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111 FORM AND SPACE

EONARDO WAS TRAINED asa sculptor and wished to be con-sidered as such. After detailing thecategories of military and civilengineering in which he excelled,in his letter of self-recommendationto Ludovico Sforza, he reminded

the Duke that: 'also 1 can execute sculpture in marble,bronze and clay.Likewise in painting, 1 can do every-thing possible as well as any other, whosoever he maybe'(eA roSzr) He clearly possessed extraordinary, innatepotentials of plastic and spatial visualization, but hisapprenticeship in a srudio primarily famed for sculpturewashighly significant in laying down the tools to realizethese potentials. His feeling for the nature of any giventhree-di rne nsional form - its solid rnass, the manipu-lations to which it can be subject, the negative shape tharsurrounds it (irs 'rnould'). the ways in which it can besectioned, dismembered and rendered transparent, themanner in which it can be represented from variousviewpoints, how it behaves at rest and in rnorion, and itsgraphic reduction to plan and elevation - was as stronglydeveloped in him as has ever been the case throughouthistory Whenever he drew anything on paper, he wasreaching out graphically for its plastic essence.

No one wasever more inventive at devising the graphicmeans to accomplish visual rhinking than Leonardo, and noonewasmore skilled at inventing presentational methodsto let the spectator see what he was thinking. Even the1ll0Stroutine diagram assumes a special kind oElife at thetouch of his pen Many of the graphic means that are nowtaken for granted in science and technology made theirfirst appearance in his rnanuscripts. And when he adopted

a stock technique, it always looked different in his hands.In exploring how he manipulated form and space on paper,1will begin with the most apparendy conventional, lar linediagrams and work through to the highly cornplex systemshe devised for the human body.

Mathematics andth e Music o f P ro po rtio nThe drawing of Hat diagrarns was the stock-in-trade ofany geol11eter.Manuscripts and books of Euclid's El ements,the classic point of reference, were illustrated with figuresof the kind that still adorn school textbooks. The greatArchimedes, according to legend, wasbutchered by invadingRoman soldiers while distractedly drawing diagrams in thesand. The vast majority of geometrical drawings over theages must have been of this transitory type, drawn andrubbed out or otherwise discarded. Geometers must haveundertaken 'experimental' drawings when they were workingthrough a problcm ", searching for a theorem. Preciselywhat these drawings looked like, we do nor know. Sayingconfidendy rhar Leonardo's Hat geometrical drawings arenovel is therefore irnpossible, but they do have a characterthat seems highly individual.

The geometry on the "Therne Sheet' (iNTRO.l) is drawnmore precisely than the majority of his diagrams, many ofwhich are freehand or combine ruled and cornpass-drawnelements with freehand improvisations. We do, however,need to rake into account that there is 'invisible' geometryon a surprising number of sheets, drawn wirh a blank srylusthat creates incised lines visible only in raking light. Such

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111.1Studies of squaring the circlePLATE 111.3 (detail)

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lines are not restricted to pages containing geometricalfigures or technical drawings. A number of sheets, as yetunquantified and largely unresearched, contain incisedgeometrical designs that are unrelated to the ink or chalkdrawings added later. Comments on some typical examplesare included in rhe 'note on incised lines' in the finalsection ofthe book (see p. 191).Even on rhe 'Theme Sheet', the diagrams have an

improvised quality. beginning with some formal moves ofa standard kind, only to be overtaken by a series of specula-tive lines rhat not only express what Leonardo is thinking,but also suggest new conjunctions in their own right. Thereis a sense of freewheeling search, undertaken with a speedand density that often make it very difficult in retrospectto tease out what he is doing. When something significantseems to be emerging - above all in his search for equiva-lences of rectilinear and curvilinear areas - he adds somequick hatching lines, so that the designated areas areendowed with an especially emphatic status. Occasionallylines are drawn as a row of dots, when he wants thern toplaya role in the construction, but wishes to leave therndifferentiated from rhe main lines of the figure. He usesletters to denote points, line ends, vertices and areas ina standard way, but very sparingly. It is only when he hasreached a point of some resolution that he adds a full suiteofletters and an accompanying note, taking us through thesequence of moves in the demonstration or proof. Notinfrequentlya diagram was lettered, but the interpretativenote was never added.

