multiplying by 6 and reversing it. It is clear also that whatever step we had taken the two first operations performed successively are always equivalent to the third, and we may thus write the equation fc2(r3) = r6. Suppose however we take another step and treble it and reverse it, and then double it and reverse it again ; we should have the result of multiplying it by six and keeping its direction unchanged. This may be written r2{r3) = k. 6. If we compare the last two formxdae with those which we previously obtained, viz. &2(— 3) = —6 and r2(— 3) = +6, we shall see that the two sets are alike, except that in the one last obtained k and r are written instead of + and — respectively. The two sets however express entirely different things. Thus, taking the second formulae of either set on the one hand, the statement is. Double and reverse the step backward 3, and you have a step forward; 6 ; on the other hand. Treble and reverse and then double and reverse any step whatever, and you have the effect of sextupling and keeping the step.' We shall find that this analogy holds good in general, that is, if we write down the effect of any number of successive operations performed upon a step, there will always be a correspond- ing statement in which this stepping is replaced by a.n operation ; or we may say, any operation which converts one step into another will also convert one operation into 38 THE COMMON SENSE OF THE EXACT SCIENCES. another where the converted operation is a multiplying by the number expressing the step and a keeping or rerersing. according as the step is forward or backward. § 12. Extension of the Meaning of Symbols. We now proceed to do something which must appa- rently introduce the greatest confusion, but which, on the other hand, increases enormously our powers. Having two things which we have so far quite rightly denoted by different symbols, and finding that we arrive at results which are uniform and precisely similar to one another except that in one of them one set of symbols is used, in the other another set, we alter the meaning of our symbols so as to see only one set instead of two. We make the symbols + and — mean for the future what we have here meant by h and r, viz. keep and reverse. We give them these meanings in addition to their former meanings, and leave it to the context to show which is the right meaning in any particular case. Thus, in the equation (—2) (— 3) = + 6 there are two possible meanings ; the — 3 and + 6, may both mean steps, in this case the statement is : Double and reverse the step backwards of 3 and you get the step forward 6. But the —3 and the +6 may also mean not steps but operations, and in this case the meaning is; triple and reverse and then double and reverse any step whatever, and you get the same result as if you had sextupled and kept the step.^' Let us now see what the reason is for saying that these two meanings can always exist together. Let us first of all take the second meaning, and frame a rule for finding the result of any number of successive operations. NUMBER. 39 First, the number which, is the multiplier in the result must clearlj be the product of all the numbers in the successive operations. Next, every pair of reversals cancel one^another, so that, if there is an even number of them, the result must be an operation of retaining. This then is the rule : Multiply together the numbers in the several operations, prefixing to them + if there is an even number of minus or reversing operations, prefixing — if there is an odd number. In the next place, suppose that many successive operations are performed upon a step. The number in the resulting step will clearly be the product of all the numbers in the operations and in the original step. If there is an even number of reversing operations, the resulting step will be of the same kind as the Original one ; if an odd number, of the opposite kind. Now let us suppose that the original step were a step backwards ; then if there is an even number of reversing operations, the resulting step will also be a step backwards. But. in this case the number of (— ) signs, reckoned independently of their meaning, will be odd ; and so the rule coincides with the previous one. If an odd number of reversing operations is per- formed on a negative step, the result is a positive step. But here the whole number of (— ) signs, irrespective of their meaning, is an even number ; and the result again agrees with the previous one. In all cases therefore by using the same symbols to mean either a ' forward ' and a ' backward ' step respectively, or ' keep ' and ' reverse ' respectively, we shall be able to give to every expression two interpreta- tions, and neither of these will ever be untrue. In the process of examiniag this statement we have 40 THE COMMON SENSE OF THE EXACT SCIENCES, shown by the way that the result of any number of successive operations on a step is independent of the order of them. Tor it is always a step whose magnitude is the prodttct of the numbers in the original step and in the operations, and whose character is determined by the number of reversals. § 13. Addition and Multiplication of Operations. We may now go on to find a rule which connects together the multiplication and the addition of steps. If I multiply separately the steps + 3 and — 7 by 4, and then take the resultant of the two steps which I so obtain, I shall get the same thing as if I had first formed the resultant of +3 and —7, and then multi- plied it by -4. In fact, + 12 — 28 = — 16, which is 4(— 4). This is true in general, and it obviously amounts to the original rule that a set of things comes to the same number in whatever order we count them. Only that now some of the counting has to be done backwards and some again forwards. But now, besides adding together steps, we may also in a certain sense add together operations. It seems natural to assume at once that by adding toge- ther + 3 and — 7 regarded as operations, we must needs get the operation —4. It is very important not to assume anything without proof, and still more import- ant not to use words without attaching a definite meaning to them. The meaning is this. If I take any step whatever, treble it without altering its character, and combine the resiUt with the result of multiplying the original step by 7 and reversing it, then I shall get the same result as if I had multiplied the original step by 4 and NUMBER, 41 reversed it. This is perfectly true, and we may see it to be true by, as it were, performing our operations in the form of steps. Suppose I take the step +5, and want to treble it and keep its character unchanged. I can do this by taking three steps of five numbers each in the same direction (viz, the forward direction) as the original step was to be taken. Similarly, if I want to multiply it by — 7, this means that I must take 7 steps of five numbers each in the opposite or backward direc- tion. Then finally, what I have to do is to take three steps forwards and seven steps backwards, each of these steps consisting of five numbers ; and it appears at once that the result is the same as that of taking 4 steps backwards of five numbers each. We have thus a definition of the sum of two operations ; and it appears from the way in which we have arrived at it that this sum is independent of the order of the operations. We may therefore now write the formulse : — a + b = b + a a (b + c) = ab + ac {a + b)c =■ ac + be ab = ba, and consider the letters to signify operations performed upon steps. In virtue of the truth of these laws the whole of that reasoning which we applied to finding a power of the sum of two numbers is applicable to the finding of a power of the sum of two operations. If it did not take too much time and space, we might go through it again, giving to all the symbols their new meanings. It is worth while, perhaps, by way of example, to explain clearly what is meant by the square of the sum of two operations. 42 THE COMMON SENSE OF THE EXACT SCIENCES. We will take for example, +5 and —3. The formula tells us that ( + 5 — 3)= is equal to (-)-6)2 + (_3)2 + 2( + 5)( — 3), This means that if we apply to any step twice over the sum of the operations + 5 and —3, that is to say, if we multiply it by 5 and keep its direction, and combine with this step the result of multiplying the original step by 3 and reversing it, and then apply the same process to the result so obtained, we shall get a step which might also have been arrived at by combining together the following three steps,: — First, the original step twice multiplied by 5. Secondly, the original step twice multiplied by 3 and twice reversed; that is to say, unaltered in direction. Thirdly, twice the result of tripling the original step and reversing it, and then multiplying by 5 and retain- ing the direction. § 14. Division of Operations. We have now seen what is meant by the multipli- cation of operations; let us go on to consider what sort of question is asked by division. Let us take for example the symbolic statement — 3( + 5) = — 15 ; and let us give it in the first place the meaning that to triple and reverse the step forward 5 gives the step backward 15. We may ask two questions upon this statement. First, What operation is it which, being performed on the step forwards 6, will give the step backwards 15 ? The answer, of course, is triple and reverse. Or we may ask this question. What step is that, which, being tripled and reversed, will give the step backward 15? The answer is. Step forwards 5. But we have only one word to describe the process by which we get the answer in these two NUMBER. 43 ' cases. In the first case we say that we divide the step — 15 by the step +5 ; in the second case we say we divide the step —15 by the operation —3. The word divide thus gets two distinct meanings. But it is very important to notice that symbolically the answer is the same in the two cases, although the interpretation to be given to it is different. The step —15 may be got in two ways; by tripling and reversing the forward step + 5, or by quintupling the backward step — 3. In symbols, (-3) ( + 5) = ( + 5) (-3) = -15. Hence the problem. Divide —15 by —3 may mean either of these two questions : What step is that which, being tripled and reversed, gives the step —15? Or, What operation is that which, performed on the step — 3, gives the step —15? The answer to the first question is, the step + 5 ; the answer to the second is the operation of quintupling and retaining direction, that is, the operation + 5. So that although the word divide, as we have said, gets two distinct meanings, yet the two different results of division are expressed by the same symbol. In general- we may say that the problem. Divide the step a by the step h, means. Find the operation (if any) which will convert b into a. But the problem, Divide the step a by the operation 6, means, Mnd the step (if any) which b will convert into a. In both cases, however, the process and the symbolic result are the same. We must divide the number of a by the number of b, and prefix to it + if the signs of a and b are alike, — if they are different. We may also give to our original equation (-3) X ( + 5) = -15 44 THE COMMON SENSE OF THE EXACT SCIENCES. its other meaning, in which both —3 and +5 are ope- rations, and —15 is the operation which is equivalent to performing one of them after the other. In this case the problem, Divide the operation —15 by the operation —3 means, Find the Operation which, being succeeded by the operation —3, will be equivalent to the operation — 15. Or generally, Divide the operation a by the operation h, means, Find the operation which, being succeeded by 6, will be equivalent to a. Now it is worth noticing that the division of step by step and the division of operation by operation, have a certain likeness between them, and a common differ- ence from the division of step by operation. Namely, the result of dividing a by i, or, as we may write it, — , when a and h are both steps or both operations, is an operation which converts h into a. This we may write in shorthand, — . = a. But when a is a step and h an operation, the result of division is a step on which the operation h must be performed to convert it into a; or, in shorthand, J. <* b . - ■= a, The fact that the symbolic result is the same in the two cases may be stated thus : — and in this form we see that it is a case of the commu- tative law. So long, then, as the commutative law is true, there is no occasion for distinguishing symboli- cally between the two meanings. But, as we shall see NUMBER. 45 by-and-by, there is occasion to deal with other kinds of steps and operations in which the commutatiTe law does not hold ; and for these a convenient notation has been suggested by Professor Cayley. Namely, ^ means the operation which makes b into a ; but '— repre- sents that which the operation b will convert into a. So that — fJ . 6 = a, but 6 . L- = a. \b h\ It is however convenient to settle beforehand that when- ever the symbol -j- is used without warning it is to have the first meaning — namely, the operation which makes 6 into a. § 16. General Results of our Extension of Terms, It will be noticed that we have hereby passed from the consideration of mere numbers, with which we began, to the consideration first of steps of addition or subtraction of number from number, and then of operations of multiplying and keeping or multiplying and reversing, performed on these steps ; and that we have greatly widened the meaning of all the words that we have employed. To addition, which originally meant the addition of two numbers, has been given the meaning of a combina- tion of steps to form a resultant step equivalent in effect to taking them in succession. To multiplication, which was originally applied to two numbers only, has been given the meaning of a combination of operations upon steps to form a resultant operation equivalent to their successive performance. 46 THE COMMON SENSE OF THE EXACT SCIENCES. "■^ We have found that the same properties which characterise the addition and multiplication of numbers belong also to the addition and multiplication of steps and of operations. And it was this very fact of the similarity of properties which led us to use our old words in a new sense. We shall find that this same process is carried on in the consideration of those other subjects which lie before us ; but that the precise similarity which we have here observed in the pro- perties of more simple and more complex operations will not in every case hold good ; so that while this gradual extension of the meaning of terms is perhaps the most powerful instrument of research which has yet been used, it is always to be employed with a cau- tion proportionate to its importance. CHAPTER n. SPACE, § 1. Boundaries take wp ifio Room. Geometet is a physical science. It deals with the sizes and shapes and distances of things. Just as we have studied the number of things by making a simple and obvious observation, and then using this over and over again to see where it would bring us ; so we shall study the science of the shapes and distances of things by making one or two very simple and obvious obser- vations, and then using these over and over again, to see what we can get out of them. The observations that we make are :— First, that a thing may be moved about from one place to another without altering its size or shape. Secondly, that it is possible to have things of the same shape but of different sizes. Before we can use these observations to draw any- exact conclusions from them, it is necessary to consider rather more precisely what they mean. Things take up room. A table, for example, takes up a certain part of the room where it is, and there is another part of the room where it is not. The thing makes a difference between these two portions of space. Between these two there is what we caU the surface of the table. We may suppose that the space all round the table 48 THE COMMOIf SENSE OF THE EXACT SCIENCES. is filled with air. The surface of the table is then something just between the air and the wood, which separates them from one another, and which is neither the one nor the other. It is a mistake to suppose that the surface of the table is a very thin piece of wood on the outside of it. We can see that this is a mistake, because any reason which led us to say so, would lead us also to say that the surface was a very thin layer of air close to the table. The surface in fact is common to the wood and to the air, and takes up itself no room whatever;' Part of the surface of the table may be of one colour and part may be of another. On the surface of this sheet of paper there is drawn a round black spot. We call the black part a circle. Fio. 1. It divides the surface into two parts, one where it is and one where it is not. This circle takes up room on the surface, although the surface itself takes up no room in space. We are thus led to consider two different kinds of room ; space- room, in which solid bodies are, and in which they move about ; and surface-room, which may be regarded ' It is certain that however smooth a natural surface may appear to be, it could be magnified to roughness. Hence, in the case of the surface of the table and the air, it would seem probable that there is a layer in which particles of wood and air are mingled. The boundary in this case of air and table would not be what we 'see and feel ' (cf. pT'^^O, nor would it correspond to the surface of the geometer. "We are, I t'fiink, compelled to consider the surface of the geometer as an ' idea or imaginary conception,' drawn from the apparent (not real) boundaries of physical objects, such as the writer is describing. Strongly as I feel the ideal nature of geometrical conceptions in the exact sciences, I have thought it unadvisable to alter the text. The distinction is made by Clifford himself (Essays I pp 306 321).--K.P. J • -tt^- SPACE. 49 from two different points of view. From one point of view it is the boundary between two adjacent portions of space, and takes up no space-room whatever. From the other point of view it is itself also a kind of room which may be taken up by parts of it. These parts in turn have their boundaries. Between the black surface of the circle and the white surface of the paper round it there is a line, the circumference of the circle. This line is neither part of the black nor part of the white, but is between the two. It divides one from the other, and takes up no surface- room at all. The line is not a very thin strip of surface, any more than the surface is a very thin layer of solid. Anything which led us to say that this line, the boundary of the black spot, was a tljin strip of black, would also lead us to say that it was a thin strip of white. We may also divide a line into two parts. If the paper with this black circle upon it were dipped into Fia. 2. water so that part of the black circle were sub- merged, then the line surrounding it would be partly in the water and partly out. The submerged part of the line takes up room on it. It goes a certain part of the way round the circum- ference. Thus we have to consider line-room as well as space-room and surface-room. The line takes up absolutely no room on the surface; it is merely the boundary between two adjacent portions of it. Still less does it take up any room in space. And yet it has a certain room of its own, which may be divided into parts, and taken up or filled by those parts. 50 THE COMMOIf SEIiTSE OF THE EXACT SCIENCES. These parts again have boundaries. Between tte submerged portion of tbe circumference and the other part there are two points, one at each end. These points are neither in the water nor put of it. They are in the surface of the water, just as they are in the sur- face of the paper, and on the boundary of the black spot. Upon this line they take up absolutely no room at all. A point is not a very small length of the line, any more than the line is a very thin strip of surface, It is a division between two parts of the line which are next one another, and it takes up no room on the line at all. The important thing to notice is that we are not here talking of ideas or imaginary conceptions, but only making common-sense observations about matters of every-day experience. The surface of a thing is something that we con- stantly observe. We can see it and feel it, and it is a mere common-sense observation to say that this surface is com- mon to the thing itself and to the space surrounding it. A line on a surface which separates one part of the surface from another is also a matter of every-day experience. It is not an idea got at by supposing a string to become indefinitely thin, but it is a thing given directly by observation as belonging to both por- tions of the surface which it divides, and as being there- fore of absolutely no thickness at all. The same may be said of a point. The point which divides the part of our circumference which is in water from the part which is out of water is an observed thing. It is not an idea got at by supposing a small particle to become smaller and smaller without any limit, but it is the boundary between two adjacent parts of a line, which is the boundary between two adjacent portions of a surface which is the boundary between two adjacent portions pf SPACE. 51 space. A point is a thing which we can see and know, not an abstraction which we build up in our thoughts. When we talk of drawing lines or points on a sheet of paper, we use the language of the draughtsman and not of the geometer. Here is a picture of a cube represented by lines, in the draughtsman's sense. Each of these so-called 'lines' is a black streak of printer's ink; of varying breadth, taking up a certain Fis. 3. amount of room on the paper. By drawing such ' lines ' suflSciently close together, we might entirely cover up as large a patch of paper as we liked. Each of these streaks has a line on each side of it, separating the black surface from the white surface ; these are true geometrical lines, taking up no surface-room whatever. Millions of millions of them might be marked out between the two boundaries of one of our streaks, and between every two of these there would be room for millions more. Still, it is very convenient, in drawing geometrical figures, to represent lines by black streaks. To avoid all possible misunderstanding in this matter, we shall make a convention once for all about the sense in which a black streak is to represent a line. When the streak is vertical, or comes straight down the page, like this I , the line represented by it is its right-hand boun- dary. In all other cases the line shall be the upjptr boundary of the streak. So also in the case of a point. When we try to represent a point by a dot on a sheet of paper, we 52 THE COMMON SENSE OP THE EXACT SCIENCES. make a black patch of irregular shape. The boundary of this black patch is a line. When one point of this boundary is higher than all the other points, that highest point shall be the one represented by the dot. When however several points of the boundary are at the same height, but none higher than these, so that the boundary has a flat piece at the top of it, then the right-hand extremity of this flat piece shall be the point represented by the dot. This determination of the meaning of our figures is of no practical use. We lay it down only that the reader may not fall into the error of taking patches and streaks for geometrical points and lines. § 2. Lengths can he Moved without Change. Let us now consider what is meant by the first of our observations about space, viz., that a thing can be moved about from one place to another without altering its size or shape. First as to the matter of size. We measure the size of a thing by measuring the distanceS^pf various points on it. Fpr example, we should measure the size of a table by measuring the distance from end to end, or the distance across it, or the distance from the top to the bottom. The measurement of distance is only possible when we have something, say a yard measure or a piece of tape, which we can carry about and which does not alter its length while it is carried about. The measure- ment is then effected by holding this thing in the place of the distance to be measured, and observing what part of it coincides with this distance. Two lengths or distances are said to be eq^ual when the same part of the measure will fit both of them. SPACE. 