rosinska - bianchini decimaltables from ms. modena bibl.estense, lat145

7/30/2019 Rosinska - Bianchini DecimalTables from ms. Modena Bibl.Estense, Lat145

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A R T Y K ' U Y

Grayna Rosiska(Warszawa)

D E C I M A L P O S I T IO N A L F R A C T I O N S * .T H E I R U SE F O R T H E S U R V E Y I N G P U R P O S E S

( FER R A R A , 1442)

So m e years ago, G iovanni B ianchini (ca. 140 0-ca . 1470) was introd uced intothe history of mathematics as the author of the tables of decimal tr igonometricfun ctio ns, the first such tables in the Wes t1 . In these tables, how eve r, the num ericalvalues were not tabulated as fractions 2 . The question that I posed then to myselfwas about the presenc e of decim al fractions, in proper sense of the word, in o therBianchini 's works t i l l now unpublished or even uncatalogued. Lastly, whilepreparing the f irst edit ion of B ianc hin i 's De arithmetica - so is entitled the tre atisethat opens his Flores Almagesti - I wondered whether i t would be possible toattr ibute the invention of decima l posit ional fraction s to Bian chini o n the gro und sof a fragment that runs as follows:

". . .omnis f igura f irmata in ordine numerorum dnott fractionemdecenariam loci f igurae immediate sequentem ad sinistram, ut puta342. Dico qu od 2 dnott duo decima unius de ce na ee t4 sunt 4 dec imaunius centenarii etc."3

Actually, as it results from the quotation, even if Bianchini uses the term"frac tio dece naria", n evertheless h e does not go as far as to consider ten th parts ofa unit4.

For tunately , how ever , the De arithmetica appears not to be Bianchini 'sult imate expression about decimal fractions, since his mathematical and astrono-mical legacy furnishes st i l l another source valuable for the subject , namely thetreatise Compositio instrument?. In this treatise, f inally, the concept of decimal

K w ar t a l n i k H i s t o r i i N au k i i Te chn i k i R . 40 : 1995 n r 4 s . 17 -3 2


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18 G. Rosiskaposit ional f ractions, together with the rules for the operations on them, are clear lyexposed .

T h e Compositio instrumenti is devoted to the construction and use of a survey-ing ins t rument . The calcu la t ions of the a l t i tudes and d is tances of inaccess ib leobje cts are based e i ther on the theorem of similar t r iangles or on the Pyth ago ras 'theorem. All is referred to the Elements.

Bianchin i descr ibes an ins t rument that a t a g lance seems to be re la ted toP t o l e m y ' s regulae (Almagest V, 12). I t is only a closer lo ok at the p rinc iple s of itsconstru ct ion and use that permits to d iscover the connect ion of the ins t rum ent withthe observat ional quadrant ( the scala mathematica at the revers of a quadrant)rather than with the regulae6. Althoug h Bianchin i do es not g ive any specif ic nam eto the ins t rument , accord ing to Domenico Fava such ins t ruments were knownunder the nam e biffc?.

The instrument deserves an exhaustive study for i tself , as well as for i ts placein the Renaissance t rad i t ion of the surveying ins t ruments . For the sake of th ispap er , how ever , I con f ine myself to the s tudy of wh at is the essenc e of B ianc hin i ' sach ieve m ent, na m ely to the study of pr inciple s of both, cons truction and use of theinstrument, the special at tention being paied to i ts scales.

In f ac t , wh i le to sca le the in s t rumen t Bianch in i in t roduces f r ac t ions o f thetenth progress, and subsequently while operating on the data read off f rom theins trum ent and expressed in decimal pos i t ional f rac t ions , nam ely w hile mult ip ly-ing, dividing and extracting roots , he follows rules used in our days, and employsthe decimal point to dist inguish the whole number par ts ( integers or "zero inte-gers") f rom f ract ions .

