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• Introduction
• References and Further Reading
• Note On the Text
• Acknowledgments
• An Inquiry Into the Original of Our Ideas of Beauty and Virtue ; In Two Treatises.
• 2 to His Excellency, John, Lord Carteret, Ii Lord Lieutenant of Ireland.
• The Preface
• Treatise I: Viz. an Inquiry Concerning Beauty, Order, &c.[1]
• Section I: Concerning Some Powers of Perception, Distinct From What Is Generally Understood By Sensation.
• Section II: Of Original Or Absolute Beauty.
• Section III: Of the Beauty of Theorems.
• Section IV: Of Relative Or Comparative Beauty.
• Section V: Concerning Our Reasonings About Design and Wisdom In the Cause, From the Beauty Or Regularity of Effects.
• Section VI: Of the Universality of the Sense of Beauty Among Men.
• Section VII: Of the Power of Custom, Education, and Example, As to Our Internal Senses.
• Section VIII: Of the Importance of the Internal Senses In Life, and the Final Causes of Them.
• ‖ 1 Treatise II: Viz. an Inquiry Concerning the Original of Our Ideas of Virtue Or Moral Good.
• Introduction
• Section I: Of the Moral Sense By Which We Perceive Virtue and Vice, and Approve Or Disapprove Them In Others.
• Section II: Concerning the Immediate Motive to Virtuous Actions.
• Section III: The Sense of Virtue, and the Various Opinions About It, Reducible to One General Foundation. the Manner of Computing the Morality of Actions.
• Section IV: All Mankind Agree In This General Foundation of Their Approbation of Moral Actions. the Grounds of the Different Opinions About Morals.
• Section V: A Further Confirmation That We Have Practical Dispositions to Virtue Implanted In Our Nature; With a Further Explication ‖ 1 of Our Instinct to Benevolence In Its Various Degrees‖; With the Additional Motives of Interest, Viz. Honour, Shame an
• Section VI: Concerning the Importance of This Moral Sense to the Present Happiness of Mankind, and Its Influence On Human Affairs.
• Section VII: A Deduction of Some Complex Moral Ideas, Viz. of Obligation, and Right, Perfect, Imperfect, and External, Alienable, and Unalienable, From This Moral Sense.

SECTION III

Of the Beauty of Theorems.

Theorems.

I. The Beauty of Theorems, or universal Truths demonstrated, deserves a distinct Consideration, ‖1 being‖ of a Nature pretty different from the former kinds of Beauty; and yet there is none in which we shall see such an amazing Variety with Uniformity: and hence arises a very great Pleasure distinct from Prospects of any further Advantage.

II. For in one Theorem we may find included, with the most exact Agreement, an infinite Multitude of particular Truths; nay, often ‖2 an Infinity‖ of Infinites: so that altho the Necessity of forming abstract Ideas, and universal Theorems, arises perhaps from the Limitation of our Minds, which cannot admit an infinite Multitude of singular Ideas or Judgments at once, yet this Power gives us an Evidence of the Largeness of the human Capacity above our Imagination. Thus for instance, the 47th Proposition of the first Book of Euclid’s Elements contains an infinite Multitude of Truths, concerning the infinite possible Sizes of right-angled Triangles, as you make the Area greater [31] or less; and in each of these Sizes you may find an infinite Multitude of dissimilar Triangles, as you vary the Proportion of the Base to the Perpendicular; all which ‖3 Infinitys of‖ Infinites agree in the general Theorem. ‖4a In Algebraick, and Fluxional Calculations, we shall ‖5b still find a greater^b‖ Variety of particular Truths included in general Theorems; not only in general Equations applicable to all Kinds of Quantity, but in more particular Investigations of Areas and Tangents: In which one Manner of Operation shall discover Theorems applicable to ‖6c infinite^c‖ Orders or Species of Curves, to the infinite Sizes of each Species, and to the infinite Points of the ‖7d infinite^d‖ Individuals of each Size.^a‖

Foundation of their Beauty.