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PLA TE 111 .2Determining the areas of figureswith straight and curved sides1509Penand ink29 x 209 cmwrndsor castle, The RoyalCollection, 19145'

In many ofhis later pages Leonardo focused heavilyon the squaring of a circle or on techniques thar mighteventually lead towards a solution. The problem was todefine a square exactly equal to that of a given circle. Ir hadbecome a c1assicproblem, in much the same way as Pierrede Ferrnat's last theorem later continued to bother mathe-maticians for centuries. Leonardo knew that Archimedes,whom he increasingly carne to revere, was deeply involvedin this and related questions of areas and volumes. Theproblem of squaring the circle was to be reshaped in rheeighteenth century by Johann Larnberr's demonstrationof the irrationality of 'IT, which can only be approxirnatedin numbers. Leonardo covered sheets wirh urgent littlediagrams in the quesr, or in the subsidiary quest to findrectilinear equivalents to figures that possessed at leastone curved side (111.2 and 111.3). Various methods weretried, ultimately in vain.If the circle could be reduced to a polygon, by the

subtracting of slices from its circumference (111.1), thenthe area of the polygon could be calculated without toomuch difficulry. Leonardo's sketch of what he called rhe'Quadrature of Archimedes' is accompanied by a noteclaiming that 'Archimedes gave the quadrature [squaring]of a srraighr-sided figure but not of rhe circle. HenceArchimedes never squared any figure with curved sides;and T have squared the circle minus the smallest portionthat the intellect can imagine, that is to saya visible poinr.'The basis of Leonardo's c1aim is unclear. He seems to

be referring to something more that the traditional methodof'exhaustion', in which a polygon is drawn with a hugenumber of sides, so thar rhe curved portions 'sliced' offaround the circumference would be of negligible area.This method leads to an infinitely perfectible approximationIr may be that he was referring to his 'rnechanical' solution,which involved the 'unrolling' of a segmented circle likean orange folded back on its peel. These rather tangibleand plastic procedures appealed greatly to hirn, but they allinvolved some measure of mathematical approximation.Ir is utterly characteristic that three-dimensionalshaping creeps in even when he is ostensibly dealing withHat geometry. The circles all have an inherent tendencyto sphericality. and in the lower centre an overt sphere


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ur .sStudies 01squaring the circlec.rsosWindsor Castle, The RoyalCollection, 12280

"'l. '1._. :.;.. ' Often occurring inabbreviated form, this is a recurrent verbal doodle, bur itbreathes an undeniable air of frustration. Here it reads,'Tell me if anything similar was ever done?' This varan tinverts what seems to be the normal pessimism of the tag,

and suggests that Leonardo hoped he really was breakingnew ground with his kind of geometryAt one juncture he was confident that he had 'cornplered'

the squaring of the circle. In the Codex Madrid I he nored:'On the night of St Andrew [30 November] I reached theend of squaring rhe circle; and at rhe end of the light ofthe candle, of the night and of rhe paper on which I waswriting, it was completed.' His confidence was misplaced.Or perhaps he just meant that he had gone as far as hecould and was calling it a day (or night).A few years later he announced that he had solved a

problem that would potentially release the solution tothe squaring of the circle (11I.2): 'Having for a long timesearched to square the angle of rwo equal curves ... now inthe year 1509 on the eve of the Calends of May [30 April]

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111.4The'golden section'111 .5Portrait of a MusicianC.1485Milan, Biblioteca Ambrosiana

I have found the solution at me 22nd hour on Sunday'The 'solution' is fair enough on its own terrns. but does nothelp to square the circle. We are presented with the nicepcture of Leonardo working late in the evenings and intothe night by the dim light of acandle, scribbling geometricaldiagrams on sheaves of paper. It is not known wherher thecold light of the next morning immediately doused hishopes or whether he carne to a slower realization thar rhesquaring of the circle remained elusive. In any event, hecontinued to nag at the question in subsequent years.Ir is revealing that he wrote down dated records of his

solutions to geometrical problems - something he did onfour occasions - whereas his notebooks contain no datedreferences to his paintings or notable successes in otherareas ofhis activities. Geometry clearly had a special statusfor Leonardo, not least in potentially ensuring his place inposreriry alongside Euclid and Archimedes.The relationship between geometrical areas thar triggered

his memoranda belonged to his all-embracing quest tomaster the principies of proportion that governed the formand functioning of all things in the visible and invisibleworld. Given his reverence for proportions that couldonly be expressed geometrically, such as \12 (the diagonalof a square 1 x 1) and the 'golden section' - rhe divisionof a line ab at esuch that ab :c b =b:ac (111.4) - it is perhapssurprising that he did not corral the squaring of the circleinto the territory of the unquantifiable.