T h e Compositio instrumenti, dedicated to the Marquess Leonel lo d 'Es te ofFerrara, the patron and fr iend of Bianchini, had to be ready not later than in theco urse of 1 450 - the year of the dea th of Le one llo . Th e analysis of the m anu scripttex t , how eve r , perm its to suppose that the t rea t ise was of fe red to Leo nel lo as ear lyas 144 2, som e m onth only af ter he succe ded his fathe r Nic colo (died in 1441) tothe rulership of Ferrara 8 . As for the copy of the treatise prese rved at the BibliotecaEstense , Modena, Cod. Lat . 145 ( a .T.6 .19) , t i l l now the unique i tem known tohistor ians, i t seems to be done twenty f ive years later 9.

Fur the rm ore , the Estense copy, a l though wr i t ten on vellum and provided withthe border decoration (for these reasons, probably, currently taken for the or iginalof fered once to Leonel lo) , conta ins fau l ts that a l ter the sense of Bianchin i ' sexposit ion. I t seems, therefore, that this copy was even not done directly fromoriginal but f rom one of the copies that possibly circulated in I taly in the secondhalf of the f if te en th centu ry. In cer tain plac es the law q uali ty of the text is surelydue to the copyist , ignorant of the subject .


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Decimal positional fractions 191. T H E P R I N C I P L E S O F T H E C O N S T R U C T I O NA N D O F T H E S C A L E O F T H E I N S T R U M E N T

In the letter of ddicac opening the treatise, Bian chini m ention s E uc lid andPtolemy as the sources of inspiration, and displays the contents of the treatisedisposed in five parts. While in parts one and five the construction of the instru-ment and its adaptation to various purposes is discussed, in parts two, three andfour the use of the instrument is show n, i llustrated with exam ples. Eac h ex am pleis provided with a diagram that interprets geom etric ally the situation resulting fr omthe sighting. The calculations are done either by the rule of three or by thePythagoras' theorem. The results of calculations given in numbers with decimalfractions, close the procedure.

According to Bianchini's description here is a matter of a large device, in thestyle evok ing the future Tyc hon ian instruments. The vertical post AB ( c a t h e c u s )is about four meters (duode cim p edes) high. Th e peculiarity of Bia nc hin i's instru-ment l ie s in two p a i r s o f bars: AQ, LQ and DE, EM . The respective use o fthem enabled the surveyor to take two sights of an object without changing theposition of the instrument itself.

While the horizontal bars AQ, DE (bases) are fixed atright angle s to the post A B , (in points A, D ), the transversalbars LQ, EM (h y po t h e n u s e ) are pivoted at the extremities(Q, E) of the horizontal bars. Each of the transversal bars isprovided either with two pinnule (du e tabelle otoni, perfo-rate) or with a sighting tube ( c an e l l a ) . During the sightingthe lower extremity of a transversal bar slides freely alongthe scale of the post10. Actually, the transversal bars are thealidads, but since Bianchini avoids the names that are notlinked with the classic tradition, he calles them not alidadebut sim ply rige or linee transversales. (I mark in m y sketchthe scales that are missing in the manuscript, and I use, atvariance with the copyist of the manuscript, the upper castletters)11.

At the very beginning of the treatise Bianchini remarksthat the way to scale the post and the bars is especiallychosen to facilitate arithmetical operations. He speaks thenof m ultiplication and of div ision but in the course of thetreatise, w he n the Pythagor as' theorem is used, the ques tionof the extraction of square roots emerges as well.

The unit of integers used in the scale of the instrumentis foot (pes). Bianchini presents the idea of the divisio n o f foo t in ten parts, and ofthe subsequent decim al su bdivision of eac h of its parts etc. , as a new de vic e. H e

Rys. 1


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20 G. Rosiskadoes not underl ine, however, his-own authorship of the procedure 1 2 . W hi le apply-ing names to the new units of length Bianchini uses "tradit ional" terms. He callsuntia one tenth of foot, minuta one tenth of untia, secunda one tenth of minuta,tertia on e tenth of secunda etc. In this wa y B ianchini presen ts the idea of decim alposit ional fra ct ion s using a m elang e of nam es originally at tr ibuted to the units oft ime, of angle and of length. The eclect ic termino logy, h ow eve r, does not al ter theclari ty of Bianchini 's exposit ion.