III. That we may the better discern this Agreement, or Unity of an Infinity of Objects, in the general Theorem, to be the Foundation of the Beauty or Pleasure attending their Discovery, let us compare our Satisfaction in such Discoverys, with the uneasy state of Mind ‖8 in which we are‖, when we can only measure Lines, or Surfaces, by a Scale, or are making Experiments which we can reduce to no general Canon, but ‖9 only‖ heaping up a Multitude of particular incoherent Observations. Now each of these Trials discovers a new Truth, but with no Pleasure or Beauty, notwithstand-[32]ing the Variety, till we can discover some sort of Unity, or reduce them to some general Canon.

Little Beauty in Axioms.

IV. Again, let us take a Metaphysical Axiom, such as this, Every Whole is greater than its Part; and we shall find no Beauty in the Contemplation. ‖10 For tho‖ this Proposition ‖11 contains‖ many Infinitys of particular Truths; yet the Unity is inconsiderable, since they all agree only in a vague, undetermin’d Conception of Whole and Part, and in an indefinite Excess of the former above the latter, which is sometimes great and sometimes small. So, should we hear that the Cylinder is greater than the inscrib’d Sphere, and this again greater than the Cone of the same Altitude and Diameter with the Base, we shall find no pleasure in this Knowledge of a general Relation of greater and less, without any precise Difference or Proportion. But when we see the universal exact Agreement of all possible Sizes of such Systems of Solids, that they preserve to each other the constant Ratio of 3, 2, 1; how beautiful is the Theorem, and how are we ravish’d with its first Discovery!

Easy Theorems.

‖12a We may likewise observe, that easy or obvious Propositions, even where the Unity is sufficiently distinct, and determinate, do not please us so much as those, which [33] being less obvious, give us some Surprize in the Discovery: Thus we find little Pleasure in discovering that a Line bisecting the vertical Angle of an Isosceles Triangle, bisects the Base, or the Reverse; or, that Equilateral Triangles are Equiangular. These Truths we almost know Intuitively, without Demonstration: They are like common Goods, or those which Men have long possessed, which do not give such sensible Joys as much smaller new Additions may give us. But let none hence imagine, that the sole Pleasure of Theorems is from Surprize; for the same Novelty of a single Experiment does not please us much: nor ought we to conclude from the greater Pleasure accompanying a new, or unexpected Advantage, that Surprize, or Novelty is the only Pleasure of Life, or the only ground of Delight in ‖13b Truth.^ab‖

Corollarys.

V. There is another Beauty in Propositions, ‖14 which cannot be omitted; which is‖, When one Theorem ‖15 contains‖ a ‖16 vast‖ Multitude of Corollarys easily deducible from it. Thus ‖17 that Theorem which gives us the Equation of a Curve, whence perhaps most of its Propertys may be deduc’d, does some way please and satisfy our Mind above any other Proposition‖: Such a Theorem also is the 35th of the 1st Book of Euclid, from which the whole Art of measuring right-lin’d Areas is deduc’d, by [34] Resolution into Triangles, which are the halfs of so many Parallelograms; and these are each respectively equal to so many Rectangles of the Base into the perpendicular Altitude: The 47th of the 1st Book is another of like Beauty, and so are many ‖18 others‖.