53 Thus we should say that two tables are equally broad, if we marked the breadth of one of them on a piece of tape, and then carried the tape over to the other table and found that its breadth came up to just the same mark. Now the piece of tape, although convenient, is not absolutely necessary to the finding out of this fact. We might have turned one table up and put it on top of the other, and so found out that the two breadths were equal. Or we may say generally that two lengths or distances of any kind are equal, when, one of them being brought up close to the other, they can be made to fit without alteration. But the tape is a thing far more easily carried about than the table, and so in prac- tice we should test the equality of the two breadths by measuring both against the same piece of tape. We find that each of them is equal to the same length of tape ; and we assume that two lengths which are equal to the same length are egual to each other. This is equiva- lent to saying that if our piece of tape be carried round any closed curve and brought back to its original position, it will not have altered in length. How so ? Let us assume that, when not used, our piece of tape is kept stretched out on a board, with one end against a fixed mark on the board. Then we know what is meant by two lengths being equal which are both measured along the tape from that end. Now take three tables. A, B, C, and suppose we have measured and found that the breadth of A is equal to that of B, and the breadth of B is equal to that of C, then we say that the breadth of A is equal to that of C. This means that we have marked off the breadth of A on the tape, and then carried this length of tape to B, and found it fit. Then we have carried the same length from B to C, and found it fit. In saying that the 51 THE COMMON SENSE OP THE EXACT SCIENCES. breadth of C is equal to that of A, we assert that on taking the tape from C to A, whether we go near B or not, it wUl be found to fit the breadth of A. That is, if we take our tape from A to B, then from B to 0, and then back to A, it will still fit A if it did so at first. These considerations lead us to a very singular con- clusion. The reader will probably have observed that we have defined length or distance by means of a measure which can be carried about without cJianging its length. But how then is this property of the measure to be tested? We may carry about a yard measure in the form, of a stick, to test our tapie with; but all we can prove in that way is that the two things are always of the same length when they are in the same place ; not that this length is unaltered. The fact is that everything would go on quite as well if we supposed that things did change in length by mere travelling from place to place, provided that (1) different things changed equally, and (2) anything which was carried about and brought back to its original position filled the same space.* All that is wanted is that two things which fit in one place should also fit in another place, although brought there by different paths ; unless, of course, there are other reasons to the contrary. A piece of tape and a stick which fit one another in London will also fit one another in New York, although the stick may go there across the Atlantic, and the tape via India and the Pacific. Of course the stick may expand from damp and the tape may shrink from dryness; Such non-geometrical cir- cumstances would have to be allowed for. But so far as the geometrical conditions alone are concerned — the ' These remarks refer to the geometrical, and not necessarily to all the physical properties of bodies. — ^K. P. SPACE. 55 mere cariying about and change of place — two things which fit in one place will fit in another. Upon this fact are founded, as we have seen, the notion of length as measured, and the axiom that lengths which are equal to the same length are equal to one another. Js it possible, however, that lengths do really change by mere moving about, without our knowing it? Whoever likes to meditate seriously upon this ques- tion will find that it is wholly devoid of meaning. But the time employed in arriving at that conclusion will not have been altogether thrown away. § 3. The Characteristics of Shape, We have now seen what is meant by saying that a thing can be moved about without altering its size; namely, that any length which fits a certain measure in one position will also fit that measure when both have been moved by any paths to some other position. Let us now inquire what we mean by saying that a thing can be moved about without altering its shape. rirst let us observe that the shape of a thing depends only on its bounding surface, and not at all upon the inside of it. So that we may always speak of the shape of the surface, and we shall mean the same thing as if we spoke of the shape of the thing. Fio. i. Let us observe then some characteristics of the sur- face of things. Here are a cube, a cylinder, and a sphere. 56 THE COMMON SENSE OF THE EXACT SCIENCES. The surface of the cube has six flat sides, with edges and corners. The cylinder has two flat ends and a round surface between them ; the flat ends being divided from the round part by two circular edges. The sphere has a round smooth surface all over. We observe at once a great distinction in shape be- tween smooth parts of the surface, and edges, and corners. An edge being a line on the surface is not any part of it, in the sense of taking up surface room ; still less is a corner, which is a mere point. But still we may divide the points of the surface into those where it is smooth (like all the points of the sphere, the round and flat parts of the cylinder, and the flat sides of the cube), into points on an edge, and into comers. For convenience, let us speak of these points respectively as smooth-points, edge-points, and corner-points. We may also put the edges and corners together, and call them rough- points. Now let us tike the sphere, and put it upon a flat face of the cube (fig. 5). The two bodies will be in con- PlG. 6. tact at one point ; that is to say, a certain point on the surface of the sjhere and a certain point on the surface of the cube are made to coincide with one another and to be the same point. And these are ooth smooth-points. Now we cannot move the sphere ever so little without separ- ating. these points. If we roll it a very little way on the SPACE. 57 face of the cube, we shall find that a different point of the sphere is in contact with a different point of the cube. Iha. 6. And the same thing is true if we place the sphere in contact with a smooth-point on the cylinder (fig. 6). Next let lis put the round part of the cylinder on the flat face of the cube. In this case there will be contact all along a line. At any point of this line, a certain point • on the surface of the cylinder and a certain point on the surface of the cube have been made to coincide with one another and to be the same point. And these are both smooth-points. It is just as true as before, that we cannot move one of these bodies ever so little relatively to the other without separating the 6^^3| if Fig. 7. points of their surfaces which are in contact. If we roll the cylinder a very little way on the face of the cube, we shall find that a different line of the cylinder is in contact with a different line of the cube. All the points of contact are changed. Now put the flat end of the cylinder on the face of the cube. These two surfaces fit throughout and mate but one surface ; we have contact, not (as before) at a point or along a line, but over a surface. Let us fix 58 THE COMMON SENSE OF THE EXACT SCIENCES. our attention upon a particular point on the flat surface of the cylinder and the point on the face of the cube with which it now coincides ; these two being smooth- FiQ. 8. points. We observe again, that it is impossible to move one of these bodies ever so little relatively to the other without separating these two points.^ Here, however, something has happened which will give us further instruction. We have all along sup- fia. 9. posed the flat face of the cylinder to be smaller than the flat face of the cube. When these two are in con- ■ In all these cases (figB. S-8) the relative motion spoken of must be either motion of translation or of tilting; one body might hare a sp,n about a vertical axis without any separation of these two points. The true distinction between the contact of smooth- points and of smooth and rough- points seems to be this : in the former case without separating two points there is only one degree of freedom — namely, spin about an axis normal to the smooth surfaces at the points in question ; in the latter case there are at least two (edge-point or smooth-point) and may be an infinite number of degrees of freedom— namely, spins about two or more axes passing through the rough-point. The reader will understand those terms better after tha chapter on Motion. — K, F, SPACE. 59 tact, let the cylinder stand on the middle of the cube, as in jBg. 8, the circle being wholly enclosed by the square. Then when we tilt the cylinder over we shall get it into the position of fig. 9. We have already observed that in this case no smooth-points which were previously in contact remain in contact. But there are two points which remain in contact ; for in the tilted position a point on the circular edge of the cylinder rests on a point on the face of the cube; and these two points were in contact before. We may tilt the cylinder as much or as little as we like — provided we tilt always in the same direction, not rolling the cylinder on its edge — and these two points will remain in contact. We learn therefore that when an edge-point is in contact with a smooth-point, it may he possihle to move one of the two bodies relatively to the other witlwut separating those two points. The same thing may be observed if we put the round or flat surface of the cylinder against an edge of the cube, or if we put the sphere against an edge of either of the other bodies. Holding either of them fast, we may move the other so as to keep the same two points in contact ; but in order to do this, we must always tilt in the same direction. If, however, we put a corner of the cube in contact with a smooth point of the cylinder, as in fig. 10, we Fia. 10. shall find that we can keep these two points in contact without any restriction on the direction of tilting. We 60 THE COMMOIT SENSE OF THE EXACT SCIENCES. may tilt the cube any way we like, and stiU keep its corner in contact with the smooth-point of the cylinder. When we put two edge-points together, it makes a difference whether the edges are in the same direction at the point of contact or whether they cross one another. In the former case we may be able to keep the same two points in contact by tilting in a particular direction ; in the latter case we may tilt in any direc- tion. So if a comer is in contact with an edge-point there is no restriction on the direction of tilting, and much more if a comer is in contact with a comer. The upshot of all this is, that in a certain sense all surfaces are of the same shape at all smooth-points ; for when we put two smooth-points in contact, the surfaces so fit one another at those points that we cannot move one of them relatively to the other without separating the points.' It is possible for two edges to fit so that we cannot move either of the bodies without separating the points in contact. For this it is necessary that one of them should be re-entrant (that is, should be a depression in the surface, not a projection), as in fig. 11 ; and here we can seethe propriety of saying that the two surfaces are of the same shape at a point where they fit in this way. The body placed in contact with the cube ' See, however, the footnote, p. 58. — K. P. SPACE. 61 is formed by joiuing together two spheres from which pieces have been sliced off. If only very small pieces have been sliced off, tlie re-entrant edge will be very sharp, and it will be impossible to bring the cube-edge into contact with it (fig. 12) ; if nearly half of each Fia. 12. Fio. 13. Fio. 14. sphere has been cut off the re-entrant edge will be wide open, and the cube will rock in it (fig. 13). There is clearly one intermediate form in which the two edges will just fit (fig. 14) ; contact at the edge will be possible, but no rocking. Now in this case, although one edge sticks out and the other is a dint, we may still say that the two surfaces are of the same shape at the edge. For if we suppose our twin-sphere body to be made of wood, its surface is not only sur- face of the wood, but also surface of the surrounding air. And that which is a dint or depression in the wood is at the same time a projection in the air. In just the same way, each of the projecting edges and corners of the cube is at the same time a dint or depression in the air. But the surface belongs to one as much as the other ; it knows nothing of the differ- ence between inside and outside ; elevation and depres- sion are arbitrary terms to it. So in a thin piece of embossed metal, elevation on one side means depression on the other, and vice versa, ; but it is purely arbitrary 62 THE COMMON SENSE OF THE EXACT SCIENCES. wMcli side we consider the right one. (Observe that the thin piece of metal is in no sense a representation of a surface ; it is merely a thin solid whose two surfaces are very nearly of the same shape.) Thus we see that the edge of wood in our cube is of the same shape as the edge of air in the twin-sphere solid ; or, which is the same thing, that the two surfaces are of the same shape at the edge. Now this twin-sphere solid is a very convenient one, because we can so modify it as to make an edge of any shape we like. Hitherto we have supposed the slices cut off to be less than half of the spheres ; let us now fasten together these pieces, and so form a solid with a projecting edge, as in fig. 16. The two solids so formed, one with a re-entrant edge from the larger pieces, the other with a projecting edge from the smaller pieces, will be found always to have their edges of the same shape, or to fit one another at the edge in the sense just explained. (j) Fio. 15. Now suppose that we cut our spheres very nearly in half. (Of course they must always be cut both alike, or the flat faces would not fit together.) Then when w-e join together the larger pieces and the smaUer pieces, we shall form solids with very wide open edges. The projecting edge will be a very slight ridge, and the re-entrant one a very slight depression. If we now go a step further, and cut our spheres actually in half, of course each of the new solids will be again a sphere; and there will be neither ridge nor SPACE. 63 depression ; the surfaces will be smooth all over. But we have arrived at this result by considering a project- (i) («) (iii) (iv) (v) (vi) (vii) OOOO' Fi9. 16. ing edge as gradually widening out until the ridge dis- appears, or by considering a re-entrant edge as gradually widening out until the dint disappears. Or we may suppose the projecting edge to go on widening otft till it becomes smooth, and then to turn into a re-entrant edge. We might represent this process to the eye-by putting into a wheel of life a succession of pictures like that in fig. 16, and then rapidly turning the wheel. We should see the two spheres, at first separate, coalesce into a single solid in (ii) and (iii), then form one sphere as at (iv), then contract into a smaller and smaller lens at (v), (vi), (vii). The important thing to notice is that the single sphere at (iv) is a step in the process ; or, what is the same thing, that a smooth-point is a particular case of an edge-point coming between the projecting and the re- entrant edges. As being this particular case of the edge-point, we say that at all smooth-points the sur- faces are of the same shape. § 4. The Characteristics of Surface Boundaries. Eemarks like these that we have made about solid bodies or portions of space may be made also about 64 THE COMMON SENSE OP THE EXACT SCIENCES. portions of surface. Only we cannot now say that the shape of a piece of surface depends wholly on that of the curve which bounds it. Still the only thing that remains for us to consider is the shape of the boundary, because we have already discussed (so far as we profit- ably can at present) the shape of the included surface. We shall find it useful to restrict ourselves still further, and only consider those boundaries which have no rough points of the surface in them. Thus on the surface of the cube we will only consider portions which are entirely included in one of the plane faces ; on the surface of the cylinder, only portions which are entirely included in one of the flat faces, or in the curved part, or which include one of the flat faces and part of the curved portion. This being so, the characteristics which we have to remark in the boundaries of pieces of surface may be sufficiently studied by means of figures drawn on paper. We may bend the paper to assure ourselves that the same general properties belong to figures on a cylinder, and to make our ideas quite distinct it is worth while to draw some on a sphere or other such surface. In fig. 17 are some patches of surface ; a square, a three-cornered piece, and two overlapping circles. For ■ 1^ 99 Fio. 17. distinctness, the part where the circles overlap is left white, the rest being made black. Attending now specially to the boundary of these patches, we observe that it consists of smooth parts and of corners or angles. Some of these corners project SPACE. 65 and some are re-entrant. The pieces of surface are not solid moveable things like the portions of space we considered before, but we can in a measure imitate our previous experiments by cutting out the figures with a penknife, so as to leave their previous positions marked by the holes. We shall then find, on applying the cut- out pieces to one another, or to the holes, that at all smooth-points the boundaries fit one another in a cer- tain sense. Namely, if we place two smooth-points in contact we cannot roll one figure on the other without separating these points ; whereas if we place a sharp- point (or angle) on a smooth-point we can roll one figure on the other without separating the points. If we attempt to put two angles together without letting the figures overlap, the same things may happen that we found true in the case of the edges of solid bodies. Suppose, for example, that we try to put an angle of the square into one of the re-entrant angles of the figure made by the two overlapping circles. If the re-entrant angle is too sharp, we shall not be able to get it in at all; this is the case of fig. 12. If it is wide enough, the square will be able to rock in it ; this is the case of fig. 13. Between these two there is an intermediate case in which one angle just fits the other ; actual contact takes place, and no rocking is possible. In this case we say that the two angles are of the same shape, or that they are equal to one another. From all this we are led to conclude that shape is a matter of angles, and that identity of shape depends on equality of angle. We dealt with the size of a body by considering a simple case of it, viz. length or distance, and by measuring a sufiicient number of lengths. in dif- ferent directions could find out all that is to be known about the size of a body. It is, indeed, also true that a 66 THE COMMON SENSE OF THE EXACT SCIENCES. knowledge of all the lengths which can be measured in a bodj would carry with it a knowledge of its shape; but still length is not in itself an element of shape. That which does the same for us in regard to shape that length does with regard to size, is angle. In other words, just as we say that two bodies are of the same size if to any line that can be drawn in the one there corresponds an exactly equal line in the other, so we say that two bodies are of the same shape, if to every angle that can be drawn on one of them there corresponds an exactly equal angle on the other. Just as we measured lengths by a stick or a piece of tape so we measure angles with a pair of compasses ; and two angles are said to be equal when they fit the same opening of the compasses. And as before, the statement that a thing can be moved about without alteriiig its shape maybe shown to amount only to this, that two angles which fit in one place will fit also in another, no matter how they have been brought from the one place to the other. § 5. The Plane and the Straight Line, We have now to describe a particular kind of surface and a particular kind of line with which geometry is very much concerned. These are the ;plane surface and the straight line. The plane surface may be defined as one which is of the same shape all over and on both sides. This pro- perty of it is illustrated by the method which is practi- cally used to make such a surface. The method is to take three surfaces and grind them do-wn until any two will fit one another all over. Suppose the three surfaces to be A, B, c ; then, since a will fit b, it follows that the SPACE, 67 space outside A is of the same shape as the space inside B ; and because b will fit o, that the space inside b is of the same shape as the space outside o. It follows there- fore that the space outside a is of the same shape as the space outside c. But since a will fit o when we put them together, the space inside a is of the same shape as the space outside c. But the space outside o was shown to he of the same shape as the space outside a ; consequently the space outside A is of the same shape as the space inside ; and so, if three surfaces are ground together so that each pair of them will fit, each of them becomes a surface which is of the same shape on both sides : that is to say, if we take a body which is partly bounded by a plane surface, we can slide it all over this surface and it will fit everywhere, and we may also turn it round and apply it to the other side of the surface and it will fit there too. This property is sometimes more technically expressed by saying that a plane is a surface which divides space into two congruent regions. A straight line may be defined in a similar way. It is a division between two parts of a plane, which two parts are, so far as the dividing line is concerned, of the same shape ; or we may say what comes to the same effect, that a straight line is a line of the same shape all along and on both sides. A body may have two plane surfaces ; one part of it, that is, may be bounded by one plane and another part by another. If these two plane surfaces have a common edge, this edge, which is called their intersection, is a straight line. We may then, if we like, take as our definition of a straight line that it is the intersection of two planes. It must be understood that when a part of the sur- face of a body is plane, this plane may be conceived as 68 THE COMMON SENSE OF THE EXACT SCIENCES. extending beyond the body in all directions. For instance, the tipper surface of a table is plane and horizontal. Now it is quite an intelligible question to ask about a point which is anywhere in the room -whether it is higher or lower than the surface of the table. The points which are higher will be divided from those which are lower by an imaginary surface which is a continua- tion of the plane surface of the table. So then we are at liberty to speak of the line of intersection of two plane surfaces of a body whether these are adjacent portions of surface or not, and we may in every case suppose them to meet one another and to be prolonged across the edge in which they meet. Leibniz, who. was the first to give these definitions of a plane and of a straight line, gave also another definition of a straight line. If we fix two points of a body, it will not be entirely fixed, but it will be able to turn round. All points of it will then change their position excepting those which are in the straight line joining the two fixed points; and Leibniz accordingly defined a straight line as being the aggregate of those points of a body which are unmoved when it is turned about with two points fixed. If we suppose the body to have a plane face passing through the two fixed points, this definition will fall back on the former one which defines a straight line as the intersection of two planes. It hardly needs any words to prove that the first two definitions of a plane are equivalent ; that is, that two surfaces, each of which is of the same shape all over and on both sides, will have for their intersection a line which is of the same shape all along and on both sides. For if we slide each plane upon itself it will, being of the same shape all over, occupy as a whole the same unchanging position (i.e. wherever there was part of SPACE, 69 tlie planes before there will be part, though a different part, of the planes now), so that their line of inter- section occupies the same position throughout (though the part of the line occupying any particular position is different). The line is therefore of the same shape all along. And in a similar way we can, without changing the position of the planes as a whole, move them so that the right-hand part of each shall become the left-hand part, and the upper part the lower ; and this will amount to changing the line of intersection end for end. But this line is in the same place after the change as before ; and it is therefore of the same shape on both sides. From the first definition we see that two straight lines cannot coincide for a certain distance and then diverge from one another. For since the plane surface is of the same