According to Bianchini ' s own words :". . .every section [of the post and of each bar, horizon tal and tranv ersalas well] , having the leng ht of one foot , has to be divided into ten equa lparts. [. ..] Th ese divisio ns w ill be called uncie. T h e uncie will be alsodivid ed [each] in ten parts, and the [units resulting fr om ] this div isionwill be called minutes. Then minutes will be divided [each] in tenparts, [the space] permitting, [. . .] and these divisions will be calledseconds. And if minutes [for the lack of the space] can't be dividedin ten parts, be divide d in five [parts], so each div ision w ill be [equalto] two seconds. Rem ark - co ncludes Bianchini - that these divisionsare always ended with ten, and this in order to faci l i tate mult ipl ica-t ions and divisions. 1 will teach it below" 1 3 .

2 . T H E S U R V E Y I N GBianch ini d escribes the use of the instrumen t for such purp oses as the determ i-

nation of the heigh t and of the distance of an inaccessible tow er, the m easurem entof the de pths of a terrain the instrum ent b eing situated in the up per pa rt of it , finallythe measurement of a dis tance between two inaccessible objects . Aa a rule thesightings have to be taken with each of al idads respectively. Exceptionally,how eve r, the upper bar is taken o ff , being rather a hinder than a help in the w ork,so the s ightings are accom plished w ith the lowe r al idad alone14 .Now we wil l look closly at the firs t example of the use of the instrument,concerning the determination of the height and of the distance of an inaccessibletower. Th e other exam ples wil l be referred to only if they supply m ore info rm ationconcerning operat ions on decimal f rac t ions .

While determining the height of the tower the measurements are taken twice,by m eans of the pinnule s i tuated at the upp er (pivoted) and low er (s l iding) ends ofeach of two al idads. The results of s ightings, expressed in feet and their decimalsubdivisions etc., are read off from the scale of the post. These results, in theexample presented by Bianchini , are respectively:


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Decimal positional fractions 21pedes .0. untie .7. minuta .4. et secunda .6.

And:pes .0. untie .8. minuta .3. et secunda ,4.15 .

After the sighting a diagram is drawn, expressing geometrically the situation.

3. TH E DEM ONSTRATIO IN QUANTITATE CONTINUAO R T H E G E O M E T R I C A L B A S E F O R C A L C U L A T I O N S

H

Rys. 2W hile the aim of the sighting was to determ ine the sides of the small triang lesgiven in the instrum ent (FA Q and G DE ), the aim of the geome trical cons iderationsis to prov e the similarity of these triangles to the large one s (FN H and G O Hrespectively), given in the field.(1) A FAQ is equiangle with FNH , and A GD E with A GO H 16.< FAQ =< FNH , < AQF=< NFH, < AFQ=< NFH< GDE=< GOH < DEG=H GO < DGE=< GHO


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22 G. Rosiskasince according to VI,4: In equiangular triagles the sides about the equal anglesare proportionals,

(2) AQ:FN=A F:H N and DE:GO =DG :HO,but since AQ =D E and FN= GO , Bianchini applies to (2) Elements VII, 9, thatin his ow n w ording run as follow s; Si fuerint quator num eri quorum prim us se-cundi tota pars fuerit quota tertius quarti, erunt permutatim p rimus tertii totapars seu partes, quota secundus quarti,17and obtains the following p roportions:(3) AF:AQ = HN:FN and DG:DE = HO:GO.

Th e further transformations lead eventually to the formula:(4) H O = F G D G : ( D G - A F)18Th e Regu la brevis, given by Bianchini in the next chapter, explains the use ofthe instrumen t as follows:

[1] Take the first sighting with the LQ.[2] Take the second sighting with ME.[3] Sub tract the result of the first sighting fro m the result of the second one.[4] Take fro m the post AB th e value corresponding to FG .[5] Multiply it by the value of the second sighting, D G .[6] Divide the product by D G - AF, and you obtain HO = FG DG : (DG-AF).Th e height of the tow er is equal to HO + OP, the quantity OP = G being takendirectly from the instrument.(By the similar procedure B ianchini finds the distance of the tower (FN = G O).Then , two sides of the right-angled triangle know n, he finds the distances FH andGH by Pythagoras' theorem).Once proportions established and the formula found, Bianchini proceeds,according to his own expression, from "continuous quantities" to "discrete num-bers"19. In a word: Bianchini replaces lines by numbers (decimal fractions) andpasses to calculations.