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Leonardo may have been drawn less instinctively toproportion expressed in numbers, but arithmetical ratiosremained of considerable importance in his theory andpractice. Numerical proportions had been seen as me key tovisual beauty in classical antiquity, as Renaissance theoristswell knew, and the theory of musical harmonics laid downby Pythagoras continued to provide the bedrock formusicians, as we have noted (see P.24). Leonardo had greatrespect for rhe sc ientia of music, even if the musician's ar tultimately took second place to painting:

111 .5

Music is not to be regarded as other than the sister ofpainting, in as much as she is dependent on hearing,second sense behind that of sight. She composesharmony from the conjunction ofher proportionalparts, which make their effect simultaneously, beingconstrained to arise and die in one or more harmonicintervals. These intervals may be said to circumscriberhe proportionality of the component parts of whichsuch harmony is composed - no differently from rhelinear conrours of the limbs from which human beautyis composed. [Urb r6r-v]

eIHis engagement with music and musicians was anythingbut casual. He was noted as a performer and a designerof instruments. As we have seen, he was a colleague ofFranchino Gaffurio, and a collaborator with composers,instrumentalists and singers on court spectacles. HisPortrai t a ja M usc an in the Biblioteca Ambrosiana in Milan(i 11.5) provides an eloquent testimony to this engagement.There are also fragmentary staves of music in his manuscriptsand, although these have been recorded, they would ontheir own do litde more than testify that Leonardo knewmusical notation.His linkage between musical proportion and the visual

arts was not limited to surface proportions. Characteristicallyhe extended the analogy into visual relations in space:

FORMANOSPACE 101


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1shall ... found my rule on a scale of 20 for 20 braccia,just as rhe musician has done for notes. Although thenotes are united and attached to each other, he hasnonetheless recognized small intervals between noteand note, designating thern as first, second, fourth andfifth, and in this manner from interval to interval hasgiven names to the varieties of raised and lowered notes.[Urb 17r]

What he means, in a passage that is not one ofhis clearest,is rhar the perspectival diminution of objects in front ofthe eye - even though they 'touch each other as ifhandin hand' - can be see n to obey a rule of ratios like thar ofmusic; that is to say, rhe rules of perspective dictate thar theapparent size of an object on a picture plane is inverselyproportional to its distance from the eye. Ir seems tharin the La st Sup per (1 NTRO.4) Leonardo contrived to accordthe widrh of the tapestries, as measured on the plane ofthe wall, the ratio 1:V,:I;3:';4.n whole numbers the ratio is12:6:4:3, and we can recognize 3:4 as rhe musical intervalof a fourth, 4:6 as a fifth, and 6:12 as an octave. To achievethis result, the tapestries in 'real life' would have to be ofdifferent actual widths.Some of the puzzling suites of numbers that make

apparently random appearances across his pages of drawingsand notes do not appear to concern addition, subtraction,rnultiplication or division (even allowing for his carelessnesswith arithmetical procedures), but are best recognized asthoughts about harmonic series. Not infrequently theyappear on sheets concerned with geometry, indicating arecurrent dialogue between the two kinds of proporrionalsystem (e.g.W 12642r).Testimony to the trouble he took to master the most

advanced theories of numerical proportions, well beyondthose required by music, is provided by the numbersquare in the Codex Arundel ( 1 1 1 . 6 ) , which was carefullyrranscribed from the Summa d e a r i thmet ic a, geometra , pr opor tio neet pr oport onaltil published in 1494 by Luca Pacioli, who washis friend and colleague in Milan in the later years of thefifteenth century and seems to have helped Leonardo withlodgings when they settled in Florence after the fall of theirpatron, Ludovico Sforza. Leonardo notes that he bought