shape on the two sides of a straight line, we may take up the surface on one side and turn it over and it will fit the surface on the other side. If this is true of one of our supposed straight lines, it is quite clear that it cannot at the same time be true of the other; for we must either be bringing over more to fit less, or less to fit more. § 6. Properties of Triangles. We can now reduce to a more precise form our first observation about space, that a body may be moved about in it without altering its size or shape. Let us suppose that oun body has for one of its faces a triangle, that is to say, the portion of a plane bounded by three straight lines. We find that this triangle can be moved into any new position that we like, while the lengths of its sides and its angles remain the same ; or we may 70 THE COMMON SENSE OF THE EXACT SCIENCES. put the statement into the form that when any triangle is once drawn, another triangle of the same size and shape can be drawn in any part of space. From this it wiU follow that if there are two triaCngles which have a side of the one equal to a side of the other, and the angles at the ends of that side in the one equal to the angles at the ends of the equal side in the other, then the two triangles are merely the same triangle in different positions ; that is, they are of the same size and shape. For if we take the first triangle and so far put it into the position of the second that the two equal sides coincide, then because the angles at the ends of the one are respectively equal to those at the ends of the other, the remaining two sides of the first triangle will begin to coincide with the remaining two sides of the second. But we have seen that straight lines cannot begin to coincide and then diverge ; and consequently these sides will coincide throughout and the triangles will entirely coincide. Our second observation, that we may have things which are of the same shape but not of the same size, may also be made more precise by application to the case of triangles. It tells us that any triangle may be magnified or diminished to any degree without altering | its angles, or that if a triangle be drawn, another triangle having the same angles may be drawn of any size in any part of space. From this statement we are able to deduce two very important consequences. One is, that two straight lines cannot intersect in more points than one ; and the other that, if two straight lines can be drawn in the same plane so as not to intersect at all, the angles they make with any third line in their plane which meets them, will be eqiial. SPACE, 71 To prove the first of these, let ab and ao (fig. 18) be two straight lines which meet at a. Draw a third line BO, meeting both of them, and the three lines then form a triangle. If we now make a point p travel along the line ab it must, in virtue of our second observation, be always possible to draw through this point a line which shall meet a o in q so as to make a triangle a p Q of the same shape as abc. But if the line ao were to meet ab in some other point D besides A, then through this point D it would clearly not be possible to draw a line so as to make a triangle at all. It follows then that such a point as D does not exist, and in fact that two straight lines which have once met must go on diverg- ing froni each other and can never meet again.' To prove the second, suppose that the lines a c and BD (fig. 19) are in the same plane, and are such as Fio. 19. never to meet at all (in which case they are called parallel), while the line a b meets them both. If we make a point p travel along b a towards a, and, as it moves, draw through it always a line making the same angle with b a that b d makes with b a, then this • This property might also be deduced from the first definition of a straight line, by the method already used to show that two straight lines cannot coincide for part of their length and then diverge. 72 THE COMMON SENSE OF THE EXACT SCIENCES. moving line can never meet A o until it wholly coincides with it. For if it can, let p Q be such a position of the moving line ; then it is possible to draw through B a line which, with A b and A c, shall form a tri- angle of the same shape as the triangle A p Q. But for this to be the case the line drawn through b must make the same angle with a b that P Q makes with, it, that is, it must be the line b d. And the three lines b d, B A, A cannot form a triangle, for B d and A c never meet. Consequently there can be no such triangle as A p Q, or the moveable line can never meet A c until it entirely coincides with it. But since this line always makes with B A the same angle that b d does, and in one position coincides with A c, it follows that A c makes with b a the same angle that b d does. This is the famous proposition about parallel lines.' The first of these deductions will now show us that if two triangles have an angle of the one equal to an angle of the other and the sides containing these angles respectively equal, they must be equal in all particulars. For if we take up one of the triangles and put it down ' Two straight lines which cut one another form at the point where they cross four angles which are eqnal in pairs. It is often necessary to dis- tinguish between the two different angles which the lines make with one another. This is done by the understanding that a b shall mean the line « (") drawn from A to b, and b a the line drawn from b to a, so that the anglfi between A B an^ D (i) is tlie angle bod, but the angle between ba and c D (ii) is the angle boa. So the angle spoken of above as made by A c with e a is not the angle CAB (which is clearly, in general, unequal to the- angle dba), but the angle c A e, where e is a point in b a produced through a. SPACE. 73 on the other so that these angles coincide and equal sides are on the same side of them, then the con- taining sides will begin to coincide, and cannot there- fore afterwards diverge. But as they are of the same length in the one triangle as they are in the other, the ends of them belonging to the one triangle will rest upon the ends belonging to the other,, so that the re- maining sides of the two triangles will have their ends in common and must therefore coincide altogether, since otherwise two straight lines would meet in more points than one. The one triangle will then exactly cover the other ; that is to say, they are equal in all respects. In the same way we may see that if two triangles have two angles in the one equal to two angles in the other, they are of the same shape. Tor one of them can be magnified or diminished until the side joining these two angles in it becomes of the same length as the side joining the two corresponding angles in the other; and as no alteration is thereby made in the shape of the triangle, it will be enough for us to prove that the new triangle is of the same shape as the other given triangle. But if we now compare these two, we see that they have a pair of corresponding sides which have been made equal, and the angles at the ends of these sides equal also (for they were equal in the original triangles, and have not been altered by the change of size), so that we fall back on a case already considered, in which it was shown that the third angles are equal, and the triangles consequently of the same shape. If we apply these propositions not merely to two different triangles but to the same triangle, we find that if a triangle has two of its sides equal it will have the two angles opposite to them also equal ; and that. 74 THE COMMON SENSE OF THE EXACT SCIENCES. conversely, if it has two angles eqnal it will liave tlie two sides opposite to them also equal ; for in each of these cases the triangle may be turned oyer and made to fit itself. Such a triangle is called isosceles. The theorem about parallel lines which we deduced from our second assumption about space leads very easily to a theorem of especial importance, viz. that the three angles of a triangle are together equal to two right angles. If we draw through A, a corner of the triangle ABO (fig. 20), a line d A e, making with the side A o Fig. 20. the same angle as b o mates with it, this line will, as we have proved, never meet b o, that is, it will be parallel to it. It will consequently make with A b the same angle as b c makes with it,' so that the three angles A b c, b A o, and boa are respectively equal to the angles e a b, b a o, and o A d, and these three make up two right angles. Another statement of this theorem is sometimes of use. If the sides of a triangle be produced, what are called the exterior angles of the triangle are formed. If, for example, the side b o of the triangle ABC (fig. 21) is produced beyond o to d, a c d is an exterior angle of the triangle, while of the interior angles of the triangle A B is said to be adjacent, and cab and A B c to be opposite to this exterior angle. It is clear that as ' The convention mentioned in the last footnote must be lemembered. SPACE. 75 each side of the triangle may be produced in two directions, any triangle has six exterior angles. The other form into which our proposition may be thrown is that either of the exterior angles of a triangle is equal to the sum of the two interior angles opposite to it. For, in the figure, the exterior angle A D, together with A c B, makes two right angles, and it must therefore be equal to the sum of the two angles which also make up two right angles with A b. § 7. Properties of Circles; Belated Circles and Triangles. We may now apply this proposition to prove an im- portant property of the circle, viz. that if we take two fixed points on the circumference of a circle and join them to a third point on the circle, the angle between the joining lines will depend only upon the first two points and not at all upon the third. If, for example, we join the points A, b (fig. 22) to o we shall show that, wherever on the ^circumference may be, the angle A c B is always one-half of A o b ; o being the centre of the circle. Let produced meet the circumference in d. Then since the triangle o A c is isosceles, the angles o a o and OCA are equal, and so for a similar reason are the angles o B o and o c b. But we have just shown that the- exterior angle A D is equal to the sum of the angles o a a and oca; 76 THE COMMON SENSE OF THE EXACT SCIENCES. and since these are equal to one another it must be double of either of them, say of o c A. Similarly the angle b o D is double of o o b, and consequently a o b is double of A c B. In the case of the first figure (i) we have taken the sum of two angles each of which is double of another, and asserted that the sum of the first pair is twice the sum of the second pair ; in the case of the second figure (ii) we have taken the difference of two angles Fig. 22. each of which is double of another, and asserted that the difference of the first pair is twice the difference of the second pair. Since therefore A c b is always half of A o b, wher- ever c may be placed in the upper of the two segments into which the circle is divided by the straight line A b, we see that the magnitude of this angle depends only on the positions of a and b, and not on the position of 0. But now let us consider what will happen if c is in the lower segment of the circle. As before, the tri- angles o A and B c (fig. 23) are isosceles, and the angles d o a and dob are respectively double of c a and o c B. Consequently, the whole angle A o b formed by making o A turn round o into the position o b, so as to pass through the position o d (in the way, that is, SPACE. 77 in which the hands of a clock turn), this whole angle is double of A B. By our previous reasoning the angle A d B, formed by joining A and b to d, is one-half of the angle A o b, which is made by turning o b towards o a as the hands of a clock move. The sum of these two angles, each of which we have denoted by A o b, is a complete re- volution about the point o ; in other words, is foiir Fio. 23. right angles. Hence the sum of the angles A D b, ao b, which are the halves of these, is two right angles. Or we may put the theorem otherwise, and say that the opposite angles of a four-sided figure whose angles lie on the circumference of a circle are together equal to two right angles. We appear therefore to have arrived at two dif- ferent statements according as the point o is in the one or the other of the segments into which the circle is divided by the straight line a b. But these statements are really the same, and it is easy to include them in one proposition. If we produce a o in the last figure to E, the angles a c b and b c e are together equal to two right angles ; and consequently b c e is equal to AD B, This angle b c E is the angle through which c b must be turned in the way the hands of a clock move. 78 THE COMMON SEIfSE OF THE EXACT SCIENCES. so that its direction may coincide witli that of A c. But we may describe in precisely tte same words tlie angle A B in fig. 22, where c was in the upper segment of the circle ; so that we may always put the theorem in these words : — If A and b are fixed points on the circumfer- ence of a circle, and c any other point on it, the angle through which c B must be turned clockwise in order to coincide with c a or AC, whichever happens first, is equal to half the angle through which o b must be turned clockwise in order to coincide with o A. We shall now make use of this to prove another in- teresting proposition. If, three points d, e, f (fig. 24) Fio. 24. be taken on the sides ot a triangle a b c, d being on b o, E on c A, p on A B, then three circles can be drawn passing respectively through A F e, b d p, c ed. These three circles can be shown to meet in the same point o. For let in the first place stand for the intersection of the two circles ape and bfd, then the angles pae and FOE make up two right angles, and so do the angles d o p and d b p. But the three angles at o make four right angles, and the three angles of the triangle ABO make two right angles : and of these six angles two pairs have been shown to make up two right SPACE. 79 angles eacli. Therefore the remainiBg pair, viz. the angles doe and doe, make up two right angles. It follows, that the circle which goes through the points E D will pass through o, that is, the three circles all meet in this point. There is no restriction imposed on the positions of the points d, e, p,' they may be taken either on the sides Fio. 25, of the triangle or on those sides produced, and in par- ticular we may take them to lie on any fourth straight line D E p ; and the theorem may he stated thus : — If any four straight lines be taken (fig. 25), one of which meets the triangle a b o formed by the other three in the points d, e, p, then the circles through the points ' If either of tlie points d, b, p, is taken on a side produced, the proof given above will not apply literally ; but the necessary changes are slight and obvious. 80 THE COMMON SENSE OF THE EXACT SCIENCES. APE, BDF, OBD meet in a point. But there is no reason why we should not take A p E as the triangle formed by three lines, and the fourth line D C B as the line which cuts the sides of this triangle. The propo- sition is equally true in this case, and it follows that the circles through abc, ecd, fed will meet in one point. This must be the same point as before, since two of the circles of this set are the same as two of the previous set; consequently all four circles meet in a point, and we can now state our proposition as follows : Given four straight lines, there can be formed from them four triangles by leaving out each in turn ; the circles which circumscribe these four triangles meet in a point. This proposition is the third of a series. If we take any two straight lines they determine a point, viz. their point of intersection. If we take three straight lines we get three such points of intersection ; and these three determine a circle, viz. the circle circumscribing the triangle formed by the three lines. Four straight lines determine four sets of three lines by leaving out each in turn ; and the four circles belonging to these sets of three meet in a point. In the same way five lines determine five sets of four, and each of these sets of four gives rise, by the proposition just proved, to a point. It has been shown by Miquel, that these five points lie on the same circle. And this series of theorems has been shown ' to be endless. Six straight lines determine six sets of five by leaving them out one by one. Each set of five has, by ' By Prof. Clifford himself in the Oxford,, Cambridge, and Dublin Messenger of Mathematics, vol. v. p. 12i. See his Mathematical Papers, pp. 61-54. SPACE. 81 Miquel's theorem, a circle belonging to it. These six circles meet in the same point, and so on for ever. Any even number (2w) of straight lines determines a point as the intersection of the same number of circles. If we take one line more, this odd number {2n+l) deter- mines as many sets of 2n lines, and to each of these sets belongs a point ; these 2n + l points lie on a circle. § 8. The Conic Sections. The shadow of a circle cast on a flat surface by a luminous point may have three different shapes. These are three curves of great historic interest, and of the utmost importance in geometry and its applications. The lines we have so far treated, viz. the straight line and circle, are special cases of these curves ; and we may naturally at this point investigate a few of the properties of the more general forms. If a circular disc be held in any position so that it is altogether below the flame of a candle, and its shadow be allowed to fall on the table, this shadow will be of an oval form, except in two extreme cases, in one of which it also is a circle, and in the other is a straight line. The former of these cases happens when the disc is held parallel to the table, and the latter when the disc is held edgewise to the candle ; or, in other words, is so placed that the plane in which it lies passes through the luminous point. The oval form which, with these'two exceptions, the shadow presents is called an ellipse (i). The paths pursued by the planets roijnd the sun are of this form. If the circular disc be now held so that its highest point is just on a level with the flame of the candle, the shadow will as before be oval at the end near the candle; 82 THE COMMON SENSE OF THE EXACT SCIENCES. but instead of closing up into another oval end as we move away from the candle, the two sides of it will con- tinue to open out without any limit, tending however to become more and more parallel. This form of the shadow is called a. parabola (ii). It is very nearly the orbit of many comets, and is also nearly represented by the path of a stone thrown up obliquely. If there were- no atmosphere to retard the motion of the stone it would exactly describe a parabola. Fio. 26. If we now hold the circular disc higher up still, so that a horizontal plane at the level of the candle flame divides it into two parts, only one of these parts will cast any shadow at all, and that will be a curve such as is shown in the figure, the two sides of which diverge in quite different directions, and do' not, as in th^case of the parabola, tend to become parallel (iii). But although for physical purposes this curve is the whole of the shadow, yet for geometrical purposes it is not the whole. We may suppose that instead of being a shadow our curve was formed by joining the luminous SPACE. 83 point by straight lines to points round tlie edge of tlie disc, and producing tliese straight lines until they meet the table. This geometrical mode of construction will equally apply to the part of the circle which is above the candle flame, although that does not cast any shadow. If we join these points of the circle to the candle flame, and prolong the joining lines beyond it, they will meet the table on the other side of the candle, and will trace out a curve there which is exactly similar and equal to the physical shadow (iv) . We may call this the anti-shadow or geometrical shadow of the circle. It is found that for geometrical purposes these two branches must be con- sidered as forming only one curve, which is called an hyperbola. There are two straight lines to which the ciu-ve gets nearer and nearer the further away it goes from their point of intersection, but which it never actually meets. For this reason they are called asymp- totes, from a Greek word meaning ' not falling to- gether.' These lines are parallel to the two straight lines which join the candle flame to the two points of the circle which are level with it. We saw some time ago that a surface was formed by the motion of a line. Now if a right line in its motion always passes through one fixed point, the surface which it traces out is called a cone, and the fixed point is called its vertex. And thus the three curves which we have just described are called conic sections, for they may be made by cutting a cone by a plane. In fact, it is in this way that the shadow of the circle is formed ; for if we consider the straight lines which join the candle flame to all parts of the edge of the circle we see that they form a cone whose vertex is the candle flame and whose base is the circle. 84 THE COMMON SENSE OP THE EXACT SCIENCES. "We must suppose tliese lines not to end at the flame but to be prolonged through it, and we shall so get what would commonly be called two cones with their points together, but what in geometry is called one conical surface having two sheets. The section of this conical surfac^/vby the horizontal plane of the table is the shadow of the circle ; the sheet in which the circle lies gives us the ordinary physical shadow, the other sheet (if the plane of section meets it) gives what we have called the geometrical shadow. The consideration of the shadows of curves is a method much used for finding out their properties, for there are certain geometrical properties which are always common to a figure and its shadow. For ex- ample, if we draw on a sheet of glass two curves which cut one another, then the shadows of the two curves cast through the sheet of glass on the table wiU also cut one another. The shadow of a straight line is always a straight line, for all the rays of light from the flame through, various points of a straight line lie in a plane, and this plane meets the plane surface of the table in a straight line which is the shadow. Consequently if any curve is cut by a straight line in a certain number of points, the shadow of the curve will be cut by the shadow of the straight line" in the same number of points. Since a circle is cut by a straight line in two points or in none at all, it follows that any shadow of a circle must be cut by a straight line in two points or in none at all. When a straight line touches a circle the two points of intersection coalesce into one point. We see then that this must also be the case with any shadow of the circle. Again, from a point outside the circle it is pos- sible to draw two lines which touch the circle j so from SPACE. 85 a' point outside either of the three curves which we have just described, it is possible to draw two lines to touch the curve. From a point inside the circle no tangent can be drawn to it, and accordingly no tangent can be drawn to any conic section from a point inside it. This method of deriving the properties of one curve from those of another of which it is the shadow, is called the method ot projection. The particular case of it which is of the greatest use is that in which we suppose the luminous point by which the shadow is cast to be ever so far away. Suppose, for example, that the shadow of a circle held obliquely is cast on the table by a star situated directly overhead, and at an indefinitely great distance. The lines joining the star to all the points of the circle will then be vertical lines, and they will no longer form a cone but a cylinder. One of the chief advantages of this kind of projection is that the shadows of two parallel lines will remain parallel, which is not generally the case in the other kind of projection. The shadow of the circle which we obtain now is always an ellipse j and we are able to find out in this way some very important properties of the curve, the corresponding properties of the circle being for the most part evident at a glance on account of the symmetry of the figure. For instance, let us suppose that the circle whose shadow we are examining is vertical, and let us take a vertical diameter of it, so that the tangents at its ends are horizontal. It will be clear from the symmetry of the figure that all horizontal lines in it ai*e divided into two equal parts by the vertical diameter, or we may say that the diameter of the circle bisects all chords parallel to the tangents at its extremities. When the shadow of this figure is cast by an infinitely distant star (which 86 THE COMMOJT SENSE OP THE EXACT SCIEXCES. we must not now suppose to be directly overhead, for then the shadow would be merely a straight line), the point of bisection of the shadow of any straight lino is the shadow of the middle point of that line, and thus we learn that it is true of the ellipse that any line which joins the points of contact of parallel tangents bisects all chords parallel to those tangents. Such a line is, as in the case of the circle, termed a diameter. Since the shadow of a diameter of the circle is a dia- meter of the ellipse, it follows that all diameters of the ellipse pass through one and the same point, namely, the shadow of the centre of the circle ; this common intersection of diameters is termed the centre also of the ellipse. Again, a horizontal diameter in the circle just con- sidered will bisect all vertical chords, and thus we see that if one diameter bisects all chords parallel to a second, the second will bisect all chords parallel to the first. The method of projection tells us that this is also true of the ellipse. Such diameters are called conjugate diameters, but they are no longer a,t right angles in the ellipse as they were in the case of the circle. Since the shadow of a circle which is cast in this way by an infinitely distant point is always an ellipse, we cannot use the same method in order to obtain the properties of the liyperbola. But it is found by other methods that these same statements are true of the hyperbola which we have just seen to be true of the ellipse. There is however this great difference be- tween the two curves. The centre of the ellipse is inside it, but the centre of the hyperbola is outside it. Also aU lines drawn through the centre of the ellipse meet the curve in two points, but it is only certain SPACE. 