4. THE OPERATIONS ON DECIMAL FRACTIONSBianchini uses two ways in the writting dow n the decimal fractions. The firstway occurs wh ile the readings are taken o ff from the decimal scale of the post:pedes .0. untie .7. minuta A. et secunda .6.,Th e same number, however, while used in the arithmetical operations, assumesa purely decimal form and is expressed as .746. seconds uderstood as 0.746 foot:


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D ecim a l p osi t i o n a l r a ct i o n s 23"T o facil itate operations reducepedem .0. u n t i a s .7. m i n u t a A . secu n -d o .6. to the last fraction, the smallest one namely to seconds. This isjust for this purpose that I did decimal divisions, for you can put theinvented similar parts [the smallest uni ts] continuously, namely: .746.Thus there wi l l be .746. seconds. A nd in the same way r educe u n t i a s.8. m i n u t a .3. et secunda .4. to the same kind, and will be .834.seconds"20.

B ianchini explains the operations on decimal fractions beginning wi th additonand subtraction. I t happens only in the examples of these operation that the succe-ssive numerals of adecimal fraction are accompagned by the names of units theyrepresent. The contrary occurs when Bianchini explains multiplication, divisionand extraction of roots. Then he abandons the metrological names for decimalplaces as such, and adecimal fraction is written as a "continuous number" (n u m e r u sc on t i n u u s ) .

A F = .746. D G = .834. F D = .834. - .746. = .88.N O = F G = F D + D G = 4 + (.834. - .746.) = 4.088.

H O = 8 3 4 m = 38.7.4.3 - V 2 1.88. 11

or: pedes .38. u n t i a .7. m i n u t a .4. secunda . 3 Bianchini rounds it to 38.7.4.3., adds the value corresponding to OP taken of f

from the instrument, and receives final ly the height of the tower.A s a rule, in the Compo s i t i o only the final results of the operations are given.That is why I had to recur to other Bianchini's treatise, namely De a r i t hme t i c a ,

while reconstructing the calculations. B esides, the "rules" of calculations one findsin the D e a r i t h m et i ca are very similar to the "rules" currently explaned in thefifteenth-century ari thmetical treatises. W ith the one exception, however, namelythe rule of signs. It was Bianchini the first to formulate it and to "prove" itgeometrically. I signal here the rule of signs, (even i f it has nothing to do with theuse of the b i f f a ) , because of its special place in Bianchini's contribution to theRenaissance mathematics22.E xample of addition:

p es .0. u n t i e .8. m in u t a 3 . et secun d a A .plus

pedes .3. u n t i e .2. m i n u t a .5. secun d a A .


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24 G. Rosiskaequal to:

pedes . untie. 0. minuta. 8. secunda. 8.Examples of subtraction: .834.

minus .746.equal to : .88.

pedes .4 .minus

pedes .0. untie.1. minu ta .4. secunda.6.equal to:

pedes.3 . untie.2. minuta.5. secunda .Exam ples of m ultiplication and of the raising to the second pow er:Multiplicaergoper .4088.

.834.1635212264327043409392 24According to B ianc hini's no tation the final result runs as follows: 34.0.9.3.9.2.Th e decimal point plays here a double role: it separates the integral part of a num berfrom the fractional one, and it marks the decim al places of the remaining fractionalunits.The raising to the second power of a num ber:feet. 92. untias .9. minuta. 0. secunda.9.gives to Bianchini an opportunity to explain the use of the decimal point inmultiplication. F irst he presents the abo ve va lue as figure continue: .92909. Theirsecond pow er is equal to .8.6.3.2.0.8.2.2.8.1. The determination ofthe integral partof the number is based on the formula 10m. 10n = 10m+n:


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Decimal positional ractions 25"C oun t then how many num erals are there in the mu ltiplied besides integers.Th ere is a fraction "90 9" w hich [is com posed] of three numerals. Th e sam e is inthe m ultiplicand beca use it is equal [to multiplied]. Both take n together co un t six

num erals. Cut off six ultime num erals fro m the product and rem ain four, namely:.8632. These are feet. The .0.8.2.2.8.1. is the rest, namely untie .0 . minuta .8.secunda .2. tertia .2. quarta .8. quinta. I.25Ex am ple of division:

387433 4 0 9 3 9 2 : 8 8264

769704

653616379352272264

8

T he result is equal to the quotient .38.7.4.3. with a remaind er of 8. A t this pointBianchini could apply to the division the "rule of zeros", and to continue itdecimally; instead of this he divides 8 by 88 in the usual way and receives "oneeleven th of a second [!] The 1/11 is added to the last unit of the decimal fr ac tio n ,and the final result expressed as .38.7.4.3 1/11 2 6. (In the same way the prod ucts arerecorded as .862.099173 ~ and .1501.027093 ^ | j - ) 2 7 .

The procedure looks like a betrayal of the decimal idea. Is this plausible thatBianchini, while dividing decimal frac tion s, becam e "blinded by an equivocationbetw een the decimal idea and the ordinary process of division " ? 2 8Although Bian chini 's examples of division are not accompanied by th e rule ofa decimal point, the determination of the integral part of the quotient implies theuse of the formu la 1 0 m : 10" = lO"" 1


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26 G. Ros iska*

* *

While an outl ine of Bianchini 's doctrine on decimal posit ional fract ions isgiven in the De arithmetica, the complete exposition of this doctrine is found inthe Compositio instruments Th e establishing w hich of the two treatises w as w rittenearlier requires further studies. From the systematical point of view the Dearithmetica seems to be antecedant to the Compositio. In fact, while the Dearithmetica signals the principle of the extention of the decimal positional num bersystem from integrals to fractions, the Compositio, at its turn, furnishes theexplan ation of the decimal do ctrine as applied to the problem s of metrology. Th emetrological fram ew ork, however, once being overpast, the universal m eaning ofthe decim al fractio ns is eventually shown .

W ha t seem s to persist am biguous in Bian chin i 's doctrine, is the place attributedin i t to com m on fractions. W hile appended at the end of decimal on es, they gen eratetogether with them quite a strange hybrid. Actually, in Bianchini 's treatise thecommon fractions do appear jus t in situations in which on e would expe ct Bianch inito point to a possibility of an infinite expression of a decim al fra ctio n (parallely towh at he had w ritten in the De arithmetica with regard to integrals).On e has not to be mistaken by B ianchini 's u se of words: numerus continuus,figure continue. W hat they me an is only a decim al frac tion as considered in term sof its smallest units. For instance .0.7.4.6. foo t is expressed " con tinuou sly" as .746.seconds. Thus, in the Compositio the "continuity" of a number seems to appearonly at the level of notation, being an effect of the om ission of m etrological n am es,(or poin ts tha t signale such na m es). In the Compositio instrumenti the "continuity"is not a qua lity inhe rent in the certain decimal e ntities (or form s), and m ean ing thepossibility of their infinite deve lopm ent (or me aning an infin ite set of fractions) .In fact, Bianchini abandoned "continuous quantities" for "discrete numbers"while he passed from geometrical proofs to calculations. Bianchini 's numericontinui are in fact discrete numbers. In this poin t Bianc hini 's decim al idea see m sto fail (even if Bianchini goes as far as to calculete values of the quarta, quinta,sexta etc).

Otherwise, the Bianchini 's realizing of the concept of decimal fractions, (heuses even the term fractio decenaria), is surely superior to the Stevin's one. Atvariance with the Simon S tevin 's "som ewha t unwieldy" notat ion, the B ianch ini 'snotation is almost as clear as the notation used in our days. This notation, how ever,onc e introduc ed into mathem atics by m eans of a treatise offere d to a Ren aissanceruler in homag e, eventually passed into oblivion. The regular use of the decim alpoin t appe ars as late as the eighteenth c eniury .

