The Mathematical Theory of Spirit by Stanley Redgrove, 1913

  MATHEMATICAL THEORY OF SPIRIT 

BEING AN ATTEMPT TO EMPLOY CERTAIN MATHEMATICAL PRINCIPLES IN THE ELUCIDATION OF SOME METAPHYSICAL PROBLEMS 

BY 

H. STANLEY REDGROVE B.Sc.(Lond.), F.C.S. 

ASST. LECTURER IN MATHEMATICS AT THE 

POLYTECHNIC, LONDON. W. 

AUTHOR OP 

'•ON THE CALCULATION OP THERMO-CHEMICAL CONSTANTS' “ MATTER, SPIRIT AND THE COSMOS " 

“ ALCHEMY : ANCIENT AND MODERN " ETC. 

59603 

ILLUSTRA TED WITH DIAGRAt^^ 

 

LONDON 

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1913 

 

BALLANTYNE & COMPANY LTD 

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WORKS BY 

MR. H. STANLEY REDGROVE, B.Sc., F.C.S. 

On the Calculation of Thermo-Chemical Con stants. (Arnold, 1909. 6s. net) Matter, Spirit and the Cosmos: Some Sugges tions towards a better Understanding of the Whence and Why of their Existence, 

(Rider, 1910. 2s. 6d. net) 

Alchemy: Ancient and Modem. Being a brief Account of the Alchemistic Doctrines, and 

their Relations, to Mysticism on the one hand, 

and to recent Discoveries in Physical Science 

on the other hand, together with some particu lars regarding the Lives and Teachings of the 

most noted Alchemists. 

(Rider, 1911. 4s. 6d. net) 

Experimental Mensuration: an elementaryText Book of Inductive Geometry. (Heinemann. 2s. 6d.) 

 

PREFACE 

In the following attempt to elucidate certain metaphysical problems by the aid of mathe matics we have constantly had in view the needs of the non-mathematical reader. The inclusion of discussions of certain elementary mathematical principles necessitated there by we consider to be justified by the un fortunate fact that a large proportion of educated people have but a very slight acquaintance with the science of Mathematics. Of course, the chapters of this book dealing more or less exclusively with the discussion of such mathematical principles must not be regarded as in any way performing the functions of a mathematical text-book. In the first place: the discussion is always  

 

subordinated to the peculiar end in view of this book, and we have invariably checked the desire to discuss points of much mathe matical importance and interest but which vii 

viii Preface 

have no particular bearing on our final aims herein. In the second place : when ever we are concerned with some elementary mathematical principle or law, the validity of which is universally admitted by mathe maticians, we give, as a rule, a geometrical  

or other illustration in preference to the formal proof of the law, partly for the reason that to many minds such illustra tions are more interesting, and, in some cases, even more convincing, than a formal proof; and partly for the reason that the reader who desires the strictly logical and formal demonstration of any of these laws can always obtain such from any good mathematical text-book covering the same ground. 

We send this book forth in the hope that, although its readers may criticise it as  speculative, they will not deem it to be  lacking in interest and utility. 

 

We should here mention that part of the first chapter of this work has already ap peared in Morning Light, a theological weekly publication, under the title “ The Law of Correspondences as an Organon of Thought.” 

Preface ix 

Our best thanks are due to Principal Sir Oliver Lodge, F.R.S., for permission to quote a rather long passage from his work  on Mathematics, and to W. G. Llewellyn, Esq., and H. F. Trobridge, Esq., for kind assistance in reading the proofs, &c. 

H. S. R. 

London: January 1912 

 

 

CONTENTS 

PrefacePAGZ 

CHAPTER I 

ON THE DOCTRINE OF CORRESPONDENCES § I. Some Popular Ideas regarding the Spiritual § 2. The Reality of the Spiritual 

§ 3. The Symbolism of the World of Matter § 4. Swedenborg’s Doctrine of Correspondences § 5. The Doctrine of Degrees 

vii 

IS 

17 

19 

21 

25 

§ 6. The Analogical Relation between Cause and Effect 27 § 7, Use and Correspondence 29 § 8. The Correspondence between the Soul and Body of 

Man 

§ 9. Appearance and Reality §10. Spiritual Perception 

§11. Correspondence and Appearance §12. On Mathematical Language 

30 33 34 36 38 

§13. Mathematics and the Philosophy of the Spiritual 40 Note on the Analysis of Consciousness 41 

CHAPTER II 

ON INCOMMENSURABLE QUANTITIES 

 

§14. Powers and Roots 

43 

§15. The Determination of Square Roots 

44 

§16. The Theorem of Pythagoras 

47 

§17. Graphical Determination of Square Roots 

51 

§18. On Incommensurability 

54 

xi 

XU Contents 

faob 

§19. Evaluation of the Base of Napierian Logarithms SS § 20. The Quadrature of the Circle 58 § 21. The Evaluation of " tt “ 60 

CHAPTER III 

ON NATURE REGARDED AS THE EMBODIMENT OF NUMBER 

§ 22. The Pythagorean Theory of Numbers 

64 

§23. §24. 

§25- 526. §27. 

Nature as the Embodiment of Number The Significance of Incommensurability The Discontinuity of Matter 

The Continuity of the Ether 

The Absolute Unit of Measurement 

CHAPTER IV 

ON NEGATIVE QUANTITIES 

66 68 70 73 75 

§ 28. Definition of a Negative Quantity 

§ 29. Examples of the Uses of Negative Quantities § 30. Addition of Negative Quantities 

§31. Subtraction of Negative Quantities 

§ 32. Geometrical Proofs of Multiplication Formulae § 33- Multiplication of Negative Quantities 

§ 34. Division of Negative Quantities 

5 35- Negative Quantities ; not Embodied in Nature § 36. Arbitrary Zero-points in Physical Measurements §37. The Reality of Negative Quantities 

 

CHAPTER V 

ON IMAGINARY " QUANTITIES 

§ 38. The Square Roots of Negative Quantities § 39, The Square Root of a Negative Quantity Physi cally Unintelligible 

§ 40. Powers of i 

$ 41, The Reality of " Imaginary " Quantities 

78 

79 

81 

82 

83 

87 

90 

91 

92 

94 

97 

98 

100 lOO 

Xlll 

fagb 

103 

104 

106 

io8 

in 

113 

114 

116 

Ii8 

119 

121 

123 

 

Contents 

§ 42. The Utility of “ Imaginary ’’ Quantities § 43- The Significance of *' Imaginary " Quantities § 44- A Mathematical Theory of Spirit 

Note on the Geometrical Representation of "Imaginary” Quantities 

CHAPTER VI 

ON THE MATHEMATICAL SOLUTION OF SOME METAPHYSICAL PROBLEMS 

§ 45. The Correspondence between Matter and Spirit § 46. The Distinction between Matter and Spirit § 47. The Origin of Matter 

$ 48. The Relativity of Experiential Knowledge § 49. The Interpretation of Appearance on the Material Plane 

§ 50. The Interpretation of Appearance on the Spiritual Plane 

§ 51. Further Implicates of the Law of Correspondences §52. Conclusion 

 

A MATHEMATICAL THEORY OF SPIRIT 

CHAPTER I 

ON THE DOCTRINE OF 

CORRESPONDENCES 

§ 1. At the present day it appears that the world’s thought is in a transition stage ; everywhere do we find old 

Some 

Popular Ideas regarding the Spiritual. 

theories rooted up, and the later ones which have been planted in their stead appear to be only just about to take root. The 

materialism which so powerfully swayed the philosophic and scientific thought of the nineteenth century has lost its power, and as. yet no other mode of thought exerts a  like sway, though it is clear that that mode which will ultimately gain the victory will be spiritual and idealistic rather than materi  

alistic. It appears that what is needed is an organon of thought whereby we may mentally pass from the realm of matter to that of spirit, such an organon of thought 15 A 

16 Mathematical Theory oj Spirit [§ 1 being essential to the construction of a satisfactory system of spiritual philosophy. There seems, however, to be a tendency either so to materialise the idea of spirit that the distinction between matter and spirit vanishes, or else so to refine and tran scendentalise the idea of spirit that one loses hold of it entirely. Both of these extremes of thought, neither of which can be regarded as satisfactory, are to be found, mixed in inextricable confusion, in the popular theo logy as well as the common superstitions, not only of the past, but also of the present time. As. examples of such super stitions we may notice two curious old Breton customs. On the eve of All Souls’ Day, it is thought by the superstitious Bretons that the dead come to visit the living. The people, therefore, gather in the church yard and sing a plaint or “ gwerz,” afterwards returning to their homes to talk of their dead; and before retiring to bed, the house wife prepares a meal of pancakes, cream 

 

and cider wherewith the spirits may refresh themselves ! The other custom is as follows : On St. John’s night the “ Tantad ” or bonfire is lighted, around which the people assemble whilst the Litany for the Dead is recited. At the conclusion they march 

§ 2] The Doctrine of Correspondences 17 around in silence, and at the third round every one throws a flint into the embers. It is supposed that the spirits of the dead come to warm fhemselves amongst the rem nants of the “ Tantad,” and whoever finds his stone turned over in the morning knows that he will die during the year! * Similar ideas regarding the spirits of the dead, namely, that they need material food and drink, and the warmth of material fires, are very widespread; whence it follows, of course, that spirits are thought of as material beings, though of an attenuated and impal pable nature. Indeed, “ matter deprived of its substance ” describes the popular con cept of a spirit as well as such a vague notion can be described. The fact that such an expression reduces to nothing and is practi cally meaningless is altogether in harmony with the common ideas on the subject. 