87 lines through the centre of the hyperbola which meet the curve at all. Of any two conjugate diameters of the hyperbola one meets the curve and the other does not. But it still remains true that each of them bisects all chords parallel to the other. § 9. On Surfaces of the Second Order. We began with the consideration of the siDiplest. kind of line and the simplest kind of surface, the straight line and the plane ; and we have since found out some of the properties of four different curved lines — the circle, the ellipse, the parabola, and the hyperbola. Let lis now consider some curved surfaces ; and first, the surface analogous to the circle. This surface is the sphere. It is defined, as a circle is, by the property that all its points are at the same distance from the centre. Perhaps the most important question to be asked about a surface is. What are the shapes of the curved lines in which it is met by other surfaces, especially in the case when these other surfaces are planes ? Now a plane which cuts a sphere cuts it, as can easily be shown, in a circle. This circle, as we move the plane further and further away from the centre of the sphere, will get smaller and smaller, and will finally contract into a point. In this case the plane is said to touch the sphere; and we notice a very obvious but important fact, that the sphere then lies entirely on one side of the plane. If the plane be moved still further away from the centre it will not meet the sphere at all. Again, if we take a point outside the sphere we can draw a number of planes to pass through it and touch the sphere, and all the points in which they touch it lie on 88 THE COMMOK' SENSE OF THE EXA.CT SCIENCES. a circle. Also a cone can be drawn whose vertex is the point, and which touches the sphere all round the circle in which these planes touch it. This is called the tangent-cone of the point. It is clear that from a point inside the sphere no tangent-cone can be drawn. Similar properties belong also to certain other sur- faces which resemble the sphere in the fact that they are met by a straight line in two points at most ; such surfaces are on this account called of the second order. Just as we may suppose an ellipse to be got from a circle by pulling it out in one direction, so we may get a spheroid from a sphere either by pulling it out so as to make a thing like an egg, or by squeezing it so as to make a thing like an orange. Each of these forms is symmetrical about one diameter, but not about all. A figure like an orange, for example, or like the earth, has a diameter through its poles less than any diameter in the plane of its equator, but all diameters in its equator are equal. Again, a spheroid like an egg has all the diameters through its equator equal to one another, but the diameter through its poles is longer than any other diameter. If we now take an orange or an egg and make its equator into an ellipse instead of a circle, say by pull- ing, out the equator of the orange or squeezing the equator of the egg, so that the surface has now three diameters at right angles all unequal to one another, we obtain what is called an ellipsoid. This surface plays the same part in the geometry of surfaces that the ellipse does in the geometry of curves. Just as every plane which cuts a sphere cuts it in a circle, so every plane which cuts an ellipsoid cuts it in an ellipse. It is indeed possible to cut an ellipsoid by a plane so that the section shall be a circle, but this must be regarded SPACE. 89 as a particular kind of ellipse, viz. an ellipse with, two equal axes. Again, just as was the case with the sphere, we can draw a set of planes through an exter- nal point all of which touch the ellipsoid. Their points of contact lie on a certain ellipse, and a cone can be drawn which has the external point for its vertex and touches the ellipsoid all round this ellipse. The ellip- soid resembles a sphere in this respect also, that when FlO. 27. it is touched by a plane it lies wholly on one side of that plane. There are also surfaces which bear to the hyperbola and the parabola relations somewhat similar to those borne to the oircle by the sphere, and to the ellipse by the ellipsoid. We will now consider one of them, a surface with many singular properties. Let A B c D be a figure of card-board having four equal sides, and let it be half cut through all along b d. 90 THE COMMON SENSE OF THE EXACT SCIENCES. SO tliat the triangles A b D, c b B can turn about the line B D. Then let holes be made along the four sides of it at equal distances, and let these holes be joined by threads of silk parallel to the sides. If now the figure be bent about the line b d and the silks are pulled tight it will present an appearance like that in fig. 27, resembling a saddle, or the top of a mountain pass. This surface is composed entirely of straight lines, and there are two sets of these straight Knes ; one set which was originally parallel to A b, and the other set which was originally parallel to A d. A section of the figure through a c and the middle point of b D will be a parabola with its concave side turned upwards. A section through b d and the middle point of A c will be another parabola with its concave side turned downwards, the common vertex of these pai'abolas being the summit of the pass. The tangent plane at this point will cut the surface in two straight lines, while part of the surface will be above the tangent plane and part below it. We may regard this tangent plane as a horizontal plane at the top of a mountain pass. If we travel over the pass, we come up on one side to the level of the plane and then go down on the other. But if we go down from a mountain on the right and go up the mountain on the left, we shall always be above the horizontal plane. A section by a horizontal plane a little above this tangent plane will be a hyperbola whose asymptotes will be parallel to the straight lines in which the tangent plane meets the surface. A section by a horizontal plane a little below will also be a hyperbola with its asymptotes parallel to these lines, but it will be situated in the other pair of angles formed by these asymptotes. If SPACE. 91 we suppose the cutting plane to move downwards from a position above the tangent plane (remaining always horizontal), then we shall see the two branches of the first hyperbola approach one another and get sharper and sharper until they meet and become simply two crossing straight lines. These lines will then have their comers rounded off and wUl be divided in the other direction and open out into the second hyper- bola. This leads us to suppose that a pair of intersecting straight lines is only a particular ease of a hyperbola, and that we may consider the hyperbola as derived from the two crossing straight lines by dividing them at their point of intersection and rounding off the comers. § 10. How to form Curves of the Third and Higher Orders. The method of the preceding paragraph may be ex- tended so as to discover the forms of new curves by putting known curves together. By a mode of expres- sion which sounds paradoxical, yet is found convenient, a straight line is called a curve of the first order, because it can be met by another straight line in only one point ; but two straight lines taken together are called a curve of the second order, because they can be met by a straight line in two points. The circle, and its shadows, the ellipse, parabola, and hyperbola, are also called curves of the second order, because they can be met by a straight line in two points, but not in more than two points; and we see that by this process of rounding off the corners and the method of projection we can derive all these curves of the second order from a pair of straight lines. 92 THE COMMON SENSE OF THE EXACT SCIENCES. A similar process enables us to draw curves of tte third order. An ellipse and a straight line taken to- gether form a curve of the third order. If now we round off the corners at both the points where they meet we obtain (fig. 28) a curve consisting of an oval and a sinuous portion called a ' snake.' Now just as when we move a plane which cuts a sphere away from the centre, the curve of intersection shrinks up into a o Via. 28. (i.) Full loop and snake. (iiO Shiunk loop and snake. (iii.) Tlie loop ha/„, and so the fact we have just stated may be written thus : — Now let us assume that the four quantities, a, h, c, d, are proportionals ; that is, that the ratios '/„ and */„ are equal to one another. It follows then that the ratios "/„ and "/^ are equal to one another. This proposition may be otherwise stated in this form ; that if a, h, c, d are proportionals, then a,M, h, d will also be proportionals : provided always that this latter statement has any meaning, for it is quite possible that it should have no meaning at all. Suppose, for in- stance, that a and b are two lengths, c and d two intervals 116 THE COMMON SENSE OF THE EXACT SCIENCES. of time, then we understand what is meant by the ratio of 6 to a, and the ratio of d to c, and these ratios may very well be equal to one another ; but there is no such thing as a ratio of c to a, or of units. We thus convert our rectangle o t into one p', of which one side, o p, contains p and the other, s, s units. Now let us apply to this rectangle the stretch -parallel to the side o s (as the figure is drawn s ^. denotes a sqiieeze). We must divide o s into s equal s parts and take r such parts, or -we must measure a length E along o s equal to r units. Thus this second stretch converts the rectangle op' into a rectangle ok', of which the side op contains p and the side OE contains r units of length, or into a rectangle containing f r square units. Hence the two stretches •?- and - applied in succession to the rectangle o T con- g s vert it into the rectangle oe'. Now this may be written symbolically thus : — •^ X - . rectangle o T = rectangle E = p r unit-rectangles. Now unit-rectangle may obviously be obtained from the rectangle o T by squeezing it first in the ratio - in the direction of o Q, ajid then in the ratio - in the di- rection o e. Now this is simply saying that o t contains 118 THE COMMON SENSE OP THE EXACT SCIENCES. g s unit-rectangles. Hence tlie operation Ex- applied to unit-rectangle must produce — of the result, of its application to the rectangle o t. That is : — P X ~ . unit-rectangle = — .jpr unit-rectangle, or, in our notation, = ^— . unit-rectangle. Hence we may say that ^ x —operating upon unity q s is equal to the operation denoted by ?~, or to multi- plying unity hj pr and then dividing the result by q s. This equivalence is termed the multiplication of frac- tions. A special case of the multiplication of fractions arises when s equals r. We then have — q r qr But the operation - denotes that we are to divide unity into r equal parts, and then take r of them ; in other words, we perform a mdl operation on unity. The symbol of operation may therefore be omitted, and we read — p _pr q~q?' This result is then expressed in words as follows : Given a fraction, we do not alter its value by multiply- ing the numerator and denominator by equal quanti- ties. From this last result we can easily interpret the operation 2 s For, by the preceding paragraph — QUANTITY. 119 p ^ r Hence — z=^, and r = «r. V , r ps , qr ■i- 4- _ ^ -^^ + -5^ • 9 s qs qa Or, to apply first the operation ^ to unity and then to add to this the result of the operation - is the same s thing as dividing unity into qs parts, taking ps ot those parts, and then adding to them q r more of the like parts. But this is the same thing as to take at once ps + qr oi those parts. Thus we may write — p , r _ ps + qr q s 2® ' This result is termed the addition of fractions. The reader will find no difficulty in interpreting addition graphically by a succession of stretches and squeezes of the unit-rectangle. We term division the operation by which we reverse the result of multiplication. Hence when we ask the ^ meaning of dividing by the fraction £ we put the question : What is the operation which, following oni the operation ^, just reverses its effect ? ^ 2 Now, 1x1=^1 x''-=PI. s q q s 2 * Suppose we take r = q, s = p. 120 THE COMMON SENSE OF THE EXACT SCIENCES. Then £x^=£i; or, to multiply unity by ^, and then by -2, is to perform the operation of dividing unity into qp parts and then taking p q oi them, or to leave unity unaltered. Hence the stretch 2. completely reverses the stretch ^ ; P .2 it is, in fact, a squeeze -which just counteracts the preceding stretch. Thus multiplying by 2 must be an operation equivalent to dividing by i-. Or, to divide by ^ is the same thing as to multiply by ^. This result is termed the division of fractions. § 8. Of Areas ; Shear. Hitherto we have been concerned with stretching or squeezing the sides of a rectangle. These opera^ tions alter its area, but leave it still of rectangular shape. We shall now describe an operation which changes its angles, but leaves its area unaltered. F H .Jy^""l. --":^ ^ ■€:-- « ^ a 1 \. r Fio. 3 1 6. i Let A B D be a rectangle, and let A b e p be a parallelogram (or a four-sided figure whose opposite sides are equal), having the same side, A B, as the rectangle, but having the opposite side, ep (equal to ab, and QUANTITY. 121 therefore to o d), somewhere in the same line as c d. Then, since d is equal to e p, the points e and f are equally distant from o and d respectively, and it follows that the triangles b o e and A d p are equal. Hence if Jhe triangle bob were cut off the parallelogram along B c and placed in the position a d p, we should have converted the parallelogram into the rectangle without changing its area. Thus the area of the parallelogram is equal to that of the rectangle. Now the area of the rectangle is the product of the numerical quantity which represents the length of A d into that quantity which represents the length of a e. a b is termed the base of the parallelogram, and A D, the perpendicular dis- tance between its base and the opposite side E p, is termed its height. The area of the parallelogram is then briefly said to be ' the product of its base into its height.' Suppose c D and a b were rigid rods capable of slid- ing along the parallel lines c d and a b. Let us imagine them connected by a rectangular elastic membrane, A b D ; then as the rods were moved along a h and c d the membrane would change its shape. It would, how- ever, always remain a parallelogram with a constant base and height ; hence its area would be unchanged. Let the rod a b be held fixed in position, and the rod c D pushed along c ^ to the position e p. Then any line, G H, in the membrane parallel and equal to a b will be moved parallel to itself into the position i j, and will not change its length. The distance through which c has moved is c E, and the distance through which ct has moved is G i. Since the triangles c b e and G b i have their sides parallel they are similar, and we have the ratio of c e to G i the same as that of b c to b g ; or, when the rectangle a b c d is converted into the 122 THE COMMO]!f SENSE OP THE EXACT SCIENCES. parallelogram a b e p, any line parallel to A B remains nnehanged in length, and is moved parallel to itself through a distance proportional to its distance from a b. Such a transformation of figure is termed a shear, and we may consider either our rectangle as being sheared into the parallelogram or the latter as being sheared into the former. Thus the area of a parallelogram is equal to that of a rectangle into which it may be sheared. The same process which converts the parallelogram a B E F into the rectangle a b o d will convert the tri- angle ABE, the half of the former, into the triangle FiQ. 37. ABC, the half of the latter. Hence we may shear any triangle into a right-angled triangle, and this will not alter its area. Thus the area of any triangle is half the area of the rectangle on the same base, and with height equal to the perpendicular upon the base from the opposite angle. This height is also termed the altitude, or height of the triangle, and we then briefly say ^ The area of a triangle is half the product of its hase into its altitude. A succession of sjiears will enable us to reduce any figure bounded by straight lines to a triangle of equal area, and thus to determine the area the figure encloses by finally shearing this triangle into a right-angled QUANTITY. 123 triangle. For example, let a b o d e be a portion of the boundary of the figure. Suppose a o joined ; then shear the triangle abc so that its yertex b falls at b' on D produced. The area a b' o is equal to the area ABC. Hence we may take a b' d e for the boundary of our figure instead of A B c D B ; that is, we have reduced the number of sides in our figure by one. By a suc- cession of shears, therefore, we can reduce any figure bounded by straight lines to a triangle, and so find its area. § 9. (y Circles and their Areas. One of the first areas bounded by a curved line which suggests itself is that of a sector of a circle, or the FiQ. 38. portion of a circle intercepted by two radii and the arc of the circumference between their extremities. Before we can consider the area of this sector it will be necessary to deduce some of the chief properties of the complete circle. Let us take a circle of unit radius and suppose straight lines drawn at the extre- mities of two diameters ab and o d at right angles ; then the circle will appear as if drawn inside a square (see fig. 39). The sides of this square will be each 2 and its area 4. Now suppose the figure composed of circle and square first to receive a stretch such that every line 124 THE COMMON' SENSE OF THE EXACT SCIENCES, parallel to the diameter a B is extended in the ratio of a : 1, and then another stretch such that every line parallel to c d is again extended in the ratio of a.: 1. Then it is obvious that we shall have stretched the square of the first figure into a second square whose sides will now he equal to 2 a. i. M yj IK l^' cf K' Pia. 39. It remains to be shown that we have stretched the first circle into another circle. Let o p be any radius and p M, PN perpendiculars on the diameters a e, c d. As a result of the first stretch the equal lengths o m and N p are extended into the equal lengths o' m' and h' p', which are such that ^^, = '^-I- = 1. Similarly M K p a ^ as a result of the second stretch m p and o n, which remained unaltered during the first stretch, are con- verted into m'p' and o'lf'; so that ?-i?^. = ^2. = I o' n' m' p' a During this second stretch o'm:' and n' p' remain un- altered. Thus as the total outcome of the two stretches we find that the triangle o p n has been changed into the triangle o' p' n'. Now these two triangles are of the same shape by what was said on p. 106, for the angles at N and n' are equal, being both right angles, and we have seen that — QUANTITY. N P 1 M 125 n' p' a m' Thus it follows that the third side o P must he to the third side o' p' in the ratio of 1 to a ; or, since o P is of unit length, o' p' must be equal to the constant quantity a. Further, since the angles PON, p' o' n' are equal, o' p' is parallel to P. Hence the circle of unit radius has been stretched into a circle of radius a. In fact, the two equal stretches in directions at right angles, which we have given to the first figure, have performed just the same operation upon it, as if we had placed it under a magnifying glass which enlarged it uniformly, and to such a degree that every line in it was magnified in the ratio of a to 1. It follows from this that the circumference of the second circle must be to that of the first as a is to 1. Or, the circumferences of circles are as their radii. Again, if the arc p q is stretched into the arc p' q' — ;that is, if o' p', o' q' are respectively parallel to o p, o Q — then the arc p' q' is to the arc p Q in the ratio of the radii of the two circles. Since the arcs p Q, p' q' are equal to any other arcs which subtend the same angles at the centres of their respective circles, we state generally that the arcs of two circles which subtend equal angles at their respective centres are in the ratio of the corre- sponding radii. Since the second figure is an uniformly magnified image of the first, every element of area in the first has been magnified at the same uniform rate in the second. Now the square in the first figure contains four units of area, and in the second figure it contains 4 a^ units of area. Hence every element of area in the first figure has been magnified in the second in the ratio of a" to 1. Thus the area of the circle in the first figure 126 THE COMMON SENSE OF THE EXACT SCIENCES, must be to the area of the circle in the second figure as 1 is to a^. Or : The areas of circles are as the squares of their radii. It is usual to represent the area of a circle of unit radius by the quantity tt ; thus the area of a circle of radius a will be represented by the quantity tt a\ If, after stretching A b to a' b' in the ratio of a to 1, we had stretched or squeezed o d to o' d' in the ratio of i to 1, where b is some quantity different from a, our square would have become a rectangle, with sides equal to 2 a and 2 b respectively. It may be shown that we Fig. 40. should have distorted our circle into the shape of that shadow of a circle which we have termed an ellipse. Furthermore, elements of area have now been stretched in the ratio of the product of a and 6 to 1 ; or, the area of the ellipse is to the area of the circle of unit radius as a 6 is to 1 : whence it follows that the area of the ellipse is represented by tt a b, where a and b are its greatest and least radii respectively. We shall now endeavour to connect the area of a circle of unit radius, which we have written tt, with the number of linear units in its circumference. Let us QUANTITY. 127 take a number of points uniformly distributed round the circumference of a circle, a b o d B F. Join them in succession to each other and to o, the centre of the circle, and draw the lines perpendicular to these radii (or the tangents) at A B D E p ; then we shall hare constructed two perfectly symmetrical figures, one of which is said to be inscribed, the other circumscribed to the circle. Now the areas of these two figures differ by the sum of such triangles as A a b, and the area of the circle is obviously greater than the area of the inscribed and less than the area of the circumscribed figure. Thus the area of the circle must differ from that of the in- scribed figure by something less than the sum of all the little triangles a a b, b /8 c, &c. Now from symmetry all these little triangles are equal, and their areas are therefore equal to one half the product of their heights, or a n, into their bases, or such quantities as a b. Hence the sum of their areas is equal to one half of the product oi an into the sum of the sides of the inscribed figure. Now the sum of the sides of the inscribed figure is never greater than the circumference of the circle. If we tate, therefore, a great number of points uniformly distributed round the circumference of "our circle, a and 7 128 THE COMMON SENSE OF THE EXACT SCIENCES. B may be brouglit as close as we please, and the nearer we bring a to b, the smaller becomes a n. Hence, by taking a sufficient number of points, we can make the sum of the triangles a a b, b /S c, &c. as small as we please, or the areas of the inscribed and circumscribed figures, together with the area of the circle which Kes between them, can be made to differ by less than any assignable quantity. In the limit then we may say that by taking an indefinite number of points we can make these areas equal. Now the area of the inscribed figure is the sum of the areas of all such triangles as A B, and the area of the triangle A o b is equal to half the product of its height o n into its base A b ; or if we write for the ' perimeter,' or sum of all the sides A B, BO, &c. the quantity p, the area of the inscribed figure will equal \ p x on. Again if f' be the sum of the sides 0/3,^87, &c. of the circumscribed figure, its area = \ p' x o b. Since the triangles a b, o b w are of the same shape, being right-angled and again equi-angled at 0, we have the ratio of b w to a b, or of their doubles A b to a j3, the same as that of w to o b. But p is obviously to p' in the same ratio as a b to a ;S ; hence p is to j?' as w to o B. By taking a sufficient number of points we can make o w as nearly equal to b as we please ; thus we can make p as nearly equal to p', and therefore either of them as nearly equal to the. circumference of the circle (which lies between them),* as we please. Hence in the limit p will equal the circumference of the circle, and n its radius, and we may state that the area^ of the inscribed and circumscribed figures, which approach nearer and nearer to the area of the circle as we in- crease the number of their sides, become ultimately • In the case of the circle the reader will recognise this intuitively. QUANTITY. 