The concept of decimal fract ions in posit ional notat ion have revived inEu rope only about a century after Bia nch ini 's death (ca 1470). Th e authors of the


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Decim al positional fractions 27mo d e r n e x p r e s s io n o f th e d e c ima l id e a , h o we v e r , a c te d wi th o u t r e fe r e n c e to th eB i a n c h i n i ' s a c h i e v e m e n t .

No te s* It wa s E. J. D i j s t e r h u i s, who postuleted the use of the exp ressio n decimal

positional fractions for fraction s written in the decim al system of positio nal n otation (inc o n tr a st w i th f r a c tio n s wr it te n in th e d e c ima l sy s te m o f n u me re a tio n ). E . J . D i j k s t e r hu i s: Simon Stevin. Science in Netherlands around 1600. R . H a s, M . G . J . M i n n e t (eds). The Hague: Martin Nijhoff 1970 p. 17.

1 G. R s i s a: Tables trigonomtriques de Giovanni Bianchini. "HistoriaMathemat i ca l Vol . 8 1981 pp . 46-55 . E a d e m: Tables of Decimal Tr igonomet r i cFunct ions from ca.1450 to ca.1550. In: From Deferent to Equant. Volume of Studies intheudies in the History of Science in the Ancient and Med ieval Near East in Honor ofE.S.Kennedy. D . i n g , G. S a 1 i b a (eds) . "Annals of the New York Academy of Sciences"1987 Vol . 500 pp .419-421 .

2 In the tables of tangent and of cosecant Bianchini avoided fractions thanks tocalculations with the radius R equal to a number of units of length expressed in the form10. Sinc e in these tables, (dressed certainly bef ore 1463, or even b efo re 1450, acco rding lyto the last discoveries), Bianchini assumed for n respectively 3 and 4, thus all quantitieswere tabula ted as in tegers with a cer ta in accuracy considered suff ic ient for as tronomicalcalcula t ions; the determinat ion, however , of the real value of the tabula ted quanti t iessupposed the knowledge of n.

3 ". . .each numeral, f ixed in the order of numbers, denotes decimal fraction [of thenum ber that] fo l low s imm ediat ly a t the lef t of the place [occupied] by th is num eral , as 342 .I say that 2 denotes two tenth of ten, and 4 are 4 tenth of hundred etc". I refe re my self tothe cr i tica l edi t ion of B ian ch ini ' sD e arithmetica, fo r thcom ming in the S tud ia Co pern icanaseries.

4 Bianchini s eem s to hold here the posi t ion that som e hunderd for ty years la ter wil l b eheld by Sim on Stevin in the De Thiende. In fact , according to Stev in ' s Preface to th is w orkth e T h ien d e ta l en ( " te n th n u mb e r s " ) a re i nt eg e rs . S imo n S t e v i n : La disrne. Enseignantfacilement expedier par nombres entiers sans rompuz, tous comptes se rencontrans auxaffaires des Hommes. Premirement de scripte en Flameng, et maintenant en Franois, parSimon Stevin de Bruges. In: L'Arithmtique. Leyd e Chr i s tophe P lan tin 1585 pp .13 2-1 48,part iculari ly pp. 139-1 40; George S a r t n: The first explanation of decima l fractions andmeasures (1585). Together with a history of the decima l idea and a facsimile of Stevin'sDisme. " Is i s" No 65 . Vol .XXII I ( l ) 1935 pp .153-244 . (Facs imi le on pp .230-238) . I t i sgeneral ly admit ted that Stevin " introduced decimal f ract ions for general purposes , andshow ed that operat ions could be perform ed as easi ly with such fract ions as with in tegers" ,M .G .J. M i n n a e r t: Simon Stevin. In: Dictionnary of Scientific Biography. V o l . 1 3 , 1 9 7 3p.4 8; E.J. D i j s t e r h u i s , op. cit . p.19, und erlines the inc om plete nes s of the St ev in 'sachievem ent: " . . .we m ay ask the quest ion as to how far Stevin can be considered to have


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28 G. Rosiskaintroduced the decimal p ositional fractions by m eans of these tenth num bers. Strictlyspeaking, h e obv iously did not introduce fractions at all; his tenth nu m bers are integers,and he even considered this feature to be their principal advan tage".