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the printed version ofhis friend's marhemarical referencebook for II9 soldi (CA J04ra; 288r). Having laid out andfilled in the table, he begins to note the names assigned tothe various rnultiples - double, triple, quadruple, quintuple- but stops at four of them, presumably because it is toographically complicated to continue. 'TeUme if ever a thingwas done: he writes under the diagram, again expressingfrustration about the perpetual gap berween ambition andachievement. On the other side of the sheet he reverts to amore conventionallisring of proportions. He also transcribesthe square in Codex Madrid l I, starting this time to namethe proportions down me left-hand side, but again does notsucceed in gerring beyond number five. Elsewhere in thesame codex (78r) he redrew another of Paciol's illustrarions,a schematic tree of 'proportions and propor rionality '.Characteristically Leonardo gave his Friend's rectilinearlayout an organic rwist by setting the numbers in round'fruirs' at the junctures of curvaceous branches.The expression of such proportionaliry in rhe visual arts

was, aswe have seen, implicit in perspective and apparentsize, and was also explicit in rhe proporcional systems hesought in the bodies ofhumans and horses. Verrocchio, hismaster, had sought to codify horse proporrions, presumablyduring his preparations for the grand bronze monumentofBartolommeo Colleoni (see p.63). Leonardo would alsohave been aware of attempts to determine the proportionaldesign of the human body, including Alberti's short treatiseD e s tatu a , which expounds an instrument for the taking ofthe necessary measurements from actual bodies. He keenlyfollowed Verrocchio's lead when he began to tackle thedesign of a huge equestrian memorial to Francesco Sforza,Ludovico's deceased father. A number of drawings testifyto his visits to Milanese srables in order to take precisemeasurements ( 1 1 1 . 7 ) . Again, a nice picture emerges ofLeonardo in action - this time arrempting patiently toget the measure of the thoroughbred horses in the stablesof Ludovco, his nobles and his military commanders.The patience of the horses is unlikely to have matchedthat of rheir measurer. That Leonardo needed a lot ofdetail is not in doubt. He breaks down the measurementsinto grad (~6 of a head), minut i (~6 of agrado) and minimi(~6 of a mnuto) .


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PlATE 111.6Square 01mult iples a dother studies nC.1496Penand ink20.5 x 16 cmt.ondon, British LibrarCodexArundel, 153r y,

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111 .7Proportional measurement01a horseC.1490Windsor Castle, The RoyalCollection, 12319

The search for human proportions occupied him moreor less throughout his career. Even his late studies of theheart valves may be seen as feeding his conviction thatevery aspect of the body was subject to proporrional rule.His search embraced the overall geometry ofhuman design,signalled in his reworking of Virruvius's formula that amanwith outstretched arms and legscan be inscribed in a circleand square (11.30). It also dealt arithmetically with the. minutiae of the srnallest components, such as the toes andfingers. At first sight, the drawing of the Vitruvian manin Venice seems to be concerned with relatively simpleproportional relationships berween the head, the bodyand the limbs, but the main vertical axis is pockmarkedwith compass points, particularly densely in the rnan's face.It seems that, in order to establish 'the proportionalityof the component parts ... from which human beauty iscornposed', Leonardo was dedicated to pushing his rneasure-ments down to the smallest practical scale. A set of relateddrawings at Windsor from around 1490 (111.9 and 111.10)

suggests thar he was planning a systematic excursus on rhesubject, either as a free-standing treatise or as part ofhis planned book 'On the Human Figure'. Other ofhisproportional srudies bear witness to his conviction tharthe drawings he made around 1490 had far from exhaustedthe problern. either in detail or in mode of analysis.The minuteness and intricacy ofhis efforts to define

human proportion may well have drawn inspiration fromVitruvius's systematic account of the proportional rulesthat govern every component in a c1assical building.Leonardo followed Vitruvius's account of the propergeometrical and arithmetical relationships in a columnbase with close attention (111.8). For him, the human faceshould be treated with no Jessscrupulous attention thana column base.No one had previously sought to define the cornplex

and interlocked internal harmonics of the body in suchdetail. He was striving to show how the dimension of onesmall part finds continual resonances either separately or in