129 equal to eacli other and to half the product of the cir- cumference of the circle into its radius. This must there- fore be the area of the circle. Hence we have the fol- lowing equality : — The area of a circle of radius a equals one half its circumference x a. But it equals also Tra' ; whence it follows that the circumference of a circle equals tt . 2 a. We may express this result in two different ways : — (i) The ratio of the circumference of a circle to its diameter (2 a) is a constant quantity tt. (ii) The number of linear units (2 7r) in the cir- cumference of a circle of unit-radius is twice the number of units of area (tt) contained by that circum- ference. The value of tt, the ratio of the circumference of a circle to its diameter, is found to be a quantity which, like the ratio of the diagonal of a square to its side (see p. 103), cannot be expressed accurately by numbers ; its approximate value is 3*14159. We have now no difficulty in finding the area of the sector of a circle, for if we double the arc of a sector we obviously double its area ; if we treble it, we treble its area ; shortly, if we take any multiple of it, we take the same multiple of its area. Hence it follows by § 6, that two sectors are to each other in the ratio of their arcs, or a sector must be to the whole circle in the ratio of its arc to the whole circum- ference. If we represent by s the area of a sector of a circle of which the arc contains s units of length and the radius a units, we may write this relation^ symboli- cally — s _ s irw^ 2 TT a * 130 THE COMMON SENSE OF THE EXACT SCIENCES. Thus we deduce s = i s x a ; or, The area of a sector is half the product of the length of its are into its radius. § 10. Of the Area of Sectors of Curves. The knowledge of the area of a sector of a circle enables us to find as accurately as we please the area of a sector whose arc is any curve whatever. Let the " arc p Q be divided into a number of smaller arcs p a, A b, BC, CD, DQ. We shall suppose that pa subtends the greatest angle at o of all these arcs. Further we shall consider only the case where the line op diminishes continuously if p be made to pass along the arc from p to Q. If this be not the case, the sector qop can always be split up into smaller sectors, of which it shall be true that a line drawn from the point o to the arc con- tinuously diminishes from one side of the sector to the other, and then for the area of each of these sectors the following investigation will hold. With o as centre de- scribe a circle of radius p to meet o a produced in p'; with the same centre and radius OA describe a circle to meet QUANTITY 131 OB in a' and OP in a; similarly circles with radius ob to meet oa in 6 and oo in b', with radius oc to meet ob in c and D in o', with radius o d to meet o c in c? and o Q in d', and finally with radius OQ to meet od in e, OA in/, and OP in q'. Then the area of the sector obviously lies between the areas of the figure bounded by op, od' and the broken line pp'aa'bb'oo'dd', and of the figure bounded by oa, oq and the broken line axbBccdDeQ,. Hence it differs from either of them by less than their difference or by less than the sum of the areas p'a, a'&, b'c, o'd, D'e. Now since the angle at pop' is greater than any of the other sectorial angles at o, the sum of all these areas must be less than that of the figure p p'/q', and the area of this figure can be made as small as we please by making the angle a o p sufficiently small. This can be achieved by taking a sufficient number of points like A,B,c,D, &c. We are thus able to find a series of circular sectors, the sum of whose areas differs by as small a quantity as we please from the area of the sector POQ; in other words, we reduce the problem of finding the area of any figure bounded by a curved line to the problem already solved of finding the area of a sector of a circle. The difficulties which then arise are purely those of adding together a very great number of quantities ; for, it may be necessary to take a very great number of points such as abcd . . . in order to approach with sufficient accuracy to the mag- nitude of the area poq. § 11. Extension of the Conception of Area. Let ABCD be a closed curve or loop, and o a point inside it. Then if a point p move round the perimeter of the loop, the line op is said to trace out the area of 132 THE COMMON SENSE OF THE EXACT SCIENCES. the loop ABCD. By this is meant that successive posi- tions of the line op, pair and pair, form together with the intervening elements of arc elementary sectors, the sum of the areas of which can, by taking the successive Pig. 43. positions sufiiciently close, be made to differ as little as we please from the area bounded by the loop. Now suppose the point o to be taken outside the loop ABCD, and let us endeavour to find the area then Fig. 44. traced out by the line op joining o to a point p which moves round the loop. Let OB and od be the extreme positions of the line OP to the left and to the right as p moves round the loop abod; then as p moves along QUANTITY. 133 the portion of the loop dab, op moves counter-clock- wise from right to left and traces out the area boimded by the arc dab and the lines od and ob. Further, as p moves along the portion of the loop bod, op moves clockwise from left to right and traces out the area doubly shaded in our figure, or the area bounded by the arc bod and the lines ob and od. It is the differ- ence of these two areas which is the area of the loop ABCD. If, then, we were to consider the latter area OBCDO as negative, the line op would still trace out the area of the loop abcd as p moves round its perimeter. Now the characteristic difference in the method of de- scribing the areas odabo and obcdo is, that in the former case o p moves eounter-clochwise round o, in the latter case it moves clockwise. Hence if we make a con- vention that areas traced out by op when it is moving counter-clockwise shall be considered positive, but areas traced out by o p when it is moving clockwise shall be considered negative, then wherever o may be inside or outside the loop, the line o p will trace out its area pro- vided p move completely round its circumference. But it must here be noted that p may describe the loop in two different methods, either going round it counter-clockwise in the order of points abcd, or clockwise in the order of points A D c b. In the former case, according to our convention, the greater area D A B o is positive, in the latter it is negative. Hence we arrive at the conception that an area may have a sign ; it will be considered positive or negative accord- ing as its perimeter is supposed traced out by a point moving counter-clockwise or clockwise. This extended conception of area, as having not only magnitude but sense, is of fundamental importance, not only in many 134 THE COMMON SEKSE OP THE EXACT SCIENCES. branches of the exact sciences, but also for its many practical applications.' Let a perpendicular o N be erected at o (which is, as we have seen, any point in the plane of the loop) to the plane of the loop, and let the length o n be taken along it containing as many units of length as there are units of area in the loop A b c d. Then o n will represent the area of the loop in magnitude; it will also represent it in sense, if we agree that o it shall always be measured in such a direction from o, that to a person standing with his feet at o and head at N the point p shall always appear to move counter-clockwise. Thus, for a positive area, N will be above the plane ; for a negative area, in the opposite direction or below* the plane. We are now able to represent any number of areas by segments of straight lines or steps per- pendicular to their planes. The sum of any number of areas lying in the same plane will then be obtained by adding algebraically all the lines which represent these areas. When the areas do not all lie in one plane the representative lines will not all be parallel. In this case there are two methods of adding areas. We may want to know the total amount of area, as, for example, when we wish to find the cost of painting or gilding a many-sided solid. In this case we add all the repre- sentative lines without regard to their direction. In many other cases, however, we wish to find some quantity so related to the sides of a solid that it can only be found by treating the lines which represent their areas as directed magnitudes. Such cases, for example, arise in the discussion of the shadows cast by ' As in calculating the cost of levelling and embanking, in the indicator diagram, &c. It was first introduced by Mobius. QUANTITY. 135 the sun or of the pressure of gases upon the sides of a containing vessel, &c. A method of combining directed magnitudes will be fully discussed in the following chapter. The conception of areas as directed magnitudes is due to Hayward. § 12. On the Area of a Closed Tangle. Hitherto we have supposed the areas we have talked about to be bounded by a simple loop. It is easy, however, to determine the area of a combination of loops. Thus consider the figure of eight in fig. 45 which has two loops : if we go round it continuously in the direction indicated by the arrow-heads, one of these loops will have a positive, the other a negative area, and therefore the total area will be their difference, or zei'o if they be equal. When a closed curve, like a figure of eight, cuts itself it is termed a tangle, and the points where it cuts itself are called hnots. Thus a figure of eight is a tangle of one knot. In tracing out the area of a closed curve by means of a line drawn from a fixed point to a point moving round the curve, the area may vary according to the direction and the route by which we suppose the curve to be described. If, however, we suppose the curve to be sketched out by the moving point, then its area will be perfectly definite for that particular description of its perimeter. We shall now show how the most complex tangle may be split up into simple loops and its whole area determined from the areas of the simple loops. We shall suppose arrow-heads to denote the direction in which the perimeter is to be taken. Consider either of the accompanying figures. The moving line o p will trace out exactly the same area if we suppose it 136 THE COMMON SENSE OF THE EXACT SCIENCES. not to cross at the knot A but first to trace out the loop A and then to trace out the loop a b, in both cases going round these two loops in the direction Fig. 45. indicated by the arrow-heads. We are thus able in all cases to convert one line cutting itself in a knot into two lines, each bounding &, separate loop, which just touch at the point indicated by the former knot. This dissolution of knots may be suggested to the reader by leaving a vacant space where the boundaries of the loops really meet. The two knots in the fol- lowing figure are shown dissolved in this fashion : — Fig. 46. The reader will now find no difficulty in separating the most complex tangle into simple loops. The posi- tive or negative character of the areas of these loops QTTAIJTITy. 137 will be sufficiently indicated by the arrow-heads on their perimeters. We append an example : — Fib. 47. In this case the tangle reduces to a negative loop a, and to a large positive loop 6, within which are two other positive loops c and d, the former of which con- EiG. 48. tains a fifth small positive loop e. The area of the entire tangle then equals 6 + c + d + e— a. The space marked s in the first figure will be seen from the second to be no part of the area of the tangle at all. 138 THE COMMON SENSE OF THE EXACT SCIENCES. § 13. On the Volumes of Space-Figures. Let us consider first the space- figure bounded by three pairs of parallel planes mutually at right angles. Such a space-figure is technically termed a * rectangular parallelepiped,' but might perhaps be more shortly described as a ' right six-face.' We may first observe that when one edge of such a right six-face is lengthened or shortened in any ratio, the other non- parallel edges being kept of a fixed length, the volume Ji'iG. 49. will be increased in precisely the same ratio. Hence, in order to make any right six-face out of a cube we have only to give the cube three stretches (or it may be squeezes), parallel respectively to its three sets of parallel edges. Let o a, o b, o o be the three edges of the cube which meet in a corner o. Let o a be stretched to o a', so that the ratio of o a' to A is represented by a ; then if the figure is to remain right all lines parallel to o a will be stretched in the same ratio. The figure has now become a six-face whose section perpendicular to o a' only is a square. Now stretch o B to o b', so that the ratio o b' to b be represented by I, and let all lines parallel to o b be QUANTITY. 139 increased in the same ratio ; the figure is now a right six-face, only one set of edges of which are equal to the edge of the original square. Finally stretch o c to o c', so that and all lines parallel to it are increased in the ratio of o o' to o c, which we will represent by c. By a process consisting of three stretches we have thus converted our original cube into a right six-face. If the cube had been of unit-volume, the volume of our six-edge would obviously be a 6 c, and we may show as in the case of a rectangle (see p. 115) that ahc = cla = hac,&c.; or the order of multiplying together three ratios is indifferent. If we term the face a' o' of our Fig. 50. right six-face its lase and o b' its height, ac will repre- sent the area of its base, and b its height, or the volume of a right six-face is equal to the product of its base into its height. , Let us now suppose a right six-face gadcebpg to receive a shear, or the face b e p g to be moved in its own plane in such fashion that its sides remain parallel to their old positions, and b and e move respectively along B p and e g. If b' e' g' p' be the new position of the face b e g p, it is easy to see that the two wedge- shaped figures B B e' b' and F G g' f' a d are exactly equal; this follows from the equality of their corre- sponding faces. Hence the volume of the sheared 140 THE COMMON SENSE OF THE EXACT SCIENCES. figure must be equal to the volume of tlie right six-face. Now let us suppose in addition that the face b' e' g' f' is again moved in its own plane into the position b" e" g" p", so that b' and b' move along b' e' and p' g' respectively. Then the slant wedge-shaped figures b' b" p" p' a and e' e" g" p' d c will again be equal, and the volume of the six-face b"e"g"p"adco obtained by this second shear will be equal to the volume of the figure obtained by the first shear, and therefore to the volume of the right six-face. But by means of two shears we can move the face b e g p to any position in its plane, b" b" g" p", in which its sides remain parallel to their former position. Hence the volume of a six- face will remain unchanged if, one of its faces, CODA, remaining fixed, the opposite face, b e g p, be moved anywhere parallel to itself in its own plane. We thus find that the volume of a six-face formed by three pairs of parallel planes is equal to the product of the area of one of its faces and the perpendicular distance between that face and its parallel. For this is the volume of the right six-face into which it may be sheared ; and, as we have seen, shear does not alter volume. The knowledge thus gained of the volume of a six- face bounded by three pairs of parallel faces, or of a so-called parallelepiped, enables us to find the volume of an oblique cylinder. A right cylinder is the figure generated by any area moving parallel to itself in such wise that any point p moves along a line p p' a,t right angles to the area. The volume of a right cylinder is the product of its height p p' and the generating area. For we may suppose that volume to be the sum of a number of elementary right six-faces whose bases, as at p, may be taken so small that they will ultimately QUANTITY. 141 completely fill the area A c b d, and whose heights are all equal to P p'. We obtain an oblique cylinder from the above right cylinder by moving the face a' o' b' d' parallel to itself anywhere in its own plane. But such a motion will only shear the elementary right six-faces, such as p p', and so not change their volume. Hence the volume of an oblique cylinder is equal to the product of its base, and the perpendicular distance between its faces. § 14. On the Measurement of Angles. Hitherto we have been concerned with quantities of area and quantities of volume ; we must now turn to quantities of angle. In our chapter on Space (p. 66) we have noted one method of measuring angles ; but that was a merely relative method, and did not lead us to fix upon an absolute unit. We might, in fact, have taken any opening of the compasses for unit angle, and determined the magnitude of any other angle by its ratio to this angle. But there is an absolute unit 142 THE COMMON SENSE OF THE EXACT SCIENCES. ■wliich naturally suggests itself in our measurement of angles, and one wliich we must consider here, as we shall frequently have to make use of it in our chapter on Position. Let A B be any angle, and let a circle of radius a be described about o as centre to meet the sides of this f lO. 52. angle in A and B. Then if we were to double the angle A B, we should double the arc A B ; if we were to treble it, we should treble the arc ; shortly, if we were to take any multiple of the angle, we should take the same multiple of the arc. We may thus state that angles at the centre of a circle vary as the arcs on which they stand. Hence if and 0' be two angles, which are subtended by arcs s and s' respectively, the ratio of 8 to 6' will be the same as that of s to s'. Now suppose 6' to represent four right angles ; then s' will be the entire circumference, or, in our previous notation, 2 tt a. We have thus — e ^ s four right angles 2 ir a' Now it is extremely convenient to choose a unit angle which shall be independent of the circle upon which we measure our arcs. We should obtain such an independent unit if we took the arc subtended by it QUANTITY. 143 equal to the radius of the circle or if we took s = a. In this case our unit equals - — of four right angles, = — of two right angles, = '636 of a right angle TT approximately. Thus we see that the angle subtended at the centre of any circle by an arc equal to the radius is a constant fraction of a right angle. If this angle be chosen as the unit, we deduce from the proportion is to ^ as s is to /, that must be to unity as s is to the radius a ; or : — s = a^. Thus, if we choose the above angle as our unit of angle, the measure of any other angle will be the ratio of the arc it subtends from the centre to the radius ; but we have seen (p. 125) that the arcs subtended from the centre in different circles by equal angles are in the ratio of the radii of the respective circles. Hence the above measurement of angle is independent of the radius of the circle upon which we base our measurement. This is the primary property of the so- •called circular measurement of angles, and it is this which renders it of such great value. The circular measure of any angle is thus the ratio of the arc it subtends from the centre of any circle to the radius of the circle. It follows that the circular mea- sure of four right angles Ls the ratio of the whole circum- ference to the radius, or equals -^*; that is, equals 2 IT. The circular measure of two right angles will then be tt, of one right angle ^, of three right O angles -— , and so on. 144 THE COMMON SE.VSE OP THE EXACT SCIEKCES. §15. On Fractional Poivers. Before we leave tlie subject of quantity it will he necessary to refer once more to the subject of powers which, we touched upon in our chapter on Number (p. 16). We there used a" as a symbol signifyinf^ the result of multiplying a by itself n times. From this defini- tion we easily deduce the following identity : — a" X a^ X a" X a" = a.'' + P + « + ', For the left hand side denotes that we are fii'st to multiply a by itself n times, and then multiply this by a", or a multiplied by itself p times, and so on. Hence we may write the left hand side — {ax ax ax a . . tow factors) X {ax ax ax a . . to p factors) X {axaxaxa . . to q factors) X {axaxaxa . . to r factors). But this is obviously equal to (a x a x a x a x ... to n+p-^q + r factors), or to a'' + '' + « + '. If b be such a quantity that b''=a,b is termed an nth root of a, and this is written symbolically i = a /a. Thus, since 8 = 2', 2 is a 3rd, or cube root of 8. Or, again, since 243 = 3^ 3 is termed a 5th root of 243. Now we have seen at the conclusion of our first chapter that we can often learn a very great deal by extending the meaning of our terms. Let us now see if we cannot extend the meaning of the symbol a". Does it cease to have a meaning when w is a fraction or negative? Obviously we cannot multiply a quantity by itself a fractional number of times, nor can we do QUANTITY. 