5 Bian chini 's t reatise was mentionned by Dom enico F a v a: La Biblioteca Estense nelsuo sviluppo storico. Co n il Catalogo dlia M ostra P ermanen te e 10 Tavole. (Modena:Libreria Editrice G.T. V incenzi e Nipoti , 1925) p.36: "Al circolo di Leo nello appartenevapure il m atema tico ed astronomo G iovanni Bianchini, che con altissime cariche presso laCo rte estense e del principe fu per qualch e tempo consigliere. Di lui sono n ote le Tabulaeastronomicae [. . .] inedita e quasi sconosciuta 'un altra sua opera latina dedicata aLeonello, che tratta dello strumento per misurare, chiamato solitamente "biffa", traman-datoci dal codice membranaceo lat . 145 (=a. T. 6.19), che dev'essere stato esemplare didedica, avendo per titolo le parole: Illu. et Ex. Principi [. . .]. The transcription of theCompositio instrumenti was recently published by Paolo G a r u t i, with a historicalIntrodu ction by G ino A r r i g h i: Giovanni Bianchini. Compositio instrumenti (Cod. Lat.145=alpha.T.6.19) delia Biblioteca Estense di Modena. A cura di Paolo Garuti eonintroduzion e di Gino Arrighi. In: "Rend iconti Classe di Lettere e Scien zeM orali e S toriche.Vol. 125(1), 1991 pp .95 -12 7 (Isti tuto Lom bardo A ccadem ia di Scienze e Lettere, Milano1992). In Gino Arrighi 's Introduction (pp. 95-107), the contribution to the developmentof the surveying instruments due to Gerbert , Leonardo Pisano, Cristofano Gherardo diDino , and Francesco di Giorgio Martini is presented. On p.103, note 10 G. A rrighi reportsthat from 16 M arch to 11 April 1987 the man uscript wa s exposed in the Estense L ibrary,Modena, at the exibition "Materiali per la storia delie matematiche nelle raccolte delleBibliotech e Estense e Universitaria di M od ena ". Un usually en ough, the publication of theCompositio instrumenti is devoided of a com m entary on dating of the Estense man uscriptas well as on the contents of Bianc hini 's treatise. Th e publication w as mea nt by the A uthorsto make av ailable the Compositio to historians of science. As for the P. Garuti 's publicationof Bian chini 's Compositio instrumenti, I share Professor A rrigh i 's opinion on it , expressedat the end of the Introduction: ".. .gli storici delia scienza e della technica hanno da esseregrati al P. Paolo G aruti O.P. che ha com piuto la trascrizione c he qui per la prima v olta vienpub blicata." In my research I served m yself of the Paolo Ga rut i 's transcription, but abov eall I used a photocopy of the Estense manuscript kindly supplied to me by the Istituto eMuseo di Storia della Scienza, Florence. While quoting the Latin text I refer to PaoloGaruti 's edition, with the exception of cases in which my reading differs from Garuti 'sone; then I try to give both readings.

A manuscript copy of a fragment of a treatise on an instrument devoted to measure-ments of inacessible objects, attributed to Bianchini (preserved in the Bodleian Library,Oxfo rd , ms . Canon . Misc . 501 ), was signaled by Lynn T h o r n d i k e : Giovanni Bianchini 'sastronomical instrument. "Scripta M athe m atica" V ol. XX I 1955 p.136: "In it the discus-sion of the measurement of inaccessible bodies is followed on fol. 2r by Canones, "A dverificandum stellas fixas.. .". L. Thorndike, ibidem, signals the treatise De mensurationererum [in]accessibilium, of which only a fragm ent is know n, preserved at the BibliothqueN ationa le, Paris, m s. lat. 727 1, f. lv . The desinit of this fragment seem s to indicate that inthe treatise the adaptation of an astronomical (observational) instrument for the surveying


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Decim al positional fractions 2 9purposes is discussed: " . . .ut sicut res qui [sic GR] in celo videntur cum ipso mensurantur ,i ta res terrene visibiles e tiam si inaccessibiles essent metir i possint ." see L. T h o r n d i k e ,op. cit . p.136. We are not interested here in Bianchini 's treatises on instruments called"aequa tor ia" .6 See, for instance, Cartography, survey, and navigation to 1400. In ; A History ofTechnology. C ha rle s S i n g e r , E .J. H 1 m y a r d , A . R . H a l l , T re v o r I. W i 1 H a m s(eds . ) Vol . I I I Oxford Clarendon Press 1957 pp.527-529.