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PlATE 111 8Narnes and proportions 01a colurnn basee 1493Penand inkOpening 012 pages,each 9 2 x 6_ 4 cmLondon, vrctor.a and AlbertMuseum, Codex Forster 1 1 1 ,44v-4sr

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PlATE 111,9Proportions of a man's torsoandlegsC 1488Penand ink147 x 217cmwindsor Castle, The RoyalCollectlon, 19130V

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PLATE 111.10Proport ions of a man's arme 1488Penand ink12.5x207cmWindsor Castle, The RoyalCollecuon, 1913H

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PLATE 111.11Studies of men climbing stairsand a running man stoppingC.1508Penand ink19 x 13.3cmWindsor Castle, TheRoyalCollection, 19038v

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rnultiples rhroughour the members of the body - in a kindof intricate visual polyphony in which harmony arises 'fromthe conjunction of ... proportional parts, which make theireffect simultaneously'. His other variation of the traditionofhuman proportions was not so much an extension oftradition as a departure. Concerned as he was with thebodyas a mobile, dynamic whole, Leonardo realized thatstatic proportions were not the entire story Whar happenswhen someone sits down, or bends rheir arm? He couldriot entertain the idea that proportionality would be Iost.Rather, the relationships would change. A number ofdrawings in the Windsor series tackle exactly this problem,seeking the proportions of the reconfigured body in itswhole and in its parts (i 11.9 and 111.].0). We can thereforethink of warching a body in motion as a kind of musicalcomposition in space and time. The man stepping upwardsalong the time-Iine ofhis axis of gravity can be seen in thislight (iII.J.l), as can rhe hammering man (iV.43), even ifin the latter case the music is more Stravinsky than Josquindes Prez.As in other areas ofhis activity, there is a complex recipro-

carien between empirical observation and thinking on paper.lndeed, since no individual can be trusred to provide anideal exernplar, the quest is ultimately driven by its premises,borh in its overall assurnptions and in rhe ratios thar aresoughr and construcred. Often the drawings thernselvesappear to be investigatory 'experirnents', in which new ratiosand conjuncrions arise during the course of rhe draftingprocedure. 1have suggesred thar detailed measurementswere raken fr om rhe drawing of rhe Vtru va n Man - rhar isto say, from rhe paper man, not from nature. The deducedratios must, in theory at least, always be confirmed innature by 'experierice', but none ofhis drawings ofhumanproportions (unlike those ofhis horses) records actuallinear measures taken from specific men. There may belost sheers recording detailed measuremenrs from actualbodies, bur the i r absence in surviving manuscripts seemssymptomatic of the main thrust of his research. 1 suspecrthar ir was predominantly paper-based, with just enoughchecking to retain its legitimate 'roots in nature'.

From Li ne to FormThe plastic impulse we have sensed at work in his Hatgeometry. either in its abstract form or as applied toproportions in nature, was given full rein when Leonardomoved into the realm of solid geometry His pice de rsis tancecarne in the set of drawings he provided for Luca Pacioli'sD e d iv ina proportione, known in two manuscript versions, thebetter of which is dated 1498, and in the book publishedwith woodcut illustrations in 1509. As such, the woodcutsare the only illustrations by Leonardo published in hisown lifetime. Pacioli's treatise, heavily dependent on Pierodella Francesca's De cinq ue corporbus regulan'bu s, expounds mebasic geometry of the five regular or 'Platonic' solids, andthe role of the 'divine proportion' (the 'golden section) intheir construction. The five bodies - me retrahedron (fourtriangular faces), cube (six squares), octahedron (eighttriangles), dodecahedron (12 pentagons) and icosahedron(20 triangles) - are the only solids thar are composedof identical faces and wholly symmetrical around theirvertices. They had long been regarded as having a specialstatus in revealing the divine perfection underlying theuniverse. Indeed, as Leonardo and Pacioli knew, Plato hadidenrified four of thern with the elements (cube =earth,icosahedron =water, octahedron =air, retrahedron = fire),with the dodecahedron seen as rhe quintessence of theuniverse as a whole. The close collaborarion betweenthe two colleagues at Ludovico's court is refiected in thefulsome compliments thar the rnarhefnatician paid toLeonardo's wide-ranging expertise and 'ineffable left hand'.For his part, Leonardo repeatedly paid direct and indirecthomage to the inspiration provided by Pacioli. A niceexample is his transcribing of a Iittle verse that Pacioliincluded in his treatise:

The sweet fruit, so attractive and refined,Has already drawn the philosophers to seekOur cause, in order to nourish the mind.[M 80v]

Pacioli extended his treatment of the five basic solids intotheir truncation and stellation. He demonstrated how thesymmetrical truncation of rhe bodies at their vertices (rhat

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PlATE 111.12Truncated and stellateddodecahedronfrom Luca Pacioli, Dedivinaproportione, Florence, PaganusPaganius, 1509(original drawing 1498)Woodcut281 x 20.4cmLondon, Vctoria and Alberl:Museum, National Art Libra ry,878.30

is to say, progressively sJicing off rheir corners) can resultin semi-regular or 'Archrmedean' solids. The basic andsemi-regular forms are rhe-n stellated (rhat is to say,pyramidsare erecred on their faces). What Leonardo provided forPacioli were brilliant graphi..: devices for letting the spectatorgrasp 'in a s ingle glance' how the forms are constructed.He produced beautiful shade.d images of the solids as if theyare material bodies suspended in space under directionalillumination, and, in a stroke of graphic genius, also displayedrhern in a skeletal or 'vacuus ' guise (111.12), with each visibleface shown as if it is a window frame through which theshape of the whole body could be perceved. Again, Leonardohas found the way to achieve the 'aha' effect, when thespectator is suddenly able to 'see' something thar has beendifficult to grasp using conventional techniques of repre-sentation. AlI that is rnissing, to complete the vivid visualdernonstration, is the rotation of the solids - somethingwe can now accomplish with compurer graphics.Leonardo, unsurprisingly, did not stop with what

Paciolihad achieved, and continued to explore problems ofstereornetry (the geometry of volumes) in an Archimedeanvein. In Codex Forster 1 (1 11.J.4) he considers a series ofsrereornetric problems on irregular and regular bodies.Ar the top of the right page he draws a transparent andsemi-shaded dodecahedron. He wishes to find a cube orrectangular figure equivalent in volume to the dodecalnedron.He begins by extracting one pentagonal face with anextended pyramid from the solidoThis pentagonal pyramidis then itself sectioned into pyramids with triangular bases,and one of rhe resulting wedges is transformed into atriangular slice with a rectangular base. Finally this slice istransformed into a rectangular body or cube, rhe area ofwhich is readily dererrnined.In rhe Codex Arundel (1 11.J.5) a related set of stereo-

merric problems is tackled with notable plastic fluency.The pyramids, looking uncannily like steeples on towers,are transmuted and turned in space as Leonardo seeks toger the rneasure of their volumes. What he is attempringto do, here as elsewhere, is to create figures of equivalentvolumes, often on the same bases. Once he has projecteda cube or box-shaped body equivalent in volume to theoriginal object, the volume of both can readily be calculated.,

111.13Lamps with revolving fountains I?)and geometryPLATEIV.l< (detail)

Once again, geometry becomes a kind of plastic manipula-tion akin to sculpture.On another of the Arundel sheets (11I.13), he takes

the famous formula of Pythagoras - 'the square of thehypotenuse of a right-angled triangle is equal to the sumof the squares of the two opposite sides' - into rhe rhirddimension. Rather rhan squares on the sides of a flattriangle, Leonardo builds cubes on the sides of a triangularbox or prisrn. The ratio of the sides of the triangle is 3:4:5,the cubes of which are 27, 64 and 125. However, the sum ofthe cubes of the shorter sides is 91, not 125, a difference of 34,as he notes in calculations below the diagram (iV.24 ). Ir isunclear how he derived the figure of 100, which he assignsro the largest of the cubes. In any event, he discovered rharPythagoras's theorem could not be automatically extrapolatedto the cubes on rhe sides of a prism. The problem is tharthe sides will not be squares with side lengths in the rato3:4:5, since the Iength of the sides perpendicular to therriangle sides will be the sarne.The rendering of three-dimensional forms on flat

surfaces, giving reality to his powers of plastic visualizaton,

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PLATE 111.14Determining the volume 01regular and irregular sol idsC 1505Pen and inkOpening of 2 pages,each 13.8 x 10.4 cmLondon, Victoria and AlbertMuseum, Codex Forster 1, sv-zr