§ 2. In many minds the terms “spirit” 

 

The Reality of the 

Spiritual. 

before that 

and “ spiritual ” are associated with no other idea than that of a discarnate personahty; but, in truth, a man is a spirit great experience which we call 

I See Dealings with /he Dead (1898), translated by Mrs. A. E. Whitehead from Le Braz’s La Legends de la Mort en Basse Bretagne. 

Mathematical Theory 0/ Spirit [§ 2 death,” equally as after. By the *« spiritual ” we understand the mental, the psychical, the ideal. Affection, thought, ideas, the things of the mind—these are Spiritual, and of such does the spiritual world consist, the spiritual world within the mind of man, and the spiritual world x>f which the inner man is a part and in which he exists. Very generally, ideal existence  appears to be regarded almost as non existence—^the things of the mental plane, such as thought and affection, are hardly admitted to be real. But as Descartes long ago made plain,®* such things, the things of the spiritual world, are firmer reahties than the things of the material world. We can doubt the existence of an external, material object corresponding to some sensation that we experience, we cannot doubt the existence of the sensation itself. Materialism would reduce all spiritual existences to terms of matter and motion; ” the brain secretes consciousness,” we are told, ” as the fiver 

 

secretes bile.” But the mere fact that we know matter and motion only in terms of consciousness, that our knowledge of matter and motion is necessarily ideal, is in itself 

* See Descartes’ Discourse on Method and Meta physical Meditations. 

§ 3] The Doctrine of Correspondences 19 

sufficient to render untenable the materialistic theory. Nor is that extreme form of ideahsm satisfactory which denies the existence of an external world, for such a world is de manded by the harmony of experience. “ Both matter and spirit are true and real ” 

we have said elsewhere, “ but on different planes of being.” • The question arises.What is the relation or connection (if any) between them ?—a question the answer to which would supply the needed organon of thought referred to above. Shall we accept the thorough-going Dualism which regards matter and spirit as conflicting elements, out of harmony with one another and ever opposed ? Not so : for this theory can never give a satisfactory explanation of the Cosmos ; for, by creating an irreconcilable antinomy be tween matter and spirit, so far from ex plaining the great Riddle of the Universe, it denies the possibility of an explanation. 

§ 3. There have, however, been some philo 

 

The Sym bolism of the World of 

Matter. 

sophers who, whilst regarding matter and spirit as quite dis tinct from one another, have held that the spiritual is prototype of the material, the material 

symbolical of the spiritual, and hence, that * Redgrove ; Matter, Spirit and the Cosmos (1910), § 86. 

20 Mathematical Theory of Spirit [§ 3 the worlds of spirit and of matter are closely related by the law of analogy. This view of the Cosmos is one of considerable antiquity. It is to be found in the writings of Plato, for whom this world was constructed after the pattern of the Archetypal, Ideal World. It is to be found in the body of literature known as the Kabalah—questionable with regard to authorship, but undoubtedly embodying the ancient traditions of Jewish philosophy—in  the works of the Neo-Platonists, and in the writings of the sixteenth- and seventeenth century theosophists and transcendentalists, Cornelius Agrippa for example, who drew mainly from these two sources ; for the doc trine necessarily follows from the emana tional theory of the origin of the Cosmos, which is the keynote of all these systems. All true mystics have caught a glimpse of it, and have more or less dimly realised and taught that Nature is a mighty parable speaking eloquently to the inward-seeing eye 

 

of the realities of the Spiritual Universe. And we may add that its truth is implied in  all true poetry, for all true poetry partakes of the spirit of mysticism. Of Peter Bell we read that : 

§ 4] The Doctrine of Correspondences 21 A primrose by a river's brim 

A yellow primrose was to him, 

And it was nothing more^ 

But in a little primrose-tuft the poet sees : 

A lasting link in Nattire's chain 

From highest heaven let down ! ® 

In all these places, however, in Plato and Plotinus, in the Zohar and Agrippa of Nettes heim, in the mystics and poets, the doctrine of the symbolism of matter is but compara tively vaguely expressed, its significance only dimly realised.® It is in the Doctrine of Correspondences of the Swedish philosopher, Emanuel Swedenborg, that we find this view of the relation between matter and spirit first explicitly stated and clearly developed to its logical conclusions. Here, for the first time, is the principle of analogy between the material and the spiritual rendered definite and exact. 

§ 4. We find the germs of this doctrine,  

which was afterwards so fully developed by Swedenborg, in his earlier philosophical works. Thus, in the Animal Kingdom 

* Wordsworth : Peter Bell. 

® Wordsworth : The Primrose oj the Rock. 

* Modern philosophical inquiry has rendered Plato’s  ideal theory, for example, quite untenable. 

22 Mathematical Theory oj Spirit [§ 4 he says, “ In our Doctrine of Representa tions and Correspondences, we shall treat of both these symbolical and 

typical representations, and of 

Correspondences, the astonishing things which occur, I will not say in the living 

body only, but throughout nature, and which correspond so entirely to supreme and spiritual things, that one would swear that the phy sical world was purely symbolical of the spiritual world: inasmuch that if we chose to express any natural truth in physical and definite vocal terms, and to convert these terms only into the corresponding spiritual terms, we shall by this means elicit a spiritual truth or theological dogma, in place of the physical truth or precept; although no mortal would have predicted that anything of the kind would possibly arise by bare literal transposition, inasmuch as the one precept, considered separately from the other, appears to have absolutely no relation to it.” ’ And, 

’ Emanuel Swedenborg : The Animal Kingdom,  

considered Anatomically, Physically, and Philosophically (translated by J. J. G. Wilkinson, 1843), vol. i. p. 451 (footnote «). Commenting on this passage Emerson says : “ The fact, thus explicitly stated, is implied in all poetry, in allegory, in fable, in the use of emblems, and in the structure of language. Plato knew of it, as is evident from his twice bisected line, in the sixth book of the Republic. Lord Bacon had found that 

§ 4] The Doctrine of Correspondences 23 again, in the Worship and Love of God he writes, “ . . . for such is the established correspondence, that by natural and moral truths, by means of the transpositions only of the expressions that signify natural things, we are introduced into spiritual truths, and vice versa. . . . For the sake of illustration, let one or two examples suffice, as first. Light reveals the quality of its object, but the quality of the object appears according to the state of the light, wherefore the object is not always 

truth and nature differed only as seal and print ; and he instanced some physical propositions, with their translation into a moral or political sense. Behmen, and all the mystics, imply this law, in their dark riddle writing. The poets, in as far as they are poets, use it; but it is known to them only, as the magnet was known for ages, as a toy. Swedenborg first put the fact into a detached and scientific statement, because it was habitually present to him, and never not seen. It was involved ... in the doctrine of identity and iteration, because the mental series exactly tallies with the material series. It required an insight that could rank things in order and series ; or, rather, it required such rightness of position, that the poles of the eye should coincide with the axis of the world. The earth had fed 

 

its mankind through five or six millenniums, and they had sciences, religions, philosophers ; and yet have failed to see the correspondence of meaning between every part and every other part. And, down to this hour, literature has no book in which the symbolism of things is scientifically opened. One would say, that, as soon as men had the first hint that every sensible object— animal, rock, river, air—nay, space and time, subsists 

24 Mathematical Theory of Spirit [§ 4 szich as it appears ; as in the case of beauties, if they are objects viewed in varied light. Now if instead of light we take intelligence, the quality of the object of which is the truth of a thing; since intelligence is univers ally allowed to be spiritual light, this con clusion follows: Intelligence discovers the truth of a thing, bid the truth of a thing appears according to the state of the intelligence ; where 

fore that is not always true which is supposed to be true. In like manner, if instead of in telligence, wisdom be called into corre 

not for itself, nor finally to a material end, but as a picture-language to tell another story of beings and duties, other science would be put by, and a science of such grand presage would absorb all faculties ; that each man would ask of all objects what they mean : Why does the horizon hold me fast, with my joy and grief, in this centre ? Why hear I the same sense from countless differing voices, and read one never quite expressed fact in endless picture-language ? Yet, whether it be, that these things will not be intellectually learned, or that many centuries must elaborate and compose so rare and opulent a soul—there is no comet, rock-stratum, fossil, fish, quadruped, spider, or fungus that, for itself, does not interest more scholars and 

 

classifiers than the meaning and upshot of the frame of things.” {Representative Men. “ Swedenborg ; or, the Mystic.”) Emerson, we should note, was opposed to Swedenborg’s theological views, hence he was not satisfied (as may be inferred from the above) with his working out in detail of the Doctrine of Correspondences. It would be beyond the limits of this work, however, to attempt a substantiation of these details. 

§ 5] The Doctrine of Correspondences 25 spondence, the object of which is good; it then follows, Wisdom manifests goodness, bitt the goodness of a thing appears according to the state of the wisdom ; wherefore that is not always good that is believed to be good.” * This, together with some twenty similar illus trations of correspondences, will also be  found in his little work entitled A Hiero glyphic Key to Natural and Spiritual Mysteries. 

§ 5. We see, therefore, that if valid, the Doctrine of Correspondences constitutes an organon of thought whereby we 

from the realm of matter 

to that of spirit, transmuting 

with a wonderful alchemy truths relating to the physical realm into those that are spiritual. If correspondence implied but an analogy of an arbitrary nature, a mere resemblance without any real connection, then the doc trine would be of little worth. But, as we shall see more fully later, correspondence implies and depends upon a very real con nection, the causal relationship of spirit to 

 

matter. It will be of interest, therefore, to examine Swedenborg’s Doctrine of Corre spondences somewhat more fully ; and, for a 

8 Emanuel Swedenborg : On the Worship and Love oj God (translated [by John Clowes], 1885) § 55 (footnote s), p. 107. 

26 Mathematical Theory of Spirit [§ 5 satisfactory discussion of this doctrine, it will be necessary, in the first place, to say  something with regard to his Doctrine of Degrees. 