7 See above, note 5 .8 Leone l lo d'Est (21. Sept . 1407 - 1. Oct . 1450), in 1442 iniciated, together with

Gu arino G uarini , the reform of the Universi ty of Fe rrara .9 In fact , the manuscript reads on f. lv (I give here my-own reading of the text):

"Imperauit autem Antoninus Adriani gener et fi l ius adoptiuus post incarnat i uerbi nat iui-ta tem annis CXLII [Garut i reads "XLII"] . Id est annis iam M.CCC.XXV ["XXV" clearlycancelled in the manuscript], ex quo tempore ipsius Almagest i fides et ueri tas comproba-tur". Thus, taking accou nt of the expression: "an nis i a m " [elapsis] and of the ca ncellat ionof "XXV" in the date "M.CCC.XXV", we obtain 1300+142 = 1442. P. G a r u t i [ed.] ,p .10 8 omits "C " that exists in the manuscript , and so he reads "X LI I" instead of "C X LI I"(despite historical evidence concerning the dates of the rul ing of Antoninus Pius). It ispos sible that Ga ruti wa s induced in this readin g by som e (vagu e) po ints abou t "C ", thathe took for a sign of suppre ssion of the cipfer "C" . As for the X X V w rit ten dow n and thencance lled, P. G a r u t i , ^ . .a nn ota t ion 1, com m ents i t: "Il X XV pare cance llato". It see m spos sible that the years that resul t from the cons idering the "X X V " as if i t wa s not can celledare the date of the Estense copy. We have then 1 325+1 42 = 1467. Certainly, how eve r, theyear 1467 can not be con sidered as the date of the t reat ise i tself.

1 0 [Ms. M od en a, B .E. La t , 145, f .3r. P. G a r u t i [ed.], pp .1 09 -1 10 : P rim o fiat una rigaseu pert ica ferrea aut otoni , vel al terius metal li , [ . .. ] Si tqu e longa ped es duo dec im et pauloplus. [ . . . ] In cuius pert ice medietate ab una eius extremitate ad al teram figetur l inea rectalongitudinis punctual i ter pedum duodecim. Que l inea si t .a.b. [ . . . ] In parte vero superiorefiat plaga per t ransversum in qua ponatur al ia riga longitudinis pedum duorum et parumplus. In cuius medio et iam signetur l inea secans l ineam perpendicularem et ab ea interse-cetur in puncto .a. ad angulum rectum; que si t l inea .a.q. longitudinis punctual i ter pedumduorum. I tem supra l ineam perpendicularem .a .b . descendendo in f inem pedum quat tuor ,punctual i ter signetur terminus .d.; que si t l inea .a.d. in cuius extremitate fiat et iam plagaper t ransversum in qua ponatur al ia riga longitudinis et iam pedum .2. et parum plus. Incuius medio signetur l inea secans l ineam perpendicularem .a.b. in puncto .d. ad angulumrec tum . Qu e l inea sit longitudinis pun ctual i ter ped um .2. Et eri t linea .d.e. Item l inea .a.d.que est pedes quattuor, l inea vero .d.b., que est pedes .8., et l inea .a.q., que est pedes .2., etl inea .d.e. , que est et iam pedes .2. , terminentur per l ineas t ransversales ut dignoscatur pesquil ibet per se.

1 1 The scheme of the instrument is on f.2v; its reproduction in P. G a r u t i , op.cit .p .106 .

1 2 In the Introduction to the Compositio Bianchini presents the t reat ise as a resul t ofthe " intel lectual peregrinat ions" of it s author. Th ese "peregrin at ions" were accom pl ished


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30 G. Rosiskain the realm of the