An Inquiry into the Original of Our Ideas of Beauty and Virtue. Francis Hutcheson

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       Of the Beauty of Theorems.

      Theorems.

      I. The Beauty of Theorems, or universal Truths demonstrated, deserves a distinct Consideration, ||¹being|| 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 ||²an 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 ||³Infinitys of|| Infinites agree in the general Theorem. ||^4aIn Algebraick, and Fluxional Calculations, we shall ||^5bstill 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 ||^6cinfinite^c|| Orders or Species of Curves, to the infinite Sizes of each Species, and to the infinite Points of the ||^7dinfinite^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 ||⁸in 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 ||⁹only|| 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 ||¹⁰take|| a Metaphysical Axiom, such as this, Every Whole is greater than its Part; and we shall find no Beauty in the Contemplation. ||¹¹For tho|| this Proposition ||¹²contains|| 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 ||¹³with|| the Base, we shall find no pleasure in this Knowledge of a general Relation of greater ||¹⁴and|| 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.

      ||^15aWe 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 ||^16bTriangle, bisects^b|| the Base, or the Reverse; or, that Equilateral Triangles are Equiangular. These Truths we ||^17calmost^c|| know Intuitively, without Demonstration: They are like common Goods, or those which Men have long possessed, which do not give such sensible ||^18dJoys^d|| 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 ||^19eTruth.^ae||

      Corollarys.

      V. There is another Beauty in Propositions, ||²⁰which cannot be omitted; which is||, When one Theorem ||²¹contains|| a ||²²vast|| Multitude of Corollarys easily deducible from it. Thus ||²³that 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 ||²⁴also|| 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 ||²⁵Book|| is another of like Beauty, and so are many ||²⁶others||.

      ²⁷In 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; ||²⁸such 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 ||²⁹Advantage|| from such [35] Manner of Deduction, ||³⁰than|| 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.

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