Swedenborg distinguished between two sorts of degrees, which are to be found in every thing that exists—“ continuous degrees,” and “ discrete degrees.” The first are such as  

gradually merge one into the other, as light merges into darkness and heat into cold. Discrete degrees, however, do not merge one into the other, but are related as end, cause and effect; and, although the end exists always in the cause, and both end and cause exist always in the effectj yet the three are perfectly distinct, or as Sweden borg puts it, “ discrete.” Taking the ex ample of affection, thought and speech: the perfect distinction between the three is clear enough; affection does not merge into thought as heat merges into cold, nor does thought merge into speech as light merges into darkness; but for affection to exist as 

 

an end, it must exist in thought; and for thought to exist as a’cause, it must exist and be manifested in speech or some other form of outward expression. Now, accord ing to Swedenborg’s philosophy, in God is the most universal End, or End of ends, and the 

§ 6] The Doctrine oj Correspondences 27 Spiritual Universe is related to the Physical Universe as cause is to effect. Therefore, since a “ discrete ” degree separates matter and spirit, they are perfectly distinct and do not merge one into the other. This dis tinction between matter and spirit is of a nature altogether different from that be tween two forms of matter, such, for example, as the solid and gaseous states. It resembles, rather, the distinction between matter and the ether of space, on the theory that matter is a sort of etheric phenomenon. Matter is, indeed, in the ultimate analysis, a spiritual phenomenon; or, as Carlyle so well puts it, “AU visible things are emblems; what thou seest is not there on its own account; strictly taken it is not there at all: Matter exists only spiritually, and to represent some Idea, and body it forth.” • And since spirit stands in the closer relationship to the One Only Real and Absolute Substance, that which is spiritual must be regarded as more real, more substantial than that which is 

 

material. 

§ 6. Now, between end, cause and effect there is an exact analogical relationship; for example, every affection has certain 

• Thomas Carlyle : Sartor Resartus, Bk. I., ch. xi. 

28 Mathematical Theory of Spirit [§ 6 thoughts which are in harmony with it and immanent in which it exists, and for every thought there are spoken 

The Analogical and written words which exactly betSeen cjuse represent it and by means of and Effect, which it is outwardly expressed. It is this relation that is called 

" correspondence.” Hence, as the spiritual universe is related to the physical universe as cause is to effect, everything of the one corresponds to everything of the other— everything physical has its correspondent on the spiritual plane, everything spiritual its correspondent on the physical plane; everything physical is a symbol, everything spiritual a prototype. We must not suppose, however, that this doctrine implies that there are no natural causes; but that such flow from spiritual causes. “ There is nothing any where in the natural world which is in order,” writes Swedenborg, “ but derives its cause and origin from the spiritual world, that is, through the spiritual from the Divine.” “ We see there 

 

fore that correspondence implies a similarity of inner relationship between the universes of spirit and matter. For, since the materialworld exists as an effect flowing from the spiritual, 

Emanuel Swedenborg: Arcana Ccelestia § 8211 (English translation, 1893, vol. x. p. 71). 

§ 7] The Doctrine of Correspondences 29 constructed after a like plan, the things of the material world stand related to one another and to the whole of this world like as do the things of the spiritual world to one another and to the whole of that world. Those things that correspond are they which, existing on “discrete” planes of being, i.e., planes which are related as cause is to effect, stand in the same relations to their respective planes of being. 

§ 7. Now, the uses or functions of any thing depend upon the relations which it bears 

Use and Corre spondence. 

to the whole of its environ ment. From this it follows that those thingsthat correspond have analogous uses or functions with 

reference to their respective planes of being ; and this supplies us with a criterion for de termining correspondences. As an example of this, let us consider the correspondence between foods and ideas. Just as various foods are necessary for the nourishment of the body, so are various knowledges or ideas 

 

necessary for the psychical nourishment of the mind. The body is built up of the foods eaten, but the body is not merely a conglomeration of foods ; consisting of the various elements of food, the body is made up of these elements woven into a new form— 

30 Mathematical Theory of Spirit [§ 8 the body is a harmonious unity, a living organism. So also is the mind built up of ideas, and yet the mind is not merely a col lection of ideas, for the ideas within the mind are also woven into a (more or less) harmonious unity—the mind is a living spiritual organism. Moreover, just as there are pseudo-foodsthat are injuriousto the body, and bring discord amongst its elements,so also there are pseudo-knowledges that poison the mind, and produce discord amongst its ele ments. We see, therefore, that the functions of foods on the physical plane are analogous to those of knowledges or ideas on the spiritual plane, or, putting it otherwise, that the part played by foods in the economy of the natural world is similar to the part played by knowledges or ideas in the economy of the spiritual world. This is precisely what is meant when it is said that ideas and foods correspond. 

§ 8. Not only can this exact analogy or  symbolism be traced between 

 

The the things of the spiritual and physical worlds in general, but 

Soul and Body also between the soul and body of Man. of man; for man considered spiritually is a microuranos—a  

Heaven in miniature—and with regard to his 

§ 8] The Doctrine of Correspondences 31 outward nature, as the old mystical philoso phers taught, a microcosm or little world. Considering, for example, the two most important organs of the body, the heart and lungs: these correspond, according to Sweden borg, to the affectional and volitional, and the intellectual sides of man’s spiritual nature respectively.” With regard to the first, the likeness in function is very evident, for just as the heart is the mainspring of man’s physical life, so is his will or love the main spring of his spiritual life. With regard to the second, we should notice that the func tion of the lungs is the cleansing and nourish ing of the blood, and it is by the knowledge of what is good and true that man is cleansed from evil desires and spiritually vitalised.” 

See “ Note on the Analysis of Consciousness ” at the end of Chapter I. 

With reference to the correspondence between the soul and body of man we may, perhaps, quote the words of the Swedish philosopher himself: “ Who is not aware,” he says, “ that affection and thought are spiritual, and therefore that all things of affection and  

thought are spiritual ? Who is not aware that action and speech are natural, and therefore that all things of action and speech are natural ? Who is not aware that affection and thought which are spiritual cause man to act and to speak ? Who may not see on these grounds what the correspondence of spiritual things with natural is ? Does not thought make the tongue speak, and affection together with thought make the 6 

32 Mathematical Theory of Spirit [§ 9 § 9. The assumption that appearance and reality are identical, that things are as they 

body act ? There are two distinct things here. I can think and not speak, and I can will and not act. And it is known that the body does not think and does not will, but that thought falls into speech, and will into action. And does not affection shine forth out of the face, and present a type of itself there ? Every one knows this. Is not affection regarded by itself spiritual, and are not the changes of the face, the looks as they are called, natural ? Who might not conclude from this that there is a correspondence ; and hence that there is a correspondence of all things of the mind with all things of the body ? And because all things of the mind are referrible to affection and thought, or what is the same thing to the will and understanding, and all things of the body to the heart and the lungs, it might have been concluded that there is a correspondence of the will with the heart and of the understanding with the lungs. The cause why these things have not been known, although they might have been known, is that man has become so external that he has chosen to acknowledge nothing except the natural. This has been the delight of his love, and therefore it has been the delight of his understanding; until it has become painful to him to elevate thought above the natural to anything spiritual separate from the natural. And so of his( natural love and its delight he could not do otherwise than think that the spiritual is a purer natural, 

 

and that correspondence is a somewhat flowing in by continuity. Yea, the merely natural man is not able to think of anything separate from the natural J such a thing to him is nothing. Another cause why these things have not been first seen and then known hitherto is. that all things of religion, all spiritual things, have been removed from the view of man by the dogma, prevailing in the whole Christian World, that the 

§ 9] The Doctrine of Correspondences 33 

are perceived, although common enough, is nevertheless one that will not stand examina tion.^® Not only does the ap- SEJrS pearance of objects depend upon the conditions under which they 

are observed, but also upon the sense-organs of the observer. Confining our attention 

Theological things which are spiritual, which Councils and certain leaders have decreed, are to be blindly believed, because as they say they transcend the understanding. Whence some have supposed the spiritual to be like a bird which flies over the air in an ether to which the sight of the eye does not reach ; when yet it is as a bird of paradise, which flies near the eye, and touches its pupil with its beautiful wings and longs to be seen. By the sight of the eye is under, stood the intellectual sight.” (Emanuel Swedenborg, Angelic Wisdom concerning the Divine Love and con cerning the Divine Wisdom, §374, translated by J. J. Garth Wilkinson, 1885.) We should like to substitute “physical” for “natural” throughout the above pas sage, as more correctly rendering Swedenborg’s meaning in modern terminology. 

The term “appearance” is here employed in its widest possible meaning, not as having reference only to visual sensations, but as implying the sum of all sensations which are in any way attributable to the 

 

external object in question. Of course, in a sense, an appearance is a reality. An appearance is real as an appearance, it is real as existing in the mind of the observer.- The distinction in question, however, is one between objective and subjective reality, implying by these terms nothing more than reality which is true for all minds and that which exists for the individual mind alone, respectively. 