19 In the search of Nature there is the like Beauty in the Knowledge of some great Principles, or universal Forces, from which innumerable Effects do flow. Such is Gravitation, in Sir Isaac Newton’s Scheme; ‖20 such also is the Knowledge of the Original of Rights, perfect and imperfect, and external; alienable and unalienable, with their manner of Translations; from whence the greatest Part of moral Dutys may be deduc’d in the various Relations of human Life.‖

It is easy to see how Men are charm’d with the Beauty of such Knowledge, besides its Usefulness; and how this sets them upon deducing the Propertys of each Figure from one Genesis, and demonstrating the mechanick Forces from one Theorem of the Composition of Motion; even after they have sufficient Knowledge and Certainty in all these Truths from distinct independent Demonstrations. And this Pleasure we enjoy even when we have no Prospect of obtaining any other ‖21 Advantage‖ from such [35] Manner of Deduction, ‖22 than‖ the immediate Pleasure of contemplating the Beauty: nor could Love of Fame excite us to such regular Methods of Deduction, were we not conscious that Mankind are pleas’d with them immediately, by this internal Sense of their Beauty.

Fantastick Beauty.

It is no less easy to see into what absurd ‖23 Attempts‖ Men have been led by this Sense of Beauty, and ‖24 a silly Affectation‖ of obtaining it in the other Sciences as well as the Mathematicks. ’Twas this probably which set Descartesi on that hopeful Project of deducing all human Knowledge from one Proposition, viz. Cogito, ergo sum; while others ‖25 with as little Sense contended‖, that Impossibile est idem simul esse & non esse, had much fairer Pretensions to the Style and Title of Principium humanae Cognitionis absolutè primum. Mr. Leibnitzii had an equal Affection for his favourite Principle of a sufficient Reason for every thing in Nature, and ‖26 brags to Dr. Clarkeiii ‖ of the Wonders he had wrought in the intellectual World by its Assistance; ‖27 but his learned Antagonist seems to think he had not sufficient Reason for his Boasting.* ‖ If we look into particular Sciences, we ‖28 may see in the Systems learned Men have given us of them,‖ [36] the Inconveniences of this Love of Uniformity. ‖29 How‖ aukardly ‖30 is Puffendorf forc’d to‖ deduce the several Dutys of Men to God, themselves, and their Neighbours, from his single fundamental Principle of Sociableness to the whole Race of Mankind?31 This Observation ‖32 might easily be extended farther, were it necessary; and‖ is a strong Proof that Men ‖33 have a Sense of Beauty in‖ Uniformity in the Sciences, ‖34 even from the Contortions of common Sense they are led into by pursuing it‖.

VI. This Delight which accompanys Sciences, or universal Theorems, may really be call’d a kind of Sensation; since it necessarily accompanys the Discovery of any Proposition, and is distinct from bare Knowledge it self 35 , being most violent at first, whereas the Knowledge is uniformly the same. And however Knowledge enlarges the Mind, and makes us more capable of comprehensive Views and Projects in some kinds of Business, whence Advantage may also arise to us; yet we may leave it in the Breast of every Student to determine, whether he has not often felt this Pleasure without any such prospect of Advantage from the Discovery of his Theorem. All ‖36 which‖ can thence be infer’d is only this, that as in our external Senses, so in our internal ones, the pleasant Sensations generally arise from those Objects which calm Reason [37] would have recommended, had we understood their Use, and which might have engag’d our pursuits from Self-interest.

VII. 37 If any alledge, “that this Pleasure in Theorems arises only at first, upon the Novelty of the Discovery, which occasions Surprize:” It must be own’d indeed that* Novelty is generally very agreeable, and heightens the Pleasure in the Contemplation of Beauty; but then the Novelty of a particular Truth, found out by measuring, as above mention’d, gives no considerable Pleasure, nor Surprize. That then which is pleasant and surprizing, is the first Observation of this Unity amidst such a great Variety. There is indeed another kind of Surprize, which adds to the Beauty of some Propositions less universal, and may make them equally pleasant with more universal ones; as when we discover a general Truth which seem’d before, upon some confus’d Opinion, to be a Falshood: as that Assymptotes always approaching should never meet the Curve. This is like that Joy, which may be very strong and violent, upon the unexpected Arrival of a small Advantage, from that Occasion from which we apprehended great Evil; but still this Unity of many Particulars in the general Theo-[38]rem, is necessary to make it pleasant, in any case.

