An Inquiry into the Original of Our Ideas of Beauty and Virtue

Section III. Of the Beauty of Theorems

36 u s e c t i o n i i i u Of the Beauty of Theorems. I. The Beauty of Theorems, or universal Truths demonstrated, deserves a distinct Consideration, 㛳1 being㛳 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 wemayfind included,withthemostexactAgreement , an infinite Multitude of particular Truths; nay, often 㛳2 an Infinity 㛳 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 㛳3 Infinitys of㛳 Infinites agree in the general Theorem. 㛳4a In Algebraick , and Fluxional Calculations, we shall 㛳5b still find a greaterb 㛳 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 㛳6c infinitec 㛳 Orders or Theorems. section iii 37 Species of Curves, to the infinite Sizes of eachSpecies,andtotheinfinite Points of the 㛳7d infinited 㛳 Individuals of each Size.a 㛳 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 㛳8 in which we are㛳, when we can only measure Lines, or Surfaces, by a Scale, or are making Experiments which we can reduce to no general Canon, but 㛳9 only㛳 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. IV. Again, letus㛳10 take㛳aMetaphysicalAxiom,suchasthis,EveryWhole is greater than its Part; and we shall find noBeautyintheContemplation. 㛳11 For tho㛳 this Proposition 㛳12 contains㛳 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 㛳13 with㛳 the Base, we shall find no pleasure in this Knowledge of a general Relation of greater 㛳14 and㛳 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! 㛳15a 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 㛳16b Triangle, bisectsb 㛳 the Base, or the Reverse; or, that Equilateral Triangles are Equiangular. These Truths we 㛳17c almostc 㛳 know Intuitively, without Demonstration:They Foundation of their Beauty. Little Beauty in Axioms. Easy Theorems. 38 treatise i are like common Goods, or those whichMen havelongpossessed,which do not give such sensible 㛳18d Joysd 㛳 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...