34 Mathematical Theory of Spirit [§ 10 for the moment to the consideration of visual percepts, as Swedenborg points out in the passage quoted in § 4, the appearance of objects depends upon the light in which they are observed. In this connection it is of interest to notice that researches in the domain of physical science have demon strated the existence of ether-waves which have no effect upon our organs of sight. The  colours of objects would appear quite dif ferent to a being whose eyes were sensitive to these ultra-violet rays, but unaffected (say) by red light rays, from what they appear to us. Indeed, one might go so far as to say that the appearances of objects depend equally upon the observer as upon the thing observed. The sense-organs of all men and women are very much the same, so that the world as it appears to each one of us does not greatly differ, but no two men or women see it exactly ahke, because no two men or women see Nature through precisely the same spectacles. 

 

§ 10. Now, according to Swedenborg’s philosophy, just as the outward man possesses physical senses whereby he may 

Perraptira. t>ecome informed of his material environment, so likewise the 

inward man possesses spiritual senses where- 

§10] The Doctrine of Correspondences 35 by he may become informed of his spiritual environment. During man’s earth-life, how ever, save for somewhat rare exceptions— under which category the present writer would include all those phenomena called “■ telepathic ”—these inward senses are opened only to the small spiritual world of his own mind. We can directly perceive our own thoughts ; we can perceive the thoughts of others only in so far as they are expressed in a form which appeals to the outward senses. We can, as it were, see our own souls; we can see only the outward forms of other men  and women. With that transition ofthe sphere of manifest activity and consciousness that we call “ death,” however, the wide spiritual world will open to our view, but simultane ously the physical world will be shut out, for just as the physical senses are affected only by that which is physical, so likewise are the senses of the spirit affected only by that which is itself spiritual. Further, the appearance of things spiritual depends upon 

 

the spiritual state of the percipient, just as  , the appearance of things material depends upon the state of the physical sense-organs. That this is actually so appears from such facts as that an unintelligent person cannot “ see ” the truth of some well-reasoned 

36 Mathematical Theory oj Spirit [§ 11 

statement, whilst fallacies appeal to him as true ; and that the evil-minded man “ sees ” good things as evil and undesirable, but evil things as good and greatly to be desired ; whilst the man who is truly wise sees good things as good, evil as evil, true things as  true, false as false. 

§ 11. It also follows from the Doctrine of Correspondences that those things that corre spond are similar in appearance. 

Correspondence spirit-world, could we per Appearance. ceive it with our spiritual sight, would appear very much like 

this world of matter, its inhabitants like men and women here. In the spiritual world, however, the outward form of man 

In perception of the things of the physical world both the inward man and the outward man are involved. Hence, differences in our percepts of the same external object arise partly on account of differences in our sense-organs, &c., and partly on account of differences of a spiritual nature. In the above example of the depentjence of percepts of physical things on the nature of the sense-organs, in which we supposed a being whose eyes were sensitive to ultra-violet light-rays, 

 

the effect of the spiritual state of the percipient is a negligible factor. In Matter, Spirit, and the Cosmos, §46, where the subject of the relativity of spiritual perception is under discussion, a case is considered in which differences in perception due to differences in the sense-organs of the percipients are negligible in com parison with differences in perception due to differences in the spiritual states of the percipients. 

§11] The Doctrine of Correspondences 37 corresponds exactly with his inward state, and the environment of each spirit is also in strict harmony with that spirit’s inward state. There, the external is the exact expression of the internal, the outward world  

as perceived is the objectification of the sub jective world within the mind. Such ideas as these regarding the nature and appear ance of the spiritual world may appear to be highly speculative, but they do follow logically from the fundamental statement of the Law of Correspondences, and if we admit the validity of Swedenborg’s seership, for which there is considerable evidence, they may be regarded as based upon ex perience. It is also of interest to note that they are in agreement with many state ments said to be made by discarnate spirits through the mediumship of various “ psy chics ”—testimony, it is true, which needs much careful and critical examination, and certainly should be received in no credulous spirit.'® 

 

The Doctrine of Correspondences states the constancy of the ratio between matter and spirit, a ratio which is inexpressible in 

15 For a critical discussion of Modern Spiritualism and Psychical Research, see the present writer’s Matter, Spirit, and the Cosmos (Rider, 1910), Ch. 11. 

38 Mathematical Theory of Spirit [§ 12 

terms of experience, for spirit perceives only that which is spiritual, and the physical senses are'affected only by the things of the physical world—a. ratio, which, from a cer tain point of view, may be said to be unreal. As Louis Claude de Saint-Martin put it, “Matter is true for matter, and never for spirit.” “ We might add. Spirit is true for spirit, and never for matter. Betweeifmatter and spirit it is as if a great gulf were fixed, and it is this gulf which the Law of Corre spondences claims to enable us mentally to bridge over. 

• • • • • 

§ 18. Without going to the extent of Prof. Max Muller’s theory " and maintaining the absolute dependence of 

thought on words, itmust, never 

theless, be confessed that lan 

guage plays a very important part in the development of thought, and that one of the necessities of any system of philosophy is an adequate language, not only by means of  

which it may be expressed, but also by means of which it may be thought out. Now, there is no language so explicit, no 

“ A. E. Waite : The Lije of Louis Claude de Saint-Martin, the Unknown Philosopher (1901), p. 185. See F. Max Muller : Three Introductory Lectures on the Science oj Thought (1888). 

§18] The Doctrine of Correspondences 39 language so univocal, no language so terse and yet so full of meaning, as the language of mathematics. Such is the nature of mathematical language that it may be said that no knowledge is worthy of being called scientific until it is expressed therein. The  justice of this statement will become ap parent if we compare the state of certain of the sciences (e.g., Chemistry) to-day with their condition but a generation ago, and consider the fact that the enormous advances which have been made are due in large measure to the application to these sciences of mathematical principles and methods. 

The thought occurs to us. Might not mathematics be advantageously employed in the domain of Metaphysics ? How great an advantage it would be if, indeed, the relation between matter and spirit could be represented mathematically, if a mathe matical organon of thought were forthcom ing whereby we could mentally pass from 

 

the physical realm - to the spiritual, and deduce, thereby, spiritual principles from natural laws ! 

§ 18. Now, such, in brief, are the objects of the present work; herein we shall attempt to formulate a mathematical organon of 

40 Mathematical Theory of Spirit [§ 13 thought whereby we can mentally pass from the realm of the physical to that of the spiritual, and by means of this 

organon of thought, to arrive at Mathematics and the 

Philosophy and establish certain spiritual or of the 

Spiritual. metaphysical principles. Whether or not we are successful in these 

attempts must be left to the reader’s decision after he has perused the volume. We have discussed Swedenborg’s Doctrine of Correspon dences at some length above (though we have by no means exhausted all of its aspects), for the reason that the results obtained in this endeavour to construct a mathematical theory of the spiritual are in remarkable agreement with the several implicates of this doctrine. To those, therefore, who regard Swedenborg’s Law of Correspondences as  valid, we offer the present volume as an attempt to cast this law into a mathematical mould, whereby it may gain in precision and useful^ness. At the same time, however, it will be our aim, herein, to establish our theory 

 

by an entirely different line of argument, so that the reader is not asked to accept Swedenborg’s doctrine before proceeding with the book. If our arguments are sound, how ever, they may fairly be regarded as affording some evidence of the vahdity of this doctrine. 

The Doctrine of Correspondences 41 

Note on the Analysis of Consciousness Swedenborg’s analysis of consciousness is dual, as distinguished from the more modern, triple analysis into Cognition (=knowing). Feeling-attitude, and Conation (=wilhng). It is interesting to note, however, that, as Prof. Stout remarks, “ Of late there has again arisen a tendency to fall back upon a dual division, bringing Feeling-attitude and Cona tion under the same head.” “ It is clear,” proceeds this eminent psychologist, “ that they are more closely akin to each other than either of them is to Cognition. It also seems clear that there is as fundamental a distinction between the bare thought of an object and the affirmation or denial of its reality as there is between Feeling-attitude and Conation. The best plan is to adopt a most comprehensive dual division into ‘ Cog nition ’ on the one hand and ‘ Interest ’ on the other. Cognition may then be sub divided under the heads. Simple Apprehen 

 

sion and Judgment; and Interest may be  subdivided under the heads Conation and Feeling-attitude.” (G. F. Stout : The Ground work of Psychology, 1905, pp. 18-19.) It must be understood, of course, that all such analyses are purely ideal. Cognition, Feeling- 

42 Mathematical Theory of Spirit attitude, and Conation cannot be actually separated, for all three exist together (though in varying degrees of intensity) in every moment of consciousness. But no more  can the organs of the hving body be separated, for with their removal they cease to be hving organs and become merely inert masses of matter. 

 

CHAPTER II 

ON INCOMMENSURABLE QUANTITIES 

§ 14. The continued product of a number with itself, that is to say, the product ob 

Powers and Roots. 

tained by continually multiply ing a number by itself, is called  a power of that number. Thus, 

3 3 3 

X 3 X 3 X 3 

is 

X 3 X 3 

called the second power of 3; is called the third power of 3 ; X 3 is called the fourth power of 

3, &c. 

The second and third powers of a 

number are termed also the square and cube of that number, respectively. The second power or square of 3 is written 3®; the third power or cube, 3®; the fourth power, 3*; and so on. The small figure (written above and to the right of the number) which ex presses how often that number is repeated to form the power, is called its index or ex 

 

ponent. In general— 

The second power or square of x is written and equals x x x ; 

The third power or cube of x is written and equals x x x x x ; 

'13 

44 Mathematical Theory of Spirit [§ 15 The fourth power of x is written and equals x x x x x x x, and so on. 

The process of determining the value of a  power is called Involution. 