145 SO a negative number of times. Hence the old mean- ing of a", where n is a. positive integer, becomes sheer nonsense when we try to adapt it to the case of n being fractional or negative. Is then ct" in this latter case meaningless? In an instance like this we are thrown back upon the results of our definition, and we endeavour to give to our symbol snch a meaning that it will satisfy these results. Now the fundamental result of our theory of integer powers is that — a^ + p + i + r+... = o" X a^ X a" -x. or X ... This will obviously be true however many quantities, '"'ilPi 5fj *■> "we take. Now let us suppose we wish to inter- - I pret o"' where — is a fraction. We begin by as- suming it satisfies the above relation, and in order to arrive at its meaning we suppose that n = p = q = r= . . . = —, and that there are m such quantities. Then n'rp+€[-\-r = mx— =1; m L L L and we find a' = a™ x a™ x a"' x ... to m factors = («™) . Thus a^ must be such a quantity that, multiplied by itself m times, it equals a\ But we have defined above (p. 144) an mth root of a} to be such a quantity that, multiplied m times by itself, it equals aK Hence we say that a™ is equal to an mth root of a' ; or, as it is written for shortness, — a'" m / I 146 THE COMMON SENSE OF THE EXACT SCIENCES. We have thus found a meaning for a" when w is a frac- tion from the fundamental theorem of powers. We can with equal ease obtain from the same theorem an iutelligible meaning for a" when w is a negative quantity. We have a" x af = a" """ ^. Now let us assume p = — w in order to interpret a - *. We find a" x a ~ " = a," - « = a« = 1 (by p. 31). Or dividing by a", a-" = ~; a"- that is to say, a - " is the quantity which, multi- T^riied by a", gives a product equal to unity. The former quantity is termed the inverse of the latter, or we may say that a - " is the inverse of a". For example, what is the inverse of 4 ? Obviously 4 must be multiplied by \ in order that the product may be unity. Hence 4 ~ ' is equal to \. Or, again, since 4 = 2^, we may say that 2 ~ ^ is the inverse of 4, or 2^. The whole subject of powers — integer, fractional, and negative — is termed the Theory of Indices, and is of no small importance in the mathematical investiga- tion of symbolic quantity. Its discussion would, how- ever, lead us too far beyond our present limits. It has been slightly considered here in order that the reader may grasp that portion of the following chapter in which fractional powers are made use of. CHAPTER IV. POSITION. § 1. All Position is Relative. The reader can hardly fail to remember instances wlien he has been accosted by a stranger with some such question as : 'Can you tell me where the 'George ' Inn lies? ' — 'How shall I get to the cathedral ? ' — 'Where is the London Eoad ? ' The answer to the question, however it may be expressed, can be summed up in the one word — There. The answer points out the position of the building or street which is sought. Practically the there is conveyed in some such phrase as the follow- ing: 'Tou must keep straight on and take the first turning to the right, then the second to the left, and you will find the ' George ' two hundred yards down the street.' Let us examine somewhat closely such a question and answer. ' Where is the ' George ' ? ' We may ex- pand this into : ' How shall I get from here ' (the point at which the question is asked) ' to the ' George ' ? ' This is obviously the real meaning of the query. If the stranger were told that the ' George ' lies three hundred paces from the Town Hall down the High Street, the information would be valueless to the questioner unless he were acquainted with the position of the Town Hall or at least of the High Street. Equally idle 148 THE COMMON SENSE OF THE EXACT SCIENCES. would be the reply: 'The 'G-eorge' lies just past tlie forty-second milestone on the London Eoad,' sup- posing him ignorant of the whereabouts of the London Eoad. Yet both these statements are in a certain sense answers to the question : ' Where is the ' George ' ? ' They would be the true method of pointing out the there, if the question had been asked in sight of the Town Hall or upon the London Eoad. We see, then, that the query. Where ? admits of an infinite number of answers according to the infinite number of posi- tions — or possible heres — of the questioner. The where always supposes a definite here, from which the desired position is to be determined. The reader will at once recognise that to ast, ' Where is the ' George ' ? ' without meaning, ' Where is it with regard to some other place ? ' is a question which no more admits of an answer than this one : ' How shall I get from the * George ' to anywhere ? ' meaning to nowhere in particular. This leads us to our first general statement with regard to position. We can only describe the where of a place or object by describing how we can get at it from some other known place or object. We determine its where relative to a here. This is shortly expressed by saying that : All position is relative. Just as the ' George ' has only position relative to the other buildings in the town, or the town itself relative to other towns, so a body in space has only position relative to other bodies in space. To speak of the position of the earth in space is meaningless unless we are thinking at the same time of the Sun or of Jupiter, or of a star — that is, of some one or other of the celestial bodies. This result is sometimes POSITION. 149 described as the ' sameness of space.' By this we only mean that in space itself there is nothing perceptible to the senses which can determine position.' Space is, as it -were, a blank map into which we put our objects ; it is the coexistence of objects in this map which enables us at any instant to distinguish one object from another. This process of distinguishing, wMch supposes at least two objects to be distinguished, is really determining a this and a that, a here and a there ; it involves the conception of relativity of position. § 2. Position may be Determined by Directed Steps, Let us turn from the question: 'Where is the ' George ' ? ' to the answer : ' You must keep straight on and take the first turning to the right, then the second to the left, and you will find the * George ' 200 yards down the street.' The instruction ' to keep straight on ' means to keep in the street wherein the question has been asked, and in a direction (' straight on ') suggested by the previous motion of the questioner, or by a wave of the hand from the questioned. Assuming for our present purpose that the streets are not curved, this amounts to : Keep a certain direction. How far? This is answered by the second instruction : Take the first turning on the right. More accurately we might say, if the first turning to the right were 160 yards distant : Keep this direction for 150 yards. Let this be represented in our figure by the step A B, where A is the position at which the question is asked. At b the questioner is to turn to the right and, according to the third instruction, he is to pass the first turning to the left at c and take the second at d. ' We shall return to this point later. 150 THE COMMON SEITSB OF THE EXACT SCIENCES. More accurately we might state the distance B D to be, say, 1 80 yards. Then we could combine our second and third instructions by saying : From B go 180 yards in a certain direction, namely, B d. To determine exactly what this direction b D is with regard to the first direction A B, we might use the following method. If the stranger did not change his direction at b, but went straight on for 180 yards, he would come to a point d'. Hence if we measured the angle d'bd between the street in which the question was asked and the first turning to the right, / a Fig. 68. we should know the direction of b D and the position of D exactly. It would be determined by rotating bd' about b through the meastired angle d'b d. If we adopt the same convention for the measurement of positive angles as we adopted for positive areas on p. 133, the angle d'b d is the angle greater than two right angles through which B d' must be rotated counter-clockwise in order to take it to the position b d. Let us term this angle d'b d for shortness )8, then we may invent a new symbol {/S} to denote the operation : Turn the direction you are going in through an angle yS counter-clockwise. POSITION. 151 If we use tHe symbol ■7r/2 to denote an angle equal to a right angle, we have the following symbolic instructions : { } = Keep straight on. { •7r/2 } = Turn at right angles to the left. { TT } = Turn right round and go back. {37r/2} = Turn at right angles to the right. Thus for a turning from a b to the left the angle of our symbolic operation will be less, for a turning from AB to the right greater, than two right angles. If the directed person had gone to d' instead of to D, he would have walked 150 yards to b and then 180 yards to d' ; he would thus have walked A b + b d', or 150 yards + 180 yards. In order to denote that he is not to continue straight on at b we introduce the opera- tor of turning, namely {^}, before the 180 yards, and read 150 + {/S} 180 as the instruction : Go 150 yards along some direction a b, and then, turning your direc- tion through an angle /3 counter-clockwise, go 180 yards along this new direction. We are now able to complete the symbolic expression of our instructions for finding the ' George.' The fourth instruction runs : Take a turning at d to the left and go 200 yards along the direction thus de- termined^ Let dg' represent 200 yards measured from D along b d produced, then we are to revolve p g' through a certain angle g'd g counter-clockwise, till it takes up the position d g. Then g will be the position of the ' George.' Let the angle g'd g be represented by y. Our final instruction may be then expressed symbolically by {y} 200. Hence our total instruction may be written symboli- cally — 150 + {/3}180 + (71200, where the units are yards. 8 152 THE COMMON SENSE OF THE EXACT SCIENCES. But we have not yet quite freed this symbolic in- struction from any suggestion of direction as determined by streets ; the first 150 yards are still to be taken along the street in which the question is asked. "We can get rid of this street by supposing its direction determined by the angle which a clock-hand must revolve through counter-clockwise, to reach that direction, starting from some other fixed or chosen direction. For example, suppose the stranger to have a compass with him, and at A let AN be the direction of its needle. Then we might fix the position of the street A b by describing it as a direction so many degrees east of north, or still to preserve our counter-clockwise method of reckoning angles, we might determine it by the angle a which the needle would have to describe through west and south to reach the position a b. We should then in- terpret the notation {a} 150: Walk 150 yards along a direction making an angle a with north measured through west. Our answer expressed symbolically is now entirely cleared of any conception of streets. For, {a} 150 + {/3}180 + {7} 200 is a definite instruction as to how to get from A to G quite independent of any local characteristics. It ex- presses the position of g with regard to a in a purely geometrical fashion, or by a series of directed steps. Expanded into ordinary English our symbols read : From a point a in a plane, take a step a b of 150 units in a direction making an angle a with a fixed direction, from B take a step b d of 180 units making an angle j8 with A B, and finally from D take a step D g of 200 units making an angle y with b d. All the angles are to be measured counter-clockwise in the fashion we have described above. posiTioif. 153 § 3. The Addition of Directed Steps or Vectors. If we now compare our figure with the symbolical instruction {a} 150 + {y8}180 + {7)200, we see that {a} 150 represents the step a b, when that step is considered to have not merely magnitude but also direction. Similarly b d and D a represent more than linear expressions for number — they are also directed steps. We shall then be at liberty to replace our symbolically expressed instruction W150 + {y8}180 + (7)200 by the geometrical equivalent AB + BD + DG, provided we understand by the segments A B, b d, d G- and the symbol + something quite different to out Fio. 54. former conceptions. We give a new and extended meaning to our quantity and to our addition. AB + BD + DG no longer directs us to add the number of units in b d to that in a b and to the sum of these the number in d a, but it bids us take a step a b in a certain direction, then a step b d from the finish of the former step in another determined direction, and finally from the finish d of this second step a third 154 THE COMMON SENSE OP THE EXACT SCIENCES. directed step, dg. The entire operation brings us from A to G. Now it is obvious that we should also have got to g had we taken the directed step ag. Hence, if we gire an extended meaning to the word 'equal' and to its sign =, using them to mark the equivalence of the results of two operations, we may write AG = AB + BD + DG, i and read this expression : — A G equals the sum of A B, B D and D G. Steps such as we considered in our chapter on Quantity, which were magnitudes taken along any one straight line, are termed scalar steps, because they have relation only to some chosen scale of quantity. We add or subtract scalar steps by placing them end to end in any straight line (see § 2 of Chapter III.) A step which has not only magnitude but direction is termed a vector step, because it carries us from one position in space to another. It is usual to mark by an arrow-head the sense in which we are to take this directed step. For example in fig. 64 we are to step from A to B, and thus the arrow-head will point towards B for the step a b. In letters this is denoted by writing A before b. The method by which we have arrived at the conception of vector steps shows us at once how to add them. Vector steps are added by placing them end to end in such fashion that they retain their own peculiar directions, and so that a point moTing continuously along the zigzag thus formed will always follow the directions indicated by the arrow-heads. This may be shortly expressed by saying the steps are to be arranged in continuous sense. The sum of the vector steps is then the single directed step which joins the start of POSITION, 155 the zigzag thus formed to its finish. In fig. 65 let ab,cd, ef, and ghhe directed steps. Then let a b be drawn equal and parallel to ah; from B draw b c equal and parallel to c d, from o draw c d equal and parallel to ef, and finally from D draw d e equal and parallel to g h. We have drawn our zigzag so that the arrow-heads all have 'a continuous sense.' Hence the directed step A E is the sum of the four given vectors. If, for example, at we had stepped o d', equal and parallel to ef, hut on the opposite side of B c to d, and then taken d'e'. equal and parallel to g h, the reader will remark at once that the arrow-heads in e c, d' and d'e' are not in continuous sense, or we have not gone in the proper direction at c. Should the vector steps all have the same direction, the zigzag evidently becomes a straight line ; in this case the vector steps are added precisely like scalar quantities ; or, when vector steps may be looked upon as scalar, our extended conception of addition takes the ordinary arithmetical meaning. We can now state a very important aspect of position 156 THE COMMOM" SEKSE OF THE EXACT SCIENCES. in a plane ; namely, if the position of G relative to A be denoted by the directed step or vector a g, it may also be expressed by the sum of any number of directed steps, the start of the first of such steps being at a and the finish of the last at g (see fig. 56). We may write this result symbolically : — AG = AB + BO + CD + DE + EF + FG. It will be at once obvious that in our example as to finding the 'George,' the stranger might have been directed by an entirely different set of instructions to Fig. 56. his goal. In fact, he might have been led to make extensive circuits in or about the town before he reached the place he was seeking. But, however he might get to G, the ultimate result of his wanderings would be what he might have accomplished by the directed step A G supposing no obstacles to have been in his way (or, *as the crow flies'). Hence we see that with our extended conception of addition any two zigzags of directed steps, a b c d e f g and a b' c' d' e' f' g (which may or may not contain the same number of com- ponent steps), both starting in A and finishing in g. POSITION. 157 must be looked upon as equivalent instructions ; or, we must take AB + BC + CD + DE + EF + PG = AG = AB' + b'o' + c'd' + D'E' + e'p' + p'g. In other words, two sets of directed steps must be held to have an equal sum, when, their starts being the same, the steps of both sets will, added vector-wise have the same finish. Now let us suppose our stranger were unconsciously standing in front of the ' George ' when he asked his question as to its whereabouts, and further let us sup- pose that the person who directed him gave him a per- fectly correct instruction, but sent him by a properly chosen set of right and left turnings a considerable distance round the town before bringing him back to the point A from which he had set out. In this case we must suppose the * George ' not to be at the point g, but at the point A. The total result of the stranger's wanderings having brought him back to the place from which he started can be denoted by a zero step; or we must write (fig. 66) — AB+BO-f-CD + DE + EF + PG4-GA = . . . (i) We may read this in words : The sum of vector steps which form the successive sides of a closed zigzag is zero. Now we have found above that — AB-|-BO + CD + DE-|-EF + FG = AG (ii) Hence, in order that these two statements (i) and (ii) may be consistent, we must have — G A equal to a g, or A G + G A = 0. This is really no more than saying that if a step be taken from A to G, followed by another from G to a, the total operation will be a zero step. Yet the result is 158 THE COMMON SENSE OF THE EXACT SCIENCES. interesting as showing that if we consider a step from A to G as positive, a step from G to A must be considered negative. It enables us also to reduce subtraction of vectors to addition. For if we term the operation denoted by a b — d c a subtraction of the vectors A b and D c, since D c + c D = 0, the operation indicated amounts to adding the vectors A b and c d, or to A B + D. Hence, to subtract two vectors, we reverse the sense of one of them and add. X- E. P S q TV Fio. 57. The result a G + g a = can at once be extended to any number of points lying on a straight line. Thus, if P Q E s T u V be a set of such points — PQ + QE + ES + ST+TU + UV + VP = 0. For starting from p and taking in succession the steps indicated, we obviously come back to p, or have per- formed an operation whose result is equivalent to zero, or to remaining where we started. § 4. The Addition of Vectors oheys the Commutative Law. We can now prove that the commutative law holds for our extended addition (see p. 5). First, we can show that any two successive steps may be interchanged. Consider four successive steps, A b, b c, c d, and d e. If at B instead of taking the step b c we took a step b h equal to c d in magnitude, sense and direction, we could then get from h to D by taking the step hd. Now let b D be joined ; then in the triangles b h D, D c b the angles at B and D are equal, because they are formed by the straight line r. D falling on two parallel lines b h POSITION. 159 and D ; also the side b d is common, and b h is equal to d D. Hence it follows (see p. 73) that these triangles are of the same shape and size, or h d is equal to b o ; and again the angles b d h and d b o are equal, or h D and b c are parallel. Thus the step h d is equal to the step B in direction, magnitude and sense. We have then from the two methods of reacting d from B, BO + CD = BD = BH + nD = CD + BO by what we have just proved. FiQ. 58. Hence any two successive steps may be inter- changed. By precisely the same reasoning as we have used on p. 11 we can show that if we may inter- change any two successive steps of our zigzag we may interchange any two steps whatever by a series of changes of successive steps; that is, the order in which vectors are added is indifferent. The importance of the geometiy of vectors arises from the fact that many physical quantities can be re- presented as directed steps. We shall see in the suc- ceeding chapter that velocities and accelerations are quantities of this character. § 5. On Mdliods of Determining Position in a Plane. It' has been remarked (see p. 99) that scalar quantities may be treated as steps measured along a 160 THE COMMON SENSE OF THE EXACT SCIENCES. straight line. In this case we only require one point on this line to be given, and we can determine the relative position of any other by merely stating the magnitude of the intervening step. A line is occasionally spoken of as being a space of one dimension; in one-dimensioned space one point suffices to determine the relative posi- tion of all others. When we consider however position in a plane, in order to determine the whereabouts of a point p with regard to another A we require to know not only the magnitude but the direction of the step A p. Hence what scalar steps are to one-dimensioned space, that x" -y'" \ >B V :r / Fio. 59. are vector steps to plane space. In order to deter- mine the direction of a step ap we must know at least one other point b in the plane. Space which requires two points to determine the position of a third is usually termed space of two dimensions. There are various methods in general use by which position in two-dimensioned space is determined. We shall men- tion a few of them, confining our remarks however to the plane, or to space of two dimensions which is of the same shape on both sides. (a) We may measure the distances between A and p and between b and p. If these distances are of POSITION. 161 scalar magnitude r and r' respectively, there will be two points corresponding to any two given, values of r and r ; namely p and p' the intersections of the two circles with centres at A and b and radii equal to r and / respectively. We may distinguish these points as being one above, and the other below ab. Only in the case of the circles touching will the two points coincide ; if the circles do not meet, there will be no point. If p moves so that for each of its positions with re- gard to A and B the quantities r and r' satisfy some defi- nite relation, we shall obtain a continuous set of points in the plane or a curved line of some sort. For example, if we fasten the ends of a bit of string of length I to Fig. 60. pins stuck into the plane of the paper at A and b, and then move a pencil about so tha.t its point p always remains on the paper, and at the same time always keeps the string a p b taut round its point, the pencil will trace out that shadow of the circle which we have called an ellipse. In this case r + / = AP4-PB= I, the constant length of the string. This relation r -t- r' = Z is an equation between the scalar quantities r, / and I, which holds for every point on the ellipse, and expresses a metric property of the curve with regard to the points A and b. If on the other hand we cause p to move so that the difference of a p and b p is a constant length (r— r' = Q, then p will trace out the curve we have termed the 1G2 THE COMMON SENSE OF THE EXACT SCIENCES. hyperbola. We can cause P to move in this fashion bj' means of a Tery simple bit of mechanism. Suppose a rod B L capable of revolving about one of its ends B : let a string of given length be fastened to the other end L and to the fixed point a. Then if, as the rod is moved round b, the string be held taut to the rod by a Fig. 61. pencil point p, the pencil will trace out the hyperbola. For since lp + pa equals a constant length, namely that of the string, and l p + pb equals a constant length, namely that of the rod, their difference or pa— pb is equal to the constant length which is the difference of the string and the rod. Fia. 62. The points A and b are termed in the cases of both ellipse and hyperbola the foci. The name arises from the following interesting property. Suppose a bit of polished watch spring were bent into the form of an ellipse so that its flat side was turned towards the foci of the ellipse ; then if a hot body were placed at one focus B, all the rays of heat or light radiated from b POSITION'. 163 which fell upon the spring would be collected, or, as it is termed, ' focussed ' at A ; hence A would be a much brighter and hotter point than any other within the ellipse (b of course excepted). The name focus is from the Latin, and means a fireplace or hearth. This property of the arc of an ellipse or hyperbola, that it collects rays radiating from one focus in the other, depends upon the fact that a p and p B make equal angles with the curve at p. This geometrical relation corresponds to a physical property of rays of heat and light; namely, that they make the same angle with a reflecting surface when they reach it and when they leave it. A third remarkable curve, which is easily obtained from this our first method of considering position, is the lemniscate of James Bernoulli (from the Latin lemniscus, a ribbon). It is ti'aced out by a point p which moves so that the rectangle under its distances from a and B is always equal to the area of a given square i'lQ. 63 (r . / = c"). If the given square is greate]' than the square on half A B, it is obvious that p can never cross between a and b ; if it is equal to the square on half A B, the lemniscate becomes a figure of eight ; while if it is less, the curve breaks up into two loops. In our figure a series of lemniscates are represented. A set of curves obtained by varying a constant, like the 164 THE COMMON SENSE OF THE EXACT SCIENCES. given square in tlie case of the lemniscate, is termed a family of curves. Such families of curves constantly occur in the consideration of physical problems. § 6. Polar Co-ordinates. (/3) The points A and b determine a line ■whose direction is A b. If we know the length a p and the angle bap, we shall have a means of finding the position of p. Let r be the number of linear units in A p and 6 the number of angular units in b a p, where r and 6 may of course be fractions. In measuring the angk we shall adopt the same convention as we have employed in discussing areas (see p. 134) ; namely, if a line at first coin, can only differ from either of them by a quantity less than 1/&. If then b be taken large enough, or the equal angles at small enough fractions of the unit angle, this dif- ference 1/6 can be made vanishingly small. In this case we may say that in the limit the angle becomes equal to n/6 and the ray o p equal to o n or o Q, which will thus be ultimately equal. Hen ce o p = o a . X"'* = o A . \*, or in words : If a ray o p of the equiangular spiral make an angle a o p with another ray o A, the ratio of o p to o A is equal to a certain number X raised to the power of the quantity which expresses the magnitude of the angle A o p in units of angle. If a and r be the numbers which express the magnitudes of o a and o p, we have r=aX'. This is termed the polar equation of the spiral. We proceed to draw some important results from a POSITION. 