Works of Art.

VIII.38 As to the Works of Art, were we to run thro the various artificial Contrivances or Structures, we should constantly find the Foundation of the Beauty which appears in them, ‖39 to be‖ some kind of Uniformity, or Unity of Proportion among the Parts, and of each Part to the Whole. As there is a ‖40 vast‖ Diversity of Proportions possible, and different Kinds of Uniformity, so there is room enough for that Diversity of Fancys observable in Architecture, Gardening, and such like Arts in different Nations; they all may have Uniformity, tho the Parts in one may differ from those in another. The Chinese or Persian Buildings are not like the Grecian and Roman, and yet the former has its Uniformity of the various Parts to each other, and to the Whole, as well as the latter. In that kind of Architecture which the Europeans call Regular, the Uniformity of Parts is very obvious, the several Parts are regular Figures, and either equal or similar at least in the same Range; the Pedestals are Parallelopipedons or square Prisms; the Pillars, Cylinders nearly; the Arches Circular, and all those in the same Row equal; there is the same Proportion every where observ’d in the same Range between the Diameters of Pillars and their Heights, their Capitals, the Dia-[39]meters of Arches, the Heights of the Pedestals, the Projections of the Cornice, and all ‖41 the‖ Ornaments in each of our five Orders. And tho other Countrys do not follow the Grecian or Roman Proportions; yet there is even among them a Proportion retain’d, a Uniformity, and Resemblance of corresponding Figures; and every Deviation in one part from ‖42 that‖ Proportion which is observ’d in the rest of the Building, is displeasing to every Eye, and destroys or diminishes at least the Beauty of the Whole.

43 IX. The same might be observ’d thro all other Works of Art, even to the meanest Utensil; the Beauty of every one of which we shall always find to have the same Foundation of Uniformity amidst Variety, without which they ‖44 appear‖ mean, irregular and deform’d. [40]

[i ]René Descartes (1596–1650), French philosopher and mathematician, published the cogito-ergo-sum principle first in his Discours de la méthode (1637) and in his Meditationes de prima philosophia (1641), meditations 2 and 3.

[ii ]Gottfried Wilhelm von Leibniz (1646–1716), German philosopher, mathematician, historian, and jurist, discovered differential and integral calculus, and developed the first binary arithmetic. Of his numerous writings and extensive correspondence, little was published during his lifetime. The principle of sufficient reason is central to his metaphysics and logic; see his Monadologie (1720). In an exchange of letters with Samuel Clarke (see note iii below), he discussed the philosophical principles of Newton’s physics, especially space and time (The Leibniz-Clarke Correspondence, ed. by H. G. Alexander, New York: Manchester University Press, 1998).

[iii ]Samuel Clarke (1675–1729), English theologian and philosopher, was a friend of Newton whose philosophical doctrine he defended against Leibniz (see note ii above).

[* ]See the Letters which pass’d between Dr. Clarke and Mr. Leibnitz, Pag. 23.

[* ]See Sect. vi. Art. 13. and the Spectator there referr’d to.

[1]A (p. 27): because

[2]D (p. 30): Multitude

[3]Omitted in D (p. 31).

[4]A (p. 21): Thus also one Fluxional Calculation shall determine the Tangents of all Algebraick Curves; of these Curves there are infinite Orders and Species possible[,] of each Species infinite Sizes, or Magnitudes of Areas, of each Size infinite Individuals, of each Individual Curve an Infinity of Points, from which Tangents may be drawn. But all these Infinitys of Infinites are exactly comprehended in the general Theorem, which fixes the Lengths of the Subtangents, or their Proportion to the Abscissa.

[5]D (p. 31): find a like

[6]D (p. 31): many

[7]D (p. 31): innumerable

[8]Omitted in D (p. 31).

[9]D (p. 31): are only

[10]A (p. 29): Because however

[11]A (p. 29): does contain

[12]Paragraph not in A (p. 29).