The number which gives rise to a power is called its root. Thus 3 is said to be the square root of 9, since 9 is the square of 3 ; 2 is said to be the cube root of 8, since 8 is the cube of 2, &c. The symbol 7 is used to denote the root of a number; thus, the square root of 9 is written ^9, or more simply 79, the 2 in generally being omitted; the cube root of 8 is written <8; similarly, a fourth root is indicated by the sign V, and so on. The symbol 7 is called the radical sign. The process of deter mining the value of a root is called Evolution ; hence involution and evolution are inverse  processes, i.e., one undoes the effect of the other. 

§ 15. Methods for the extraction of square and certain other roots will be found in any text-book of Arithmetic. Here, 

 

shall be concerned only with 

Square Roots, square roots. Now, in the case of certain numbers no difficulty 

whatever is experienced in determining exactly their square roots. Thus, for ex ample, ^4 = 2 exactly, 79=3 exactly, &c.. 

§15] Incommensurable Quantities 45 or, as an example of a larger number,, it is found that 77569=87 exactly, &c. Such numbers, however, are the exception rather than the rule; it is found that the exact square roots of the vast majority of whole numbers (all whole numbers, in fact, other than those which can be obtained by multiplying an integer by itself) cannot be exactly expressed either as integers or  

as the ratios between pairs of integers, i.e., either as whole numbers or as vulgar or decimal fractions (terminating or recur ring), the decimals obtained in the attempt to extract such roots by the usual arith metical process never terminating and never recurring. 

This may be proved as follows: 

A whole number can be obtained by multiplying together two vulgar fractions * only in the case in which the denominator of one fraction is a factor of the numerator of the other, and the denominator of the other a factor of the numerator of the first (the 

 

fractions both being expressed in their lowest terms); that is to say,— x —equals a whole 

1 It should be borne in mind that a decimal fraction (so long as it terminates or recurs) can always be con verted into a vulgar fraction ; so that the above proof holds good with reference to such decimal fractions. 

46 Mathematical Theory of Spirit [§ 15 number only if b is a factor of x and y a factor of «; as in the case, for example, of x 

which equals 6 exactly, since i6 -? 8 =2, 154-5=3, and 2x3=6. Now, in the case of a fraction multiplied by itself, if this holds good it is clear that the denominator of the fraction must be a factor of its own numer^itor; thus, referring to the algebraical illustration above, since, in this case, fl=%, b—y and b is a factor of x, it is clear that b is a factor of a; the fraction, therefore, reduces to a  whole number. Consequently, the product obtained by multiplying a fraction by itself cannot be a whole number. Therefore, the root of a whole number cannot be a fraction. But clearly 73, ^5, &c., are not whole numbers, whence it follows that ^2, s/3, 5/5, &c., cannot be expressed exactly either as whole numbers or as fractions. Such roots as these are termed surds, or irrational quantities. 

It is possible, however, by the usual arith  

metical processes for extracting roots, to obtain the approximate value of the root of any number expressed as a decimal (which can, of course, be converted into a vulgar fraction if desired) to any required degree of accuracy. Thus, for example, it is found 

§16] Incommensurable Quantities that, correct to six decimal places {i.e.^ one millionth part of a unit) 5/2=1.414214; we know, therefore, that 5/2 is greater than 1.41421, but less than 1.41422. For most purposes the approximation 1.4142 is sufficiently accurate. Similarly in the case of other roots approximate values can be obtained. 

§ 16. The square root of any number can always be represented, by a simple geo metrical construction, as a 

oVpySra“ straight line, the length of which, measured in terms of some other 

straight fine, shall be equivalent to the surd in question, as we shall show later. It should be noted, however, that the like can not be done in the case of irrational cube roots. 

The discovery of the mathematical principle or law employed in the graphical representa tion of irrational square roots is attributed to Pythagoras ; it is, therefore, known as the Theorem of Pythagoras. This theorem states  

that in any right-angled triangle the square on the hypotenuse {i.e., the side opposite the right angle) is equal in area to the sum of the squares on the two other sides. Thus, in Fig. i, for example, we find by actual measurement 

that: 

c 

48 Mathematical Theory of Spirit [§ 16 

AB (hypotenuse) = 3.94in.,therefore,sq. on AB = 15.52 sq. in.’ AC=2.24in., ,, sq. onAC= 5.02 sq. in. 

BC“3.24in., ,, sq.onBC = io.5osq.in. 

Therefore, sum of sqq. on AC and BC = i5.52sq.in. = sq, on AB. 

As a furtherillustration of the truth of this  

’.These values are given correct to two decimal places only, since the original measurements were not carried beyond a greater degree of accuracy. By carrying out a large number of like measurements on various right-angled triangles, we could experimentally or inductively prove the approximate truth of Pytha goras’s theorem, which is all that is necessary for practical purposes. 

§16] Incommensurable Quantities 49 theorem, we show below how the two squares on the sides of any right-angled triangle at right angles to one another, can be divided up into five pieces, which can be arranged so as to cover exactly the square on the hypotenuse. 

Draw any right-angled triangle, ABC, the right angle being at C. Draw a square on each of its sides. Produce the sides of the square on the hypotenuse so as to cut the sides of the other squares in D and E re spectively, the point E being in the side of the larger square {see Fig. 2). At E draw a straight fine perpendicular to BE, cutting the adjacent side of the square in F. Cut out the two smaller squares, and divide them into five pieces along the lines AD, BE, EF. It will be found that if these five pieces are correctly arranged (as indicated by the dotted lines in Fig. 2) they will exactly cover the square on the hypotenuse, showing that the area of this square is equal to the sum of their areas, and, therefore, equal to the sum of 

 

the areas of the two squares which they together make up.® 

The proof of the Theorem of Pythagoras 

• There are several other methods of dividing up the squares whereby the truth of the theorem may be verified. 

50 Mathematical 'Theory of Spirit [§ 16 was probably the chief aim of the first book of Euclid’s Elements of Geometry. The the 

 

orem constitutes proposition 47 of this book, to which we refer readers who are not ac quainted with Euclid’s proof.* 

* Euclid aimed at proving the absolute, not approxi mate, truth of the laws of geometry, and, therefore, adopted the deductivemethod. It is possible, however. 

§ 17] Incommensurable Quantities 51 § 17. Now, let us draw a right-angled triangle, making the two sides which are „ ,. , at right angles to one another Determination exactly I in. long {see Fig. 3). of Square Then, clearly, the area of the square on the hypotenuse 

equals i sq. in. -h i sq. in., i.e., 2 sq. in. 

 

(That thjs is actually so can readily be seen by dividing the square on the hypotenuse 

to question the absolute truth of the fundamental assumptions of the Euchdean geometry ; though their practical or experimental truth is universally admitted. 

52^ Maihematical Theory of Spirit [§ 17 

into four equal triangles, each of which is one-half of a one-inch square, as shown by the dotted fines in the diagram). It follows, therefore, that the length of the hypotenuse of this triangle is inches. Similarly we can represent any other irrational square 

root. Thus, to draw a line ^3in. in length.— Draw a right-angled triangle, making the two sides which are at right angles to one 

 

another i in. and in. long respectively. (To draw the side which is to be in. long, proceed as above, constructing first a right angled triangle in which the sides at right angles to one another are each i in. long. The hypotenuse of this triangle is then used 

§ 17] Incommensurable Q,uantities 53 as one side of the new triangle, as in Fig. 4-) Then, clearly the area of the square on the hypotenuse of this triangle equals i sq. in. + 2 sq. in., i.e., 3 sq. in., whence it follows that the length of its hypotenuse is in. Thus, in Fig. 4, AB=BC = i in. Therefore, AC*= \/2 in. Also, AD = 1 in., whence DC = 5/3 in. It will be seen from this example that having drawn a line \/x units in length, we can always draw a hne +1 units in length, by making the first line one of the sides of a right-angled triangle of which the side at right angles to it is i unit in length ; it follows, therefore, that any and every square root can be represented by this method. 

The Theorem of Pythagoras, therefore, supphes us with a graphical method for determining the approximate values of ir rational square roots. All that is necessary  

is to draw a right-angled triangle such that the area of the square on its hypotenuse measured in terms of some convenient unit 

 

le.g., a square inch, or a square centimetre) shall be the number whose square root it is desired to extract; then, the length of the hypotenuse measured in terms of the corre sponding hnear unit will give the square root required. Results correct to two places of 

54 

Mathematical Theory of Spirit [§ 18 

decimals can easily be obtained method. 

§ 18. We must bear in mind, 

by this 

that in actual practice there is, ofhowever, 

course, a 

limit to the accuracy with which 

drawn, and there is 

a limit to the accuracy with which 

it can be measured. Since the ratio between  the lengths of the hypotenuse of a right angled triangle and one of the other sides is found, in the majority of cases, to be a surd,® it follows that, if we were able to draw such a right-angled triangle with absolute accuracy, then, the more accurately we measured the length of its hypotenuse, our unit being the length of one of the other sides of the triangle or some length commensurable therewith, the more decimal places would make their appearance, and this would continue for ever, 

® There are certain exceptional right-angled triangles in which the hypotenuse is commensurable with the other sides, as, for example, the right-angled triangle in which the two sides at right angles to one another are 

 

3 and 4 units in length respectively. For, since 5/3’’+4® =V 9 + 16=V25 =5, 

the hypotenuse of this triangle is exactly 5 units in length. The fact, that if the sides of a triangle are in the ratio 3:4:5, then the greatest angle of the triangle is a right angle, was known to the ancient Egy’ptians and employed by them in the erection of their temples. 