175 consideration of this spiral. The reader will at once obserTO that the ratio of any pair of rays o p and o q is equal to the ratio of any other pair which include an equal angle, for the ratio of any pair of rays depends only on the included angle. Further, if we wanted to multiply the ratio of any two quantities p and q by the ratio of two other quantities r and s we might proceed as follows : Find rays of the equiangular spiral o p, o Q, E, s containing the same number of linear units as p, g, r, s contain units of quantity (see p. 99), and let d be the angle between the first pair, ^ the angle between the second pair. Then ^ = X^and^ = X*; OP OE whence it follows that ^ x — = \» x \* = \«+*, OP OE or is equal to the ratio of any pair of rays which include an angle 6 + ^. Thus if the angle qot be taken equal to ^, and o t be the corresponding ray of the spiral, - — = X*+*, and is a ratio equal to the pro- duct of the given ratios. Hence to find the product of ratios we have only to add the angles between pairs of rays in the given ratios, and the ratio of any two rays including an angle equal to the sum will be equal 9 176 THE COMMON SENSE OF THE EXACT SCIENCES. to the required product. Thus the equiangular spiral enables us to replace multiplication hy addition. This is an extremely valuable substitution, as it is much easier to add than to multiply. Since ^ divided by — = x« divided by \*= \»-* OP OB, it is obvious that we may in like fashion replace the division of two ratios by the subtraction of two angles. A set of quantities like the angles at the pole of an equiangular spiral which enables us to replace multiplication and division by addition and subtrac- tion is termed a table of logarithms. Since the equi- angular spiral acts as a graphical table of logarithms, it is frequently termed the logarithmic spiral. § 10. On the Nature of Logarithms. Since in the logarithmic spiral o p = o A x X*, where 6 is equal to the angle A p, we note that as increases, or as the ray o p revolves round o, o p is equally mul- tiplied during equal increments of the vectorial angle A p. When one quantity depends upon another in such fashion that the first is equally multiplied for equal increments of the second, it is said to grow at logarithmic rate. This logarithmic rate is measured by the ratio of the growth of the first quantity for unit increment of the second quantity to the magnitude of the first quantity before it started this growth. Let us endeavour to apply this to our equiangular spiral. Suppose aob, boo, cod &c. to be as before the triangles by means of which we construct it (see fig. 69), the angles at o being all equal and very small. A.long o B measure a length o a' equal to o A ; along o 0, a length b' equal to o b ; along o d, a length c' POSITION. 177 "equal to o o, and so on. Then a'b, b'o, c'd, &c., will be tlie successive growths as a ray is turned succes- sively from A to o B, from o b to o 0, and so on. Join A a', b b', c', &c. Now the triangles A o b, boo, COD, &c., are all of the same shape ; so too are the isosceles triangles a o a', b o b', o o c', &c. Hence the differences of the corresponding members of these sets, A a'b, b b'o, c c'd, &c., must also be of equal shape, and thus their corresponding sides proportional. It follows then that the lengths a'b, b'o, c'd, &c., are in the same ratio as the lengths a' A, b'b, o'c, &c., or again as the lengths oa, OB, oc, &c. Wlience we deduce that — — ^ = — =&e OA ~ OB ~ 00 ~ Or, the growth a'b is always in a constant ratio to the growing quantity o a. Now, if the angles at o be very small, the line A a' will practically coincide with the arc of a circle with centre o and radius equal to o a. Hence (see p. 143) A a' will ultimately equal o A x the angle a o a', while the angle at a' will ultimately be equal to a right angle. Further, the ratio of a'b to a a' remains the same for all the little triangles a a'b, b b'o, c o'd, &c. It is in each case the ratio of the base to the perpendicular when we look upon these triangles with regard to the equal angles aba', bob', o d c', &c. Now these are the angles of the triangles which give the spiral its name. Let any one of them, and therefore all of them, be equal to a. By definition the cotangent of an angle (see p. 166) is equal to the ratio of the base to the perpendicular. 178 THE COMMON SENSE OF THE EXACT SCIEKCES. Hence , a'b a'b cota = — ; = ^ 7j A A o A X angle a o a or — = ansrle A o a' x cota, OA Now a'b denotes the growth for an angle aoa', supposed very small ; whence it follows that the loga- rithmic rate, or the ratio of the growth to the growing quantity for unit angle, is equal to cota. Thus the logarithmic rate for the growth of the ray of the equi- angular or logarithmic spiral, as it descrihes equal angles about the pole, is equal to the cotangent of the angle which gives its name to the spiral. Let us suppose o a to be unit of length, then, since o p = o A X X'-, the result o p of revolving the ray o a through an angle 6 equal to unity will be \, or X is the result of making unity grow at logarithmic rate cota, Now let us denote by the symbol e the result of making unity grow at logarithmic rate unity during the description of unit angle. Then e will have some definite numerical value. This value is found, by a pro- cess of calculation into which we cannot enter here, to be nearly equal to 2'718, This means that, if while unit ray were turned through unit angle it grew at loga- rithmic rate unity, its total growth (1"718) would lie between eight and nine-fifths of its initial length. Since e is the result of turning unit ray through unit angle, and since the ray is equally multiplied for equal multi- ples of angle, ef must represent the result of turning unit ray through 7 unit angles. Hitherto we have been concerned with unit ray growing at logarithmic rate unity ; now let us suppose unity to grow at logarithmic rate y ; then it grows 7 times as much as if it grew at logarithmic rate unity, or the result of turning unit ray POSITION. 179 througli unit angle, while it grows at logarithmic rate 7, must be the same as if we spread I/7 of this rate of growth over 7 unit angles ; that is, as if we caused unity to grow at logarithmic unity for 7 unit angles, or eT. Hence &'' denotes the result of making unit ray grow at logarithmic rate unity while it describes 7 unit angles, or again of making unit ray grow at loga- rithmic rate 7 while it describes a unit of angle. Let us inquire what is the meaning of & when 7 is a commensurable fraction equal to s/i, s and t being integers. Let x be the as yet unknown result of turn- ing unit ray through an angle equal to 7 while it grows at unit logarithmic rate; then a;' will be the result of turning unit ray through t angles equal to 7 while it grows at unit rate ; but t angles equal to 7 form an angle containing s units, or this result must be the same as the result of turning unity through an angle s while it grows at logarithmic rate t. Thus we have »' = e'. That is, a; is a i-th root of e>, or, as we write it, equal to e'/' = ev. Thus gi, if 7 be a commensurable fraction, is the result of causing unit ray to grow at logarithmic rate unity through an angle equal to 7, or as we have seen at logarithmic rate 7 through unit angle. Now let us suppose it possible to find a commen- surable fraction 7 equal to cota; then the result of making unity grow at logarithmic rate cota as it is turned through unit angle must be e^. But we have seen (see p. 178) that it is equal to \. Hence A, = eT. Further, the result of making unity grow at loga- rithmic rate cota as it is turned through an angle Q is \' ; or, \9 = ev«. 180 THE COMMON SENSE OF THE EXACT SCIENCES. Thus we may write o P = o A . X* = A •. e''*, or with otir previous symbols, r = a. ev». This is therefore the equation to our equiangular spiral expressed in terms of the quantity e. If we take a spiral in which a is the unit of length, and in which cota or 7 is also unity, we find r = e*. The symbol e* is then read the exponential of 6, and is termed the natural logarithm of r. It is denoted symbolically thus : — = log^r. The quantity e is termed the base of the natural system of logarithms. Our spiral would in this case form a graphical table of natural logarithms. Eeturning to the equation r = a . eT*, let us suppose y so chosen that e'>'=10 ; then 7 will re- present the angle through which unit ray must be turned in order that, growing at unit logarithmic rate, it may increase to ten units. Again taking a to be of unit length we find r=e>*=10*. is in this case termed the logarithm of r to the base 10, and this is symbolically expressed thus : — = logio «•• The spiral obtained in this case would form a graphical table of logarithms to the base 10. Such logarithms are those which are usually adopted for the purposes of practical calculation. Natural logarithms were first devised by John Napier, who published his invention in 1614.' Loga- • Logarithmorum Canonis Descripiio. 4to. Edinburgh, 1614. POSITION, 181 ritlims to the base 10 are now used in all but the simplest numerical calculations which it is needful to make in the exact sciences; their value arises solely from the fact that addition and subtraction are easier operations than multiplication and division. § 11. T/ie Cartesian Method of- Determining Position. (7) In order to determine the position of a point p, in space of two dimensions, we may draw the line b a b', joining the given points A B and another line c A c' at right angles to this through a. These will divide the plane into four equal portions termed quadrants. Let Pj M be a line drawn from the point p, (the position of B' C Pa V, n' ^ M V. J ?♦ c' Fio. 71. which relative to A we wish to determine), parallel to c A and meeting b'a b in m. Then we may state the following rule to get from a to p, : Take a step A m from A on the line b'a b, and then a step to the left at right angles to this equal to M p,. Now a step like a m may be taken either forwards along a b or back- wards along A b'. Precisely as before (see p. 100) we 182 THE COMMON SENSE OF THE EXACT SCIENCES. shall take + A m to mean a step forwards along A B, and —AM to mean a step am' hackwards along ab' througli the same distance a m. Let us use the letter * to denote the operation, which we have represented hy (7r/2) on p. 151. Thus applied to unit step it will signify : Step forwards in the direction of the previous step and from its finish unit distance, and then rotate this unit distance through a right angle counter-clockwise about the finish of the previous step. The operator i placed before a step, thus i.MP,, will then be interpreted as follows: Step from m in the direction A b a distance equal to the length m p„ and then rotate this step M p, about m counter-clockwise through a right a.ngle. We are thus able to express symbolically the position of p, relative to A, or the step A Pj, by the relation AP, = AM 4 i.MPi. If we had to get to a point P4 in the quadrant b a c', instead of to p„ we should have, instead of stepping for- wards from M, to step haehwards a distance m P4, and then rotate this through a right angle counter-clock- wise. The step backwards jvould be denoted by insert- ing a — sign as a reversing operation (see p. 39), and we should have A P4 = A M — i . M P4. Next let us see how we should get to a point like p^ in the quadrant c A b', where Pj is at a perpendicular distance PjM' from ab'. First, we must take a step, A m', backwards ; this is denoted by —am'; secondly, we must step forwards from m' a distance m'p^; since this step is forwards, it will be towards A ; thirdly, by applying the operation i to this step, we rotate it about POSITION. 183 m' couuter-clockwise through a right angle, and so reach Pj. Hence A Pj = — A m' + i . m' Pj. Finally, if we wish to reach P3 in the quadrant b'a c', we must step backwards a m', and then still further backwards a step h' Pj, and lastly rotate this step counter-clockwise through a right angle. This will be expressed by A Pg = — A m' — ■!! . m' p„. Now let us suppose Pp Pj, P3, P4, to be the four corners of a rectangular figure whose centre is at a and whose sides are parallel to b A b' and c a 0'. Let the number of units in A M be x, and the number in M p, be y, then we may represent the four steps which determine the positions of the p's relative to A as follows : — A'P^ = X + ly k-p^ = - X + iy AP3 = — X — iy AF^ = X — iy. Here x and y are mere numbers, but, when we represent these numbers by steps on a line, the '^-numbers are to be taken on a certain line at right angles to that line on which the se-numbers are taken. Thus the moment we represent our x and y numbers by lengths, they give us a means of determining posi- tion. The quantities x and y might thus be used to deter- mine the position of a point, if we supposed them to carry with them proper signs. Our general rule would then be to step forwards from a along a b a- distance x, and then frooi the end of « a distance forwards equal to y ; rotate this step y about the end of x counter- clockwise through a right angle, and the finish of y will then be the point determined by the quantities x, y. 184 THE COMMON SENSE OF THE EXACT SCIENCES. If SB or y be negative, the corresponding forwards must be read: Step forwards a negative quantity, that is, step backwards. Thus : — Pj, or position in the quadrant B A C is determined by x, y. Pj . . . . cab' • . — », 2/' Pj . . . . b'ac' . . —x,—y. P4 .... o'ab . . X, —y. The quantities x and y are termed the Cartesian co- ordinates of the point p, this method of determining the Fig. 72. position of a point having been first used by Descartes. B A b' and c A c' are termed the co-ordinate axes of x and y respectively, while A is called the origin of co- ordinates. For example, let the Cartesian co-ordinates of a point be (—3, 2). How shall we get at it from the origin a ? If p be the point, we have ap= —S + i,2. Hence we must step backwards 3 units ; from this point step forwards 2 and rotate this step 2 about the ex- tremity of the step 3 through a right angle counter- clockwise ; we shall then be at the required point. If p be determined by its Cartesian co-ordinates a? and y, we might find a succession of points, p, by always POSITION. 185 taking a step y related in a certain invariable fashion to any step x which has been previously made. Such a succession of points p, obtained by giving X every possible value, will form a line or curve, and the relation between x and y is termed its Cartesian equation. As an instance of this, suppose that for every step X, we take a step y equal to the double of it. Then we shall have for our relation y = 2x, and our instructions c — 1 — I — 1 — I — I — p-i — I f I ' I 1 'T'T"' i "i — n y ^1 _ ,/. __ f. — — .^ 1 i _--^_ _II qia: ::ii:i|i|::::::r -^i A. _ Fio. 73. to reach any point p of the series are x + i.2x. Suppose the quadrant bag divided into a number of little squares by lines parallel to the axes, and let us take the sides of these squares to be of unit length. Then if we take in succession x=l, 2, 3, &c., we can easily mark off our steps. Thus : 1 along A b and then 2 to the left ; 2 along A B and 4 to the left ; 3 along A b and then 6 to the left ; 4 along a b and then 8 to the left ; 5 along AB and then 10 to the left, and so on. It will be obvious (by p. 106) that our points all lie upon a 186 THE COMMON SENSE OF THE EXACT SCIENCES. straight line througli A, and however many steps we take along a b, followed by double steps perpendicular to it, we shall always arrive at a point on the same line. If we gave x negative values we should obtain that part of the line which lies in the third quadrant b'ac'. Hence we see that y = 2x is the equation to a straight line which passes through a. Let us take another example. Suppose that the rectangle contained by y and a length of 2 units, always contains as many units of area as there are square units in x^. Our relation in this case may be expressed by 2 1/ = x^, and we have the following series of steps from a to points of the series : — 2» 1+i.l, 2+i.2, 3 + i. 4 + i.8, 5^-^.V, 6 + i.l8,&c. We can by means of our little squares easily follow out the operations above indicated ; we thus find a series of points like those in the quadrant b A of the figure. If however we had taken x equal to the negative quantities —1, —2, —3, —4, —6, —6, &c., we should have found precisely the same values for y, because we haveseenthat ( — a) x (— a)= a^ = ( + «) x ( + a). These negative values for x give us a series of points like those in the quadrant b' a of the figure. It is impossible that any points of the series should lie below b a b', because both negative and positive values for x give when squared a positive value for the step y, so that no possible iB-step would give a negative 2/-step. The series of points obtaired in this fashion are found to lie upon a curve which is one of those shadows of a circle Vv'hich we have termed parabolas. Hence we may say that 2y = x^ is the equation to a parabola. POSITION. 187 This method of plotting out curves is of great value, and is largely used in many branches of physical inves- tigation. For example, if the differences of successive SB-steps denote successive intervals of time, and y-steps the corresponding heights of the column of mercury in a barometer above some chosen mean position. ] '. \ _i 1 ' 1 I -- 3 N -1 - -I - _ i. ~l ! '' ^j -t r^ ' • il : : '" _4_ I ■ - -t n ± _i_ T- - ^ r \ i^ i 1 ■; 1 1 1 1 rr T T ; • ^ *■, r •" 1 ' b: ». y. L "^ _ : : :a_u. _^ .. Ti ^ -B Pig. 74. the series of points obtained will, if the intervals of time be taken small enough, present the appear- ance of a curve. This curve gives a graphical repre- sentation of the variations of the barometer for the whole period during which its heights have been plotted out. Barometric curves for the preceding day are now given in several of the morning papers. Heights cor- responding to each instant of time are in this case 188 THE COMMON- SENSE OP THE EXACT SCIENCES. generally registered automatically by means of a simple pliotograpliic apparatus. The plotting out of curves from their Cartesian equations, usually termed curve tracing, forms an ex- tremely interesting portion of pure mathematics. It may be shown that any relation, which does not in- volve higher powers of le and y than the second, is the equation to some one of the forms taken by the shadow of a circle. § 12. Of Complex Numbers, We shall now return to our symbol of operation i, and inquire a little closer into its meaning. Let the point p be denoted as before by a m +i . m p, so that we c / ^ ; 1 t c ^ 1> If a Fia. 75. should read this result : Step from A to M along A B, and from M to p' along the same line (where M p' = m: p), finally rotate M p' about m counter-clockwise through a right angle ; M p' wjU then take up the position m p. Now let M q' be taken equal to a p', then a M + i . m q' will mean : Step from a to m and then from m perpendicular POSITION. 189 to A M to the left through a distance, M q', equal to a p'. Since however mq' = ap' = am: + m:p=mp + pq', pq' must be equal to a m and we can read our operation AM: + i.(M:p + p q'), which denotes two successive steps at right angles to A M, namely m p followed by the step' p q'. Suppose now we wished to rotate this latter step through a right angle coanter-clockwise, we should have to introduce before it the symbol i, and m p + i . p q' would signify the step M p followed by the step p q at right angles to it to the left. Now p q' is equal to A M, and hence the result of this operation must bring us to Q, a point on a c which might have been reached by the simple operation + i . A Q. Thus we may put + i . A Q, = AM + i . {m. p + i . T? q) = am; + i.MP + i. i. pq; or, since the quantities a q, am, m p, and p q here merely denote numerical magnitudes, and since as such A Q = M p and A M = p Q, we must have = AM. + i . i . AM, or — AM = i .i . AM. Thus the operation i is of such a character that repeated twice it is equivalent to a mere reversor, or, as we may express it symbolically, -1 = i\ This may be read in words : Turn a step counter- clockwise through a right angle, and then again counter-clockwise through another right angle, and we have the same result as if we had reversed the step. Now we have seen (p. 144) that if x be such a quantity that multiplied by itself it equals a, x is termed the square root of a, and written V*. He nce s ince •i^rr: — 1, WO may write i=V - 1. 190 THE COMMON SENSE OF THE EXACT SCIENCES. This symbol is completely unintelligible so far as quantity is- concerned; it can represent no quantity conceivable, for the squares of all conceivable quantities are positive quantities. For this reason V— 1 is some- times termed an imaginary quantity. Treated however as a symbol of operation V—l has a perfectly clear and real meaning ; it is here an instruction to step forwards a unit length and then rotate this length counter-clock- wise through a right angle. , Any expression of the form x + v—l y is termed a complex number. Let p be any point determined by the step a p = AM + V — Imp, and let r, x, y be the numerical values of the lengths a p, am, and p m. It follows from the . right-angled triangle p a m that r^ = x^ + y^. The quantity r is then termed the modulus of the complex number x + V — l y. Further let the angle map contain 6 units of angle; then ■ a FVL y /.AM X sin.6 = — = -, cost' = — = -, A p r A p r or y = r sin^, x =r cos0. The angle is termed the argument of the com- plex number. Here r and 6 are the polar co-ordinates of p, and we are thus able to connect them with the Car- tesian co-ordinates ; they are respectively the modulus and argument of the complex number which may be formed from the Cartesian co-ordinates. Since r is merely numerical we may write the complex number X + V — ly in the form r . (cos^ -|- V—l sin^), or as the product of its modi|Jus and the operator C0s9 + V—l sin^, POSITION. 191 Hence we wliieh depends solely on its argument 6. may interpret the step A p = r . {cos6 + V— i sin^) as obtained in the following fashion: Rotate unit length from A B through an angle 6, and then stretch it in the ratio of r ; 1. The latter part of this operation will be signified by the modulus r, the former by the operator (cos^ + V— 1 sin^). Thus if ad be of unit length and lying in a b, w.e may read — A p = r , (cos5 + V— I smO) . a d, and we look upon our complex number as a symbol denoting the combination of two operations performed on a unit step a d. Starting then from the idea of a complex niiraber as denoting position, we have been led to a new opera- tion represented by the symbol cos0 + V — 1 sin^. This is obviously a generalised form of our old symbol V— 1. The operator cos9 + V— 1 sin^ applied to any step bids us turn the step through an angle 6. We shall see that this new conception has important results. 192 THE COMMON SENSE OF THE EXACT SCIENCES, § 13. On the Operation which turns a Step through a given Angle. Suppose we apply the operator (cos0 + V—1 sin^) twice to a unit step. Then the symbolic expression for this operation will be (cos0 + V^ sin0) (cos0 +V—1 sin^), or (cos^ + V — 1 sin^)^ But to turn a step first through an angle 6 and then through another angle d is clearly the same operation as turning it by one rotation through an angle 26, or as applying the operator cos2^ +V — 1 sin2^. Hence we are able to assert the equivalence of the operations expressed by the equation — (cos0 + V^ sin^)" = cos2^ + V^ sin20. In like manner the result of turning a step by n operations through successive angles equal to must be identical with the result of turning it at once through an angle equal to n times 0, or we may write (cos0 + V^ sin^)" = cosw^ + V^-l sinnd. This important equivalence of operations was first ex- pressed in the above symbolical form by De Moivre, and it is usually called after him De Moivre's Theorem, We are now able to consider the operation by means of which a step A p can be transformed into another A Q. "We must obviously turn a p about A counter-clockwise till it coincides in position with a Q ; in this case P will fall on p', so that a p' = a p. Then we must stretch a p' into A Q ; this will be a process of multiply- ing it by some quantity p, which is equal to the ratio of A Q to A p'. POSITION. 