[13]C (p. 33), D (p. 33): Truth. Another kind of Surprize in certain Theorems increases our Pleasure above that we have in Theorems of greater Extent; when we discover a general Truth, which upon some confused Notion we had reputed false: as that Asymptotes always approaching should never meet the Curve. This is like the Joy of unexpected Advantage where we dreaded Evil. But still the Unity of many Particulars in the general Theorem is necessary to give Pleasure in any Theorem.

[14]A (p. 29): which cannot be omitted; which is this [Omitted in D (p. 33).]

[15]A (p. 29): shall contain

[16]D (p. 33): great

[17]D (p. 33): there are some leading, or fundamental Propertys, upon which a long Series of theorems can be naturally built

[18]D (p. 34): others in higher Parts of Geometry

[19]No new paragraph in A (p. 30).

[20]D (p. 34): What is the Aim of our ingenious Geometers? A continual Inlargement of theorems, or making them extensive, shewing how what was formerly known of one Figure extends to many others, to Figures very unlike the former in Appearance.

[21]A (p. 31): Advantage in Life

[22]A (p. 31): besides

[23]A (p. 31): Whimsys

[24]C (p. 35), D (p. 35): an Affectation

[25]D (p. 35): pleaded

[26]D (p. 35): boasts

[27]Without footnote in A (p. 31). Omitted in D (p. 35).

[28]D (p. 35): see

[29]A (p. 32): Dr. Cumberland has taken a great deal of needless Pains to reduce the Laws of Nature to one general practical Proposition; and how

[30]D (p. 35): does Puffendorf

[31]A (p. 32): As if they had not been better drawn, each respectively, from their immediate Sources, viz. Religion, Self-Love, and Sociableness.

[32]Omitted in D (p. 35).

[33]C (p. 35), D (p. 35): perceive the Beauty of

[34]A (p. 32): notwithstanding the Contortions of Common Sense they may be led into by pursuing it

D (p. 36): since they are led into unnatural Deductions by pursuing it too far

[35]D (p. 36) adds note: *Aristotle (Ethic. Nicom. I. ro. c. 3. [NE, X 2, 1174a 4–8]) justly observes, that we have certain natural Propensitys to certain Actions, or to the Exercise of certain natural Powers, without a View to, or Intention of, obtaining those Pleasures which naturally accompany them. Πεϱὶ πολλὰ σπουδὴν ποιησαίμεϑα Ἄν, καὶ εἰ μηδεμίαν ἐπιϕέϱοι ἡδονήν, οἱ̑ον ὀϱα̂ν, μνημονεύειν, εἰδέναι, τὰς ἀϱετὰς ἔχειν· εἰ δ’ ἐξ ἀνάγκης ἕπονται τούτοις ἡδοναὶ, οὐδέν διαϕέϱει· ἑλοίμεϑα γὰϱ Ἄν ταυ̑τα, καὶ εἰ μὴ γένοιτ’ Ἄν ἀπ’ αὐτω̑ν ἡδονή. [Translation: “Also there are many things which we should be eager to possess even if they brought us no pleasure, for instance sight, memory, knowledge, virtue. It may be the case that these things are necessarily attended by pleasure, but that makes no difference; for we should desire them even if no pleasure resulted from them.” (Aristotle, Nicomachean Ethics, with an English translation by H. Rackham, Cambridge, Mass.: Harvard University Press, 1975, p. 588.)]

[36]A (p. 33): that

[37]Paragraph not in A (p. 33), C (p. 35), D (p. 36).

[38]A (p. 33): wrongly numbered VI. In C (p. 37), D (p. 37): numbered VII.

[39]A (p. 33): to be constantly

[40]C (p. 37), D (p. 37): great

[41]Not in A (p. 34).

[42]D (p. 38): the

[43]A (p. 34): numbered VII.

[44]A (p. 34): shall appear