§19] Incommensurable Quantities 55 the decimal never terminating and never  recurring; that is to say, the hypotenuse could never be measured in terms of this unit with absolute accuracy. We see, there fore, that, as a rule, the hypotenuse of a  right-angled triangle is incommensurable with the other sides of the triangle. It should be  noted, however, that a length is never in itself incommensurable, though it may be  incommensurable with some other length. <v/2, 5/3, which represent the ratio between two quantities which are in commensurable with one another, are called incommensurables. Consequently, we can  define a surd or irrational quantity as a root 

which involves an incommensurable. § 19. Irrational quantities are not the only incommensurables that occur in Mathematics. A very important incommensur 

Evaluation of the Base of 

Napierian 

logarithms. 

 

able is the base oj natural loga rithms,^ which is always denoted by the letter e. e can be repre sented as the sum of an infinite 

series, that is to say, a series which continues for ever, namely : 

III I . 

I ” -j“ -4" “1“ “t" ... I 1X2 1x2x3 IX2X3X4 

« Called also Napierian logarithms, from Napier, their discoverer. A discussion of the nature and uses of 

56 Mathematical Theory of Spirit [§ 19 the series being continued for ever, each term being derived from the preceding one in a similar manner. The sum of this series is equivalent to e, the sum of an infinite series being defined as the hmiting value which the sum of its terms continually approaches as more terms are taken into account, but which it never exceeds. It can easily be  shown that the sum of this series hes between 2 and 3, and the approximate value of e can be calculated to any required degree of accuracy. To six places of decimals, its value is found to be 2.718282, an approxima tion which is easily arrived at by adding up the first eleven terms of the above series, the sum of the twelfth and following terms being less than .000001. Thus : 

logarithms is beyond the confines of the present work. It must suffice here to say that the logarithm of any number to a given base is the power to which that base must be raised to give the number in question. That is to say, if a’‘= x,n is said to be the logarithm of x to the base a. Thisis written log^x=«. Thus logjj9, =2, because 3^=9. In practical work logarithms calculated to base 

 

10 are always employed. Logarithms to base e may be calculated by means of certain infinite series; these may then be converted to base 10. In this manner tables of logarithms to base 10 are constructed, and by their aid many mathematical .calculations are very much simplified. Readers unacquainted vath the subject and desirous of further information may consult a text book on practical mathematics. 

§19] Incommensurable Quantities 

I =i 

i = i 

I 

-i. =0.5 

1.2 

^-^=0.1666667 

=0.0416667 

~a.L4.5=0-00^3333 

I .5.6=0.0013889 

=0-0001984 

------- 5----- =0.0000248 

i.a.3.4-5-6.7.8 

1 

=0.0000028 I-2.3.4-3.O.7.8.9 

* =0.0000003 

I.3.3.4.3.6.7.8.9. IO 

57 

Therefor.e e =2.718282 (correct to 

6 decimal places). 

The incommensurable denoted by e has the remarkable property that any power of it can be represented by an infinite series similar to that for e, but in which the numera tors of the fractions of which the series con sist are successive powers of the index of this power. Thus, for example : 

 

e’ = i4-^4--^+—2—4-------- 3---------- 4 . . . I 1x2 1x2x3 IX2X3X4 

and, in general: 

« , ■ --’f . 1 e —I -i----1--------- 4-----------------4---------------------- 4- . • . . I 1x2 1x2x3 IX2X3X4 

58 Mathematical Theory of Spirit [§ 20 § 20. Another most important incommen surable is the ratio between the diameter and circumference of a circle, i.e., the 

The Quadra- quantity by which the length of ture of the diameter of a circle must be Circle. multiplied to obtain a length equal to that of the circumfeience. It can be shown that this ratio, which is in variably denoted by the Greek letter tt (/)/),’ is of constant value, i.e., no matter what the size of a circle may be, the length of its diameter divided into the length of its circumference, both being measured in terms of the same unit, always gives the same quotient, namely, the quantity denoted by the letter v. It can also be shown that tt times the square on the radius of any circle is equivalent to the area of the circle. This  latter fact was discovered by Archimedes, and may be proved as follows : 

Suppose the circumference of the circle in Fig. 5 to be divided up into a large num ber (w) 'of equal parts, of which AB is one 

 

such part. Let straight hues, OA and OB, be drawn from 0, the centre of the circle, to the points A and B respectively. Then, if AB be very small, that is to say, if n be 

’ The length of its circumference is the /erimeter of a circle, and tt is the Greek letter corresponding to p. 

§ 20] Incommensurable Quantities 59 very great, the figure OAB may be regarded as a plane triangle, of height equal to the radius of the circle (r), and base equal in length to the line AB. Now, the area of a triangle equals half the product of the lengths 

of its base and height, so that the area of  

the figure OAB equals half the product of r and the length of the fine AB. Further, the length of the whole circumference equals TT times the diameter, i.e., therefore, the length of the hne AB equals 2-rrr-^n ; so that the area of the figure OAB equals 

60 Mathematical Theory of Spirit [§ 21 But n such figures make up the 

whole of the circle. Therefore, the area of the circle equals -nr®. It must be under stood, of course, that all lengths above are measured in terms of the same linear unit, in which case aU areas will be obtained in terms of the corresponding superficial unit. Since n cancels out in the above proof, n may be supposed to be infinitely great, in which case the distinction between the figure OAB and the corresponding plane triangle vanishes; so that the above formula for the area of a  circle holds, not approximately, but exactly. It follows from the above that the determina tion of the correct value of fr is necessary in order to perform the quadrature or “ squar ing ” of the circle, i.e., to draw a square equal in area to a given circle; or, to express the area of a circle in terms of a given square as the unit—one of the great geometrical problems of antiquity. 

§ 21. Archimedes was one of the first to attempt the solution of this problem in a  

 

scientific manner. By consider 

ing the circumference of a circle 

as intermediate in length be 

tween the perimeters of two polygons con taining the same number of sides, one inscribed within the circle, the other circum- 

§21] Incommensurable Quantities 61 scribed about it, he concluded that the value of this ratio lay between 34^ and 3^, i.e., between 3.1408 and 3.1429. The ancient Egyptians appear, at one time, to have used the value =3.16, and the Hebrews, apparently, were content with the value 3.® With the development of Higher Mensura tion or Trigonometry during the past three centuries, better methods for the evaluation of TT have been devised, involving the use of various infinite series. The simplest of these is due to Gregory ; he showed that; 

^ = i -l+ s -|+i - . . . (ad infinitum) 

More convenient than this, however, are the series due to Euler and Machin. It can be shown from a consideration of these series that the value of t is incommensurable, and, therefore, the exact quadrature of the circle remains for ever impossible.* By 

« See The First Book of Kings, ch. vii. v. 23. 

» Correctly speaking, this proves only the impossi  

bility of exactly “ squaring the circle ” by an arith metical method, and it might stiU be thought that there may be a method of geometrical quadrature. For reasons which we cannot give here, however, mathe maticians regard this also as impossible ; but there are several methods of approximately-accurate geometrical quadrature. See Prof. Felix Klein : Lectures on Mathematics (Ziwet, 1894), pp. 51-57- 

62 Mathematical Theory of Spirit [§ 21 means of such series, however, the approxi mate value of TT can be calculated to any required degree of accuracy, so the circle can be “ squared ” as accurately as may be desired. In 1873 Mr. Shanks carried the computation of the approximate value of TT to over seven-hundred decimal places. For most purposes the value 3.14159, which is correct to five decimal places, is sufficiently accurate, and, indeed, there are no purposes for which as many as ten decimal places are required. 

The problem of “ the squaring of the circle ” has also, at times, attracted the attention of many persons with little knowledge of mathematics, but with considerable confi dence in their powers to obtain the exact value of IT, with the production of alleged solutions of the utmost absurdity. The value of v correct to at least 2 decimal places can  easily be obtained by direct measurement. Thus, the diameter of a cylinder can be  measured^ by means of calipers, in the ordinary 

 

way; and the length of its circumference can be determined by wrapping wire, string, or a strip of paper around it: the ratio be tween these two measurements will give the value of TT. Thus, in an actual case, it Was found that the diameter of a certain cylinder 

§21] Incommensurable Quantities 63 measured 7 in., whilst its circumference was 22 in. From these measurements, since -f times the length of the diameter equals that of the circumference, the value of 

obtained is 224-7=31=3.1429, which is correct to the second place of decimals.^® In spite of this fact, however, some of the " circle-squarers ” (as distinguished from the mathematicians who have scientifically in vestigated the subject) have suggested values which have erred in the second or even the first decimal place. 

The accuracy with which the value of w is deter mined by such a method depends, of course, partly on the accuracy with which the cylinder is made and partly on the accuracy of the measurements. 

 

D 

CHAPTER III 

ON NATURE REGARDED AS THE 

EMBODIMENT OF NUMBER 

§ 82. The theory that the Cosmos has its origin and explanation in Number is one that is inseparably connected with 

The P^ba- the name of Pythagoras, though 

gorean Theory J » & oi Numbers, at the present time it is extremely difficult to decide to what extent 

the theories ascribed to him were his own, and not due to his disciples; for Pythagoras, confining his teaching exclusively to the oral method, has left to posterity no account of his theory of the Cosmos, what writings are attributed to him being extremely meagre and most probably the work of his disciples. 

TJie theory in question is one for which it is not difficult to account if we take into consideration the nature of the timesin which  

it was formulated. The Greek of the period, looking upon Nature, beheld no picture of harmony, uniformity and fundamental unity. The outer world appeared to him rather as a discordant chaos, the mere sport and 64 

§ 22] The Embodiment of Number 65 

plaything of the gods. The theory of the uniformity of Nature—that Nature is ever  hke to herself—the very essence of the modern scientific spirit, had yet to be born of years of unwearied labour and unceasing delving into Nature’s innermost secrets. Only in Mathematics—in the properties of geometri cal figures, and of numbers—^was the reign of law, the principle of harmony, perceiv able. Even at this present day when the marvellous has become commonplace, that property of right-angled triangles we have already discussed in the preceding chapter comes to the mind as a remarkable and notable fact: it must have seemed a stupen dous marvel to its discoverer, to whom, it appears, the regular alternation of the odd and even numbers, a fact so obvious to us  that we are inchned to attach no importance to it, seemed, itself, to be something wonder ful. Here in Geometry and Arithmetic, here was order and harmony unsurpassed and un surpassable. What wonder then that Pytha 

 

goras concluded that the solution of the mighty riddle of the Universe was contained in the mysteries of Geometry ? What wonder  that he read mystic meanings into the laws of Arithmetic, and believed Number to be  

the explanation and origin of all that is ? 

66 Mathematical Theory of Spirit [§ 23 We say, There can be no wonder at all. The Pythagorean theory was the natural out come of an erroneous view-point, itself result ing from the fact that philosophers' as yet had not deigned to investigate the phenomena of external Nature, but apphed themselves solely to the more abstract studies. 

§ 23. Whilst criticising the Pythagorean theory of the Cosmos, however, we must not 

Nature as the Embodiment of Number. 

forget that there is undoubtedly a considerable element of truth in it. It is this—to whatever part of the material world we 

turn, we are confronted with the ubiquity of Number : the forms of crystals, beautiful in their symmetry; the combinations of the atoms, and even, it appears, the constitu tion of the atoms themselves—all are ex pressible in terms of number. In the bio logical sciences it is true that Mathematics does not, as yet, play so important a part as iq the physical sciences, but even here  its importance is being recognised. The 

 

arrangement of the leaves about the stem in various plants, to take but one fact out of many, is found to follow a mathematical law and may be expressed as a numerical ratio. In brief, the whole material world is the embodiment of number. As says Dr. 

§ 24] The Embodiment of Number 67 Ennemoser, “ The signification of the Pytha gorean numeral theory is, that numbers contain the elements of all things, and even of the sciences. It was clearly seen that everything in nature may be reduced to numeral conditions.” ” We cannot,” he remarks later, “ be blind to the fact that, in such a theory of numerals, a real and pro found signification may be contained, and not alone an idle speculation or fantastical subtlety. For, through the wonderful pro gress of modern chemistry, the old axiom that determined numerical conditions govern the material world has gained an unexpected signification. Stoichiometry shows indisput ably in the combination of atoms, a regularity of number as strictly observed by God in the minutest forms, as in the majestic nature of the heavens.” * More recent re searches in the domains of Chemistry and Physics (Dr. Ennemoser wrote in 1843) fully confirm the validity of this argument. 

§ 24. The question of incommensurables,  

however, arises. If number be all-potent in Nature, how comes it that there are ratios which numbers fail to represent exactly ? In 

1 Joseph Ennemoser : The History 0/ Magic (trans lated by William Howitt, 1854), vol. i. pp- 394, 399 and 400. 

es Mathematical Theory of Spirit [§ 84 order to reply to this question we must look a httle more closely into the signifi-- 

The Signifi cance of In commen surability. 

cation of incommensurabihty. Briefly, the reason why numbers fail to represent exactly the ratios between certain lengths, such as the diameter and circum 

ference of a circle, is that, whereas length is continuous, number is essentially discon tinuous. As Sir Oliver Lodge puts it, “Nu merical expression is more Uke a staircase than a slope: it necessarily proceeds by 

steps : 

it is discontinuous.” * This is ap 

parent with regard to the integers or whole numbers, and a httle consideration will con vince us that the whole numbers are not really connected by the fractions inter mediate in value between them and rendered continuous thereby; for between any two fractions, however slightly they may differ, another fraction, differing from both of these two less than they differ from one another, can always be inserted; and between each 

 

pair of these fractions others can be inserted likewise, and so on ad in-finitum. It is clear, 

* Sir Oliver Lodge : Easy Mathematics {1910), p. 187. (We are much indebted to Sir Oliver Lodge’s lucid treatment of the subject at present under discussion.) 

§ 24] The Embodiment of Number 69 therefore, that between any two fractions whatever, an infinite number of other frac tions can always be inserted, so that by the insertion of fractions the integers can never be rendered continuous. Sir Ohver Lodge gives an excellent illustration of this fact. He says: “ Look at the divisions on a foot rule ; they represent lengths expressed nu merically in terms of an arbitrary length taken as a unit: they represent, that is to say, fractions of an inch; they are the terminals of lengths which are numerically expressed; and between them lie the un marked terminals of lengths which cannot be so expressed. But surely the subdivision can be carried further. . . . Why not divide originally into tenths and then into hun dredths, and those into thousandths, and so on ? Why not, indeed ? Let it be done.. 

It may be thought that if we go on dividing like this we shall use up all the interspaces and have nothing left but numerically ex pressible magnitudes. Not so, that is just a 

 

mistake; the interspaces will always be infinitely greater than the divisions. For the interspaces have all the time had evident breadth, indeed they together make up the whole rule; the divisions do not make it up, do not make any of it, however numerous 

70 Mathematical Theory of Spirit [§ 25 they are. For how wide are the divisions ? Those we make, look, when examined under the microscope, like broad black grooves. But we do not wish to make them look thus. We should be better pleased with our handi work if they looked like very fine lines of unmagnifiable breadth. They ought to be  really lines—length without breadth ; the breadth is an accident, a clumsiness, an un avoidable mechanical defect. They are in tended to be mere divisions, subdividing the length but not consuming any of it. All the length lies between them; no matter how close they_are they have consumed none of it; the interspaces are infinitely more ex tensive than the barrierswhich partition them off from one another ; they are hke a row of compartments with infinitely thin walls. 

“ Now all the incommensurables he in the interspaces; the compartments are full of them, and they are thus infinitely more numerous than the numerically expressible magnitudes.” ’ It follows, therefore, as Sir 

 

Ohver Lodge points out, that the chances are infinity to one against any two given lengths being commensurable with one another. 

§ 85. Whilst, however, this is true of mathematical lengths—lengths in the ab- * Loc. cit., pp. 189-191. 

§ 25] The Embodiment of Number 71 stract—for mathematical length is continuous, the question arises whether it is hkewise 

The Dis continuity of Matter. 

true of lengths as embodied in material existence. From a dis tance a powder appears to be con tinuous, closer examination shows 

that it consists of separated particles. And it appears that this is true of all material objects : all material things have a grained structure; the most compact and dense forms of matter consist of separated mole cules, and these molecules of separated atoms. It follows, therefore, that no material length (provided that all forms of matter are ulti mately reducible to one and the same unit *) can really be incommensurable with any other material length, with the possible excep tions noted below. In the right-angled tri angle in Fig. 3, for example, however carefully 

* Otherwise-it might happen that one sort of atom might be incommensurable with some othersort. Even admitting the above it might still be argued that, as the size of any material body depends not so much upon 

 

the actual size of its constituent particles as upon the amplitude of their motion, the dimensions of a body consisting of one sort of material substance may be incommensurable with the dimensions of a body con sisting of some other sort of material substance or of the same material substance at a different temperature. The main contention above, however, will, we think, be admitted. 

72 Mathematical Theory of Spirit [§25 it may have been drawn, the hypotenuse simply cannot be exactly 5/2 times one of the other sides ; for all the sides of this triangle are material lengths ; they are thin, narrow prisms of carbon, and each one of them must consist of a definite whole number of carbon atoms, for not less than one atom of any material element can exist. It follows, therefore, that the ratio between  the hypotenuse and either side of this triangle if worked out as a decimal, did we know the number of carbon atoms in each line, would ultimately terminate or recur, although it might be an alarmingly long decimal; for the reason, that an integer, no matter how great it be, is always commensurable with every other integer, and as we have seen above, the lengths of all the sides of the triangle in question, if measured in terms of the length ® of the carbon atom as unit, would be expressible as integers. And this argument holds good of all other material lengths and other quantities, for the most 

 

perfect material line conceivable consists of a row of discontinuous atoms. Moreover, if the electronic theory of matter be true, it must be impossible for actual lengths incom- 

® I.e., the apparent length, the amplitude of the motion of the atom being taken into account. 

§ 26] The Embodiment of Number 78 mensurable one with the other to exist in a physical sense on the surface of, or within, an atom itself; for according to this theory the atoms are composed of separated electrons (or units of negative electricity), just as matter is composed of separated atoms. 

§ 26. The discontinuity of matter neces sitates the assumption of a perfectly con tinuous medium, an absolute 

Ethei^ plenum, filling all space; for if there were not such a medium, 

all action would be action at a distance, action across empty space, which is unthink able. This medium modern science calls the “ Ether of Space.” It follows, there fore, that the theory, sometimes expressed, that the Ether is atomic in structure, re duces to a contradiction in terms. Or, if it be argued that the Ether should be defined as the medium whereby light, radiant heat, &c., are transmitted, and that this medium is atomic in structure and therefore discon tinuous, then we must postulate an ether 

 

beyond the Ether to fill up the interspaces between the particles of the latter. How ever, the phenomena connected with the transmission of light and other etheric phe nomena are in better agreement with the view that the medium whereby hght is 

74 Mathematical Theory of Spirit [§ 26 transmitted is of a continuous nature. Here, then, in the ether, incommensurabihty ac quires a real physical significance. We can conceive of a prism of ether of which the cross-section is a right-angled triangle with a hypotenuse incommensurable with its other sides ; we cannot conceive of a similar material prism.. We can conceive of a perfect cylinder of ether, but a material cyhnder must neces sarily deviate somewhat from geometrical exactitude. We can conceive of lengths incommensurable one with the other existing in a physical sense in the ether; but when we reach the ether we have transcended the world of matter in the more restricted use of the term. So long as we confine our attention to the world of matter, so long do we find the reign of number supreme. In a  certain sense, the realm of the continuous is representative of Unity, whilst the realm of the discontinuous, the world of matter, is representative of the diversity of Number. For iii the realm of the continuous, one only 

 

exists—itself, and although we may consider all numbers as existing’ here potentially, they have not yet been born into actuahty; for although we may consider the realm of the continuous as consisting of diverse parts representative of aU numbers and quantities 

§ 27] The Embodiment of Number 75 (including incommensurables), there is nothing to distinguish or divide one part from another. Accepting the doctrine of modern science that the discontinuous originates from the continuous, the world of matter from the Ether, we may say that the birth of matter is the birth of numbers from potentiahty into actuality. 