193 Expressing this symbolically, if and giving it a stretch equal to p r. Thus we see that any relation between complex numbers may be treated either as an algebraical fact relating to such numbers, or as a theorem concerning operations of turning and stretching unit steps. (iii) We may consider what answer the above identity gives to the question : What is the ratio of two directed steps A Q and a p ? Or, using the notation sug- gested on p. 45, we ask : What is the meaning of the symbol . — ^ ? A step like a p (or A q) which has magnitude, direction, and sense is, as we have noted, termed a vector. We therefore ask: What is the ratio of two vectors, or what operation will convert one into the other ? The answer is : An operation which is the product of a turning (or spin) and a stretch. Now the stretch is a scalar quantity, a numerical ratio by which the scalar magnitude of a p is con- nected with that of aq. The stretch therefore is a scalar operation. Further, the turning or spin converts the direction of A p into that of A Q, and it obviously takes place by spinning a p round an axis perpendi- cular to the plane of the paper in which both ap and AQ lie. Thus the second part of the operation by which we convert A p into A q denotes a spin (counter-clockwise) through a definite angle about a certain axis. The amount of the spin might be measured by a step taken along that axis. Thus, for instance, if the spin were through 6 units of angle, we might measure 6 units of length along the axis to POSITION 195 denote its amount. We may also agree to take this length along one direction of the axis ('out from the face of the clock ') if the. spin be counter-clockwise, and in the opposite direction (' behind the face of the clock ') if the spin be clockwise. Thus we see that our spinning operation may be denoted by a line or step having magnitude, direction, and sense ; that is, by a vector. We are now able to understand the nature of the ratio of two vectors; it is an operation consisting of the pro- duct of a scalar and a vector. This product was termed by Sir William Hamilton a quaternion, and made the foundation of a very powerful calculus. Thus a quaternion is primarily the operation which converts one vector step into another. It does this by means of a spin and a stretch. If we have three points in plane space, the reader will now understand how the position of the third with regard to the first can be made identical with that of the second by means of a spin and a stretch of the step joining the first to the third, that is, by means of a quaternion.' § 14. Relation of the Spin to the Logarithmic Growth of Unit Step. Let us take a circle of unit radius and endeavour to find how its radius grows in describing unit angle about the centre. Hitherto we have treated of growth only in the direction of length ; and hence it might be supposed that the radius of a circle does not * grow ' at all as it revolves about the centre. But our method of adding vector steps suggests at once an obvious extension of our conception of growth. Let a step A p become • The term 'stretch' must be considered to include a squeeze or a stretch denoted by a scalar quantity p less than unity. 196 THE COMMON SENSE OP THE EXACT SCIENCES. A Q as it rotates about a througli the angle p A Q, tlien if we marked off 1 q a distance A p' equal to A p, p' q would be the scalar growth of A p ; that is, its growth FiQ. 78. in the direction of its length. But if A p be treated as a vector: (see p. 153) AQ = AP + PQ, or the directed step p q must be added to A p in order to convert it into a q ; p q may be thus termed the directed growth of A p. If we join p p', we shall have p Q equal to the sum of p p' and p' q. Now if the angle pap' be taken very small p p' will be ultimately perpendicular to AP, and this part of the growth pq might be represented by V — 1. pp'. Hence we are led to represent a growth perpendicular to a rotating line by a scalar quantity multiplied by the symbol a/— 1. We can now consider the case of our circle of unit radius. Let o p be a radius which has revolved through Fis. 79. an angle 6 from a fixed radius o A, and let o Q be an adjacent position of p such that the angle Q o p is very small. Then p q will be a small arc sensibly coincident POSITION. 1 97 •witli tlie straight line p q, and the line p q will be to all intents and purposes at right angles to o p. Hence to obtain o Q we must take a step p q at right angles to OP. This we represent hjV—1 qp. Siuoe the radius of the circle is unity the arc qp, which equals the ra,dius multiplied by the angle Q o p (see p. 143), must equal the numerical value of the angle q o p. Or the growth of p is given by V - 1 x angle Q o p. Now according to our definition of growing at logarithmic rate (seep. 176), since op is equally multiplied in de- scribing equal angles about o, it must be growing at logarithmic rate. What is this logarithmic rate for unit angle ? It must equal — ^ divided by the ratio of the angle P Q / — ■ Q p to unit angle = j = v — 1 since o p ^ *= OP X angleQOP is unity. Thus o p is growing at logarithmic rate V — 1 as it describes unit angle ; that is to say, the result of turning OP through unit angle might be symbolically expressed by e'*'—^. Hence the result of turning op through an angle 6 must be e'^—^^. We may then write OP = OA .e'^-'^. Drop p M perpendicular to o a and produce it to meet the circle again in p', then by symmetry m p=ii p', and we have OP = OM + V— Im p. op' = OM — V — 1 MP'. Now since o p and o p' are of unit magnitude, . OM . . PM cosy = — = M, smO = — = p M. Also the angle p'om equals the angle mop, but, according 0-A (^'"^^ vri ^f-i f^]P 198 THE COMMON SENSE OF THE EXACT SCIENCES. to our convention as to the measurement of angles, it is of opposite sense, or equals — 6. Thus we must write p' = A . e-'^-i^ Substituting their values, we deduce the symbolical results e-V=:,e ^ cosl^-V^ sin^P^^ Further, o p — p' = 2V— 1pm o p + o p' = 2 M ; that is. e V-i9 _ e-^-ie = 2V-I sin^l ^... e^-i9 + e-^=i0=2cos5 y' These values for cos5 and sin^ in terms of the ex- ponential e were first discovered by Euler. They are meaningless in the form (ii) when cos^ and sin5 are interpreted as mere numerical ratios ; but they have a perfectly clear and definite meaning when we treat each side of the equation in form (i) as a symbol of operation. Thus cos^ + V— 1 sin^ applied to unit step directs us to turn that step without altering its length through an angle 6 ; on the other hand, e '^~^^ applied to the same step causes it to grow at logarith- mic rate unity 'perpendicular to itself, while it is turned through the angle 6. The two processes give the same result. § 15. On the Multiplication, of Vectors. We have discussed how vector steps are to be added, and proved that the order of addition is in- dififerent ; we have also examined the operation denoted POSITION. 199 by the ratio of two vectors. The reader will naturally ask : Can no meaning be given to the product of two vectors ? If both the vectors he treated as complex numbers, or as denoting operations, we have interpreted their product (see p. 193) as another complex number or as a resultant operation. Or again we have interpreted the product of two vectors when one denotes an ope- ration and the other a step of position ; the product in this case is a direction to spin the step through a certain angle and then stretch it in a certain ratio. But neither of these cases explains what we are to understand by the product of two steps of position. Let A p, A Q be two such steps : What is the meaning of the product ap.aq? Were ap and aq merely ■ ip Fio. 80. scalar quantities then their product would be purely scalar, and we should have no diflSculty in interpreting the result A p . p q as another scalar quantity. But when we consider the steps A p, p Q to possess not only Q n A ^ » FiQ. 81. magnitude but direction, the meaning of their product is by no means so obvious. If A Q were at right angles to A p (see fig. 81), we should naturally interpret the product ap.aq as the 10 200 THE COMMON SENSE OF THE EXACT SCIENCES, area of the rectangle on a q and A p, or as the area of the figure Q A p e. Now let us see how this area might be generated. Were we to move the step A Q parallel to itself and so that its end a always remained in the step A p, it would describe the rectangle Q A p e while its foot A described the step A p. Hence if A p and a q are at right angles we might interpret their product as follows : The product A p . A Q bids us move the step a q parallel to itself so that its end A traverses the step a p ; the area traced out by A Q during this motion is the value of the product a p . aq. It will be noted at once that this interpretation, although suggested by the case of the angle Q A p being a right angle, is entirely independent of what that angle may be. If Q A p be not a right angle the area traced out according to the above rule would be the parallelo- gram on A p, A Q as sides. Hence the interpretation we have discovered for the product A p . A Q gives us an intelligible meaning, whatever be the angle Q A p. There is, however, a difiSculty which we have not yet solved. An area is a directed quantity (see p. 134), and its direction depends on how we go round its perimeter. Now the area q a p u will be positive if we go round its perimeter counter-clockwise, or from a to p ; that is, in Af^ Q, _ K Fig. 82, the direction of the first step of the product or in the direction of motion of the second or moving step. Thus the product A p . A q will be the area Q A p e taken with the sign suggested by the step a p. The product a q. ap POSITION. 201 will be formed by causing the step A p to move parallel to itself along a q, and it is therefore also the area of the parallelogram on A Q and a p ; but it is to be taken with the sign suggested by a q, or it is the area PAQE. By our convention as to the sign of areas, PAQB = — QAPE, or AQ.AP=— AP.AQ, Hence we see that, with the above interpretation, the product of two vectors does not follow the commutative law (see p. 45). If we suppose the angle Q A p to vanish, and the vector A Q to become identical with a p, the area of the enclosed parallelogram will obviously vanish also. Thus, if a vector step be multiplied by itself, the product is zero ; that is, AP. AP = (ap)^ = 0. If we take a series of vector steps, a, y3, 7, 8, &c. then relations of the following types will- hold among them: a\ =0, ^62 = 0, 72 ^0, B^ = 0, &c. a^ = - ^a, ay = - ja, fiy = — 7^, By = — y 8, &c. A series of quantities for which these relations hold was first niade use of by Grassmann, and termed by him alternate units. The reader will at once observe that alternate units have an algebra of their own. They dispense with the commutative law, or rather replace it by another in which the sign of a product is made to alternate with the alternation of its components. Their consideration will suggest to the reader that the rules of arithmetic. 202 THE COMMON" SENSE OF THE EXACT SCIENCES. which he is perhaps accustomed to assume as neces- sarily true for all forms of symbolic quantity, have only the comparatively small field of application to scalar magnitudes. It becomes necessary to consider them as mere conventions, or even to lay them aside entirely as we proceed step by step to enlarge the meaning of the symbols we are employing. Although 2 X 2=0 and 2 x 3= —3 x 2 may be sheer nonsense when 2 and 3 are treated as mere numbers, it yet becomes downright common sense when 2 and 3 are treated as directed steps in a plane. Let us take two alternate units a, y8 and interpret the quantity aa + 6 /S, where a and h are merely scalar Fio. 83. magnitudes. If o A be the vector a, a a signifies that we are to stretch o A to o a' in the ratio of 1 to a. To this o a' we are to add the vector o b' derived from o b by giving it the stretch h. Hence if a' p = o b' the vector o p represents the quantity a a + h ^, which is termed an alternate number. Let o Q represent a second alternate number a' a -\- b' ^, obtained by adding the results pf applying two other stretches a' and V to the POSITION. 203 alternate units a and ;S. In the same way we might obtain, by adding the results of stretching three alternate units {a, /3, 7) , alternate numbers with three terms (of the form aa + fe/3 + c7), and so on. If we take the pro- duct of as many alternate numbers as we have used alternate units in their composition, we obtain a quantity called a determinant, which plays a great part in the modern theory of quantity. We shall confine ourselves here to the consideration of a determinant formed from two alternate units. Such a determinant will be represented by the product o p . o Q, which according to our convention as to the multiplication of vectors equals the area of the parallelogram on p, 6q as sides, or (by p. 122) twice the triangle qop. Through Q draw c Q a" parallel to b, and d q b" parallel to A, then o a" = a' a and b" = 6' y3. Join b' q, then twice the triangle b'q p equals the parallelo- gram b" p. Hence, adding to both these the parallelo- gram a' b" we have the parallelogram a' b" together with twice the triangle b'q p equal to the parallelogram b'a', or to twice the triangle b'op. But the triangle b'o p equals the sum of the triangles o Q b', b'q p, and OPQ. It follows then that the parallelogram a'b" must equal tvrice the triangle p Q together with twice the triangle o Q b'. Now twice the latter equals b' a". Hence the difference of the parallelograms a' b" and b' a" is equal to twice p q. The parallelogram a' b" is obtained from the parallelogram a b by giving it two stretches a and V parallel to its sides, and therefore its area equals a V times the area a b. Similarly b' a" equals h a' times the area A b ; but the area A B itself is a^. Thus we see that the identity P . o'^Q = a' b" — b' a" 204 THE COMMON SENSE OP THE EXACT SCIENCES, may be read {aa + 6y3) (a'o + h' ^) = {aV - I a') a^. Or, the determinant is equal to tlie parallelogram on the alternate units magnified in the ratio of 1 to ab' — h a'. It obviously vanishes it ah' —ha' = 0, or if ajb = a'jV. Iq this case p and q lie, by the property of similar triangles, on the same straight line through o, and therefore, as we should expect, the determinant p . o Q is zero. The reader will find little diflBculty in discovering like properties for a determinant formed from three alternate units. In this case there will be a geometrical relation between certain volumes, which may be ob- tained by stretches in the manner explained on p. 139.' We have in this section arrived at a legitimate interpretation of the product of two directed steps or vectors. We find that their product is an area^ or ac- cording to our previous convention (see p. 134), also a directed step or vector whose direction is perpendicular to the plane which contains both steps of the product* § 16. Another Interpretation of the Product of Two Vectors. The reader must remember, however, that the result of the preceding paragraph has only been obtained hy means of a convention ; namely, by adopting the area of a certain parallelogram as the interpretation of the vector ' I have to thank my friend Mr. J. Eose-Innes for euggesting the intro- duction of thfi above remarks as to determinants. I may, perliaps. be allowed to add that by treating the alternate tnits, like Giassmann, as points, and the alternate number as their loaded centroid, a determinant of the second order is represented geometrically by a length, and we thus obtain for one of the fourth order a geometrical interpretation as a volume. POSITION. 205 product. Only as long as we observe that convention will our deductions with regard to the multiplication of vectors be true. We might have adopted a different convention, and should then have come to a different result. It will be instructive to follow out the results of adopting another convention, if only by so doing we can impress the reader with the fact that the. funda- mental axioms of any branch of exact science are based rather upon conventions than upon universal truths. Suppose then that in interpreting the product A p . A Q we consider A p to be a directed step which IB' Fig. 84. represents the area D b f G. This area will be perpen- dicular to the direction of A p, and we might assume as our convention that the product A p . a q shall mean the volume traced out by the step A Q, moving parallel to itself and in such wise that its end A takes up every possible position in the plane d e p G. This volume will be the portion of an oblique cylinder on the base d b p G intercepted by a plane parallel to that base through q. We have seen (p. 141) that the volume of this cylinder is the product of its base into its height, viz. the per- pendicular distance a h between the two planes. Now let r and p be the scalar magnitudes of A p and a q 206 THE COMMON SENSE OF THE EXACT SCIENCES. respectively, and 6 = the angle paq. Then ah = p cos5, and the volume = ap.aq = rp cos^, for r re- presents the number of units of area in d B p G. Hence, since a volume is a purely numerical quantity having only magnitude and no direction, we find that with this new convention the product of two vectors is a purely scalar quantity, or our new convention leads to a totally different result from the old. Further, since r and p are merely numbers, rp=p r, and thus ap.aq = rpcosO — pr cosd = aq.ap, if A Q be treated as the directed step which represents an area containing p units of area. Thus in this case the vector product obeys the commutative law, which again differs from our previous result. We can then treat the product of two vectors either as a vector and as a quantity not obeying the commutative law, or as a scalar and as a quantity obeying the commutative law. We are at liberty' to adopt either convention, provided we maintain it consistently in our resulting investiga- tions. The method of varying our interpretation, which has been exemplified in the case of the product of two vectors, is peculiarly fruitful in the field of the exact sciences. Each new interpretation may lead us to vary our fundamental laws, and upon those varied funda^ mental laws we can build up a new calculus (algebraic or geometric as the case may be). The results of our new calculus will then be necessarily true for those quantities only for which we formulated our funda- mental laws. Thus those laws which were formulated for pure number, and which, like the postulates of Euclid with regard to space, have been frequently supposed to be the only conceivable basis for a theory of quantity, are found to be true only within the limits POSITION. 207 of scalar magnitude. When we extend our conception of quantity and endow it with direction and position, we find those laws are no longer valid. We are com- pelled to suppose that one or more of those laws cease to hold or are replaced by others of a different form. In each case we vary the old form or adopt a new one to suit the wider interpretation we are giving to quan- tity or its symbols. § 17. Position in Three-Dimensioned Space. Hitherto we have been considering only position in a plane ; very little alteration will enable us to consider the position of a point p relative to a point A as deter- mined by a step A p taken in space. We may first remark, however, that while two points A and B are sufficient to determine in a plane the position of any third point p, we shall require, in order to fix the position of a point p in space, to be given three points A, B, c not lying in one straight line. If we knew only the distances of p from two points A and b, the point p might be anywhere on a certain circle which has its centre on the line ab and its plane perpendicular to that line ; to determine the position of p on this circle, we require to know its distance from a third point c. Thus position in space requires us to have at least three non-collinear points (or such geometrical figures as are their equivalent) as basis for our determination of position. Space in which we live is termed space of three dimensions ; it differs from space of two dimen- sions in requiring us to have three and not two points as a basis for determining position. Three points will fix a plane, and hence if we are given three points a, b, o in space, the plane through 208 THE COMMON SENSE OP THE EXACT SCIENCES. them will be a definite plane separating all space into two halves. In one of these any point p whose position we require must lie. We may term one of these halves below the plane and the other above the plane. Let p n be the perpendicular from p upon the plane ; then if we know how to find the point n in the plane A b c,-the position of p will be fully determined so soon as we have settled whether the distance p n is to be measured above or below the plane. "We may settle by convention that all distances above the plane shall, be considered positive, and all below negative. Further, the position of the point n, upon which that of p, depends, may be M Fig. 85. determined by any of the methods we have employed to fix position in a plane. Thus if N M be drawn perpendicular to a b, we have the following instruction to find the position of p : Take a step a m along A b, containing, say, x units ; then take a step M N to the right I and perpendicular to A B, but still in its plane, contain- ing, say, y units ; finally step upwards from N the distance N p perpendicular to the plane A b c, say, through ^ units. We shall then have reached the same point p as if we had taken the directed step A p. If a; had been negative we should have had to step backwards from a; ii y had been negative, perpendicular to a b only to the left ; if z had been negative, perpendicular to the plane but POSITION. 209 downwards. The reader -will easily convince himself that by observing these rules as to the sign of x, y, z he could get from a to any point in space. Let i denote unit step along A b, j unit step to the right perpendicular to A b, but in the plane a b c, and h unit step perpendicular to the plane a b c upwards, from foot to head. Then we may write xv ■= X . i + y .j + z .Tc, where x, y, z are scalar quantities possessing only magnitude and sign ; but *, 3, Jt are vector steps in three mutually rectangular directions. t / ^> / A / / /n N FiQ. 86. The step a p may be regarded as the diagonal of a solid rectangular figure (a right six-face, as we termed it on p. 138), and thus we shall get to the same point p by traversing any three of its non-parallel sides in succession starting from a. But this is equivalent to saying that the order in which we take the directed steps x.i, y .j, and z .h is indifferent. The reader will readily recognise that the sum of a number of successive steps in space is the equivalent to the step which joins the start of the first to the 210 THE COMMON SENSE OF THE EXACT SCIENCES. finish of the last ; and thus a number of propositions concerning steps in space similar to those we hare proved for steps in a plane may be deduced. By dividing all space into little cubes by three systems of planes mutually at right angles, yve may plot out sur- faces just as we plotted out curves. Thus we shall choose any values we please for x and y, and suppose the magnitude of the third step related in some constant fashion to the previous steps. For example, if we take the rectangle under » and some constant length a, always equal to the differences of the squares on x and y, or symbolically if we take a a = x^—y^, we shall reach p by taking the step AF = x.i + y .j + i- , h. a The series of points which we should obtain in this way would be found to lie upon a surface resembling the saddle-back we have described on p. 89. The above relation between 2, x, and y will then be termed the equation to a saddle-back surface. We cannot, however, enter fully on the theory of steps in space without far exceeding the limits of our present enterprise. § 18. On Localised Vectors or Rotors. Hitherto we have considered the position of a point p relative to a point a, and compared it with the position of another point q relative to the same point A. Thus we have considered the ratio and product of two steps A p and a q. We have thereby assumed either that the two steps we were considering had a common extremity a, or at least were capable of being moved parallel to themselves POSITION. 211 till they had such a common extremity. Such steps are, as we have remarked, termed vector steps. Suppose, however, that instead of comparing the position of two points p and q relative to the same point A, we compared their positions relative to two different points A and B. The position of p relative to A will then be determined by the step ap and the position of Q relative to b by the step b q. Now it will be noted that these steps a p and b q have not only direction and magnitude, bat have themselves position in space. The step A p has itself position in space relative to the step b q. It is no longer a step Fio. 87. merely indicating the position of p with regard to A, but taken as a whole it has itself attained position when considered with regard to the step b q. This localising, not of a point p relative to a point a, but of a step A p with regard to another step b q, is a new and important conception. Such a localised vector is termed a rotor from the part it plays in the theory of rotating or spinning bodies. Let us try and discover what operation will convert the rotor b q into the rotor A p ; in other words : What A P I is the operation , ' ? In order to convert B Q into ^ 1 BQ 212 THE COMMON SENSE OF THE EXACT SCIENCES. A p we must mate the magnitude and position of b Q the same as that of A p. Its magnitude may he made the same hy means of a stretching operation which stretches b Q to a p. This stretch, as we have seen in the case of a quaternion (see p. 195), may he represented hy a numerical ratio or a mere scalar quantity. Next let c D he the shortest distance hetween the rotors A p / -