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PLATE 111.15Oetermining the volume ofpyramida l and rectangu lar sol idse 1505Pen and mk218 x 14 7 cmLondon, Brttish Library,Codex Arundel, 182V

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was an empharic goal in his rechniques of drawing andpainring. He continually stressed the need to convey relievo(relief ar plasriciry). In drawing, rhis meant exploring evermore effective ways to exrend the paradoxical ability ofline on a Bar surface to describe form and space in rhreedimensions. He knew intellectually that nothing wasbounded by actual outlines, bur line was rhe necessaryvehicle of drawing and he worked rirelessly to make itplunge into depth and push forward from the surface ofthe paper. From the first, Leonardo demonsrrated anability to do this even in cursory sketches, setting up a fewsuggestive clues to denote perspective and foreshortening,and explo iting pen srrokes of varied pressure to 'sculpr'the forms. The plastic effect could then be enhanced byhatched shading or, more rarely, with rhe addition of atranslucent wash of ink applied with the brush.Until rhe late 1490S his favoured technique ofhatching

in pen, metalpoint and chalk consisted of parallel strakesrunning fram top Ieft to bottom right (the natural directionfor a Ieft-hander), often graded with extraordinary subtletyto convey the passage oflight acrass complex forms. Themetalpoint drawing of the Pr ojile of a W arror (i 11.16) isa virtuoso demonstration of what can be achieved withoutlines of curvaceous plasricity and parallel hatching of analmosr unbelievable delicacy It was indeed a demonstration-piece. We know that ir was based on a relief sculpture thatVerrocchio had pravided for the Hungarian king, MatthiasCorvinus, and we can sense Leonardo raking on the challengeto create relieve in rhe absence of any actual relief Ir is as ifhe isprecociously rehearsing the arguments in his 'Paragone',the extended comparison of the arts rhat he was to composein Milan more than 10years later:The majar cause of wonder rhat arises in painting issomething detached frorn the wall or other fiar surface,deceiving subtle judgements with this effect, as it is notseparated fram the surface of the wall. In this respecrrhe sculptor makes his works so that they appear whatthey are ... [whereas painting involves a] subtle investi-gation that concerns the true quality and quantity oflight and shade, which nature produces by herself inrhe works of the sculptor.

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PLA TE 11 1.16Bust 01awarrior in profile(1478Metalpoint on cream preparedpaper , with some whiteheightening, partially oxidized285'x 20,7cmLondon, British Museum,inv. no. 1895'9'15'474

Ir is more virtuous to remake the effecr rhrough knowledgeand rechnique rhan simply to take advantage of the facrthat it happens.Parallel hatching served Leonardo well for some 2 5 years

or more. However, in the late 1490S he discovered anorhermethod, probably having seen woodcuts by rhe grearGerman artist, Albrechr Drer. The first signs of rhe newharching - in which the strokes of rhe pen curve or hookaround the shaded parts of rhe form, as if embracing ir -occur in his mechanical drawings fram araund 1500 (111.17)The rechnique of short hooked strakes is particularly usefuIwhen shading thin forms, such as rods and axles. Afrer1500 we see curved hatching gradually insinuaring itselfinto other arcas ofhis draughtsmanship. Ir did not suppresshis parallel hatching, which remained especially effecrivefor rhe shading ofbackgraunds, but was used more andmore frequently when he wanted forms to look trulyrounded. The rows of curved lines increasingly encircle theform like a kind of graphic corset, tracking its convexityand tending to tres pass even into the lit portions. Thisharching acts as a kind ofdescriptive geometry' on thesurface of convex features. Particularly potent examplesoccur in Leonardos later anatomical studies, such as hispowerful drawing of the tongue in isolation ( iV . 1 S ) and thecompelling represenrations of the heart (1.32). The rotundwater vessels in rhe Codex Leicester are shaded in this way(i1.20 and 111.23). The curved hatching conveys a strongsense of a united mental, perceptual and physical mouldingof forrn, as ir is virtually sculpred on and into the surfaceof rhe paper. We can vividly feel the plastic impulse ofhisvisualizarion rransmitting itself into the physical reality ofthe motion of the pen in his hand.


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LECT Form Space

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