§ 27. Before concluding this chapter there is one other thought that it may be worth while to follow out. There is 

very arbitrary about 

Measurement, number if it be regarded as a  ratio merely. The number ex 

pressing some length when the unit of measurement employed is the foot is quite different from the number expressing the same length when some other unit of measure ment, say, the inch, or the yard, or the metre, is used instead; and the same holds good for all other measurements. The length of a  thing in feet, its weight in pounds, &c., all such are purely arbitrary numbers. And 

 

if we employ the so-called “ absolute units,” the case is not fundamentally altered. The metre, for example, was supposed to be  exactly the io,ooo,oooth part of the dis tance from the earth’s equator to its pole when the French Government made it the 

76 Mathematical Theory of Spirit [§ 27 compulsory unit of length in i8oi, but more recent determinations have shown it to differ shghtly from this. Supposing, however, that the metre were exactly the io,ooo,oooth part of the distance from the earth’s equator to its pole, it would still be a purely arbitrary unit on any other planet, and the same can be said of all the other so-called “absolute units.” But if all the varying forms of matter arise from some one common unit, if all material bodies are systems of one and precisely the same sort of particle, then from the properties of this ultimate particle (the electron, it may be) could be derived a system of genuinely absolute units for all material measurements whatever. Thus, its diameter would constitute the unit of length, its inertia the unit of inertia, and so on. For practical purposes, of course, these units would be most inconveniently small, but that is not the point—for practical purposes the metric system of weights and measures does very w'ell. The point is rather that all the 

 

manifold differences between the various things of the material world reduce, in the ultimate analysis, to differences in the number of ultimate particles composing them, and differences in the arrangement and motion of these particles, all of which are expressible 

§ 27] The Embodiment oj Number mathematically. From this point of every material thing is capable of represented by numbers which have a 

77 

view, being more 

than arbitrary meaning, and the doctrine that all material Nature is the embodiment of Number acquires a real significance. 

 

CHAPTER IV 

ON NEGATIVE QUANTITIES 

§ 28. A negative quantity may be defined briefly as a quantity less than nothing. Such quantities are indicated by means 

Definition of a Negative Quantity. 

of the minus sign (-), ordinary or positive quantities {i.e.^ quan tities greater than nothing) being 

indicated by the plus sign (-t-), which is, how ever, generally omitted. For example, —2 (minus two) means a quantity as much below zero as 2, i.e., 4-2 (plus two), is above zero; and, in general, -x means a quantity as much below zero as % or +x is above zero. Suppose, for instance, we were asked to find the result of subtracting 6 from 2, i.e., the value of 2 —6. It is clear that, since 2 subtracted from 2 gives zero, and since 6 is greater than 2 by 4, the result of sub tracting 6 from 2 will be 4 less than zero, 

 

i.e., -4. It may here very possibly be  objected by the reader that such an operation as the above is impossible, and that a nega tive quantity is an absurdity. In a certain 78 

§ 29] Negative Quantities 79 sense, as we shall see later, this is true; but in another sense it is not. Very many cases occur in which we must consider the subtraction of a quantity from one less than itself as possible, and in which it is found convenient to employ quantities less than zero. A few illustrations, however, will make the whole matter perfectly plain. 

§ 29. (fl) of a debt. 

Examples of the Uses of Negative 

Quantities. 

to £500 he 

Take in the first case the idea Suppose the whole of a man’s belongings amount in value to 

and the whole of his debts to £300, then, clearly, he is worth £500 -£300, i.e., £200. If, however, his debts also amount must be worth £500 -£500, i.e.. 

nothing, amount 

But suppose, now, that his debts to £700. Clearly it would not be 

correct in this case to say merely that he is worth nothing, for he is now worse off than in the second case considered above. In that case his assets exactly balance his liabihties, but now he has a balance of £2qo  

on the wrong side : he is worth £200 less than nothing ; in other words, he is worth £5oo-;£7oo, i.e., £-200. 

(6) Secondly, take the idea of direction. Suppose, for example, three towns. A, B, C, to lie so that town A is situated 4 miles due £ 

80 Mathematical Theory of Spirit [§ 29 south of town B, and town C four miles due north of town B. Then considering the position of each town with reference” to town B, whose position is taken as the zero, if we call that of town C +4 miles, since A and C lie on opposite sides of B, we must call the position of town A -4 miles. 

(c) A further example of the use of negative quantities is afforded by the measurement of temperature. On the Fahrenheit thermo meter (commonly used in England) the zero,-point indicates a certain temperature which was the lowest attainable by Fahren heit; whilst on the Centigrade thermometer (invented by Celsius, and employed on the Continent and for scientific work generally) the zero-point indicates the temperature at which ice melts. Temperatures far below either of these zero-points, however, are attainable, and to indicate these negative quantities are employed. Thus a tempera ture of -100° C.^ means a temperature as 

1 1° C. represents a temperature of i degree on the  

Centigrade scale and, similarly, 1° F. represents a temperature of l degree on the Fahrenheit scale. On both scales the temperatures at which ice melts and at which water boils when the pressure of the atmosphere is equivalent to a column of mercury 760 mm. high are taken as the fixed points. On the Centigrade scale these two points are numbered 0° and 100° re spectively, whilst on the Fahrenheit scale they read 

§ 30] Negative Quantities 81 much below the Centigrade zero as ioo° C. (the boiling-point of water) is above it. 

§ 30. We must now turn our attention to the consideration of the addition of nega 

Addition 

of Negative Quantities. 

off by the 

tive quantities. It is clear that if a man contracts a debt re ceiving nothing in return (e.g., a gambling debt) he is the worse amount of this debt. We see, 

therefore, that the addition of a negative quantity is equivalent to the subtraction of the corresponding positive quantity. Thus 

-4 added to 6, i.e., 6+ (-4), gives 2, as also does +4 subtracted from 6, i.e., 6 -(-+-4). Therefore, 6+(-4), 6-(+4) and 6-4 are equivalent expressions ; and, in general: 

x + (-y)=%-(-t-y)=A;-y 

32° and 212°. It follows therefore that a rise or fall in temperature of 5° on the Centigrade scale is equivalent to a rise or fall of 9° on the Fahrenheit scale. Allowing for the difference in the zero-points, the formulae for converting measurements on one scale into measure ments on the other are, therefore, C_5(F=J£), F = 2£+32- 

 

9 5 

where C and F are corresponding temperatures on the Centigrade and Fahrenheit scales respectively. 2 The symbol “ = ” is here used, as frequently, in the sense of “ is identical with,” that is to say, “is equal to, no matter what values be given to the alge braical symbols employed.” 

82 Mathematical Theory of Spirit [§ 31 Further, since it is clear that all a man’s debts go to decrease the balance on the right side of his accounts or to increase the balance on the wrong side, whilst the effect of all his gains is precisely the opposite of this, we see that to find the sum, or total value, of a number of quantities, some of which are positive and some negative, we must first of all find the numerical sum of all the positive quantities and the numerical sum of all the negative quantities separately, and then calculate the difference between them prefixing the sign of the greater. 

Thus : 

(a) 18+6-15-3+2-16=26-34 

= -8 

whilst 

(6) -18-6 + 15+3-2+16=34-26 , =8 {ix., +8) 

In the case of a number of negative quantities, their sum wiU, of course, be their numerical 

 

sum with the negative sign prefixed. Thus, for example, —6 —5 —10 = —21, &c. 

§ 31. The illustration of a debt will also help to make plain the result of the sub traction of negative quantities. For if we remove or take away a man’s debts, this 

§ 32] Negative Q^antities 83 is precisely the same as if we gave the man -the amount of his debts. We see, therefore, that the subtraction of a nega 

tive quantity is equivalent to 

Quantities, the addition of the corresponding positive quantity. Thus,-4 sub 

tracted from 6, i.e., 6 -(-4), gives 10, as also does 4-4 added to 6, i.e.^ 6 4-(4-4), or 64-4. Therefore, 6-(-4), 64-(-l-4) and 64-4 are equivalent expressions, and, in general: 

—^+(+y) = ^+y 

In the case of the subtraction of a negative quantity from a negative quantity the same rule holds good, but as we now have to add a positive quantity to a negative one the result will be the difference between the two quantities with the sign of the greater pre fixed. Thus, for example, 

and 

-6-(-4)= -64-4= -2 -6-(-8) = -64-8=4-2. 

§ 32. We must now turn our attention to a consideration of the multiplication of 

 

negative quantities. Let us con 

Oeometrical Proob of 

Multiplication Formnln. 

sider the several results of multi plying the sum of two quantities, (i.) by itself, and (ii.) by the difference between the two  

quantities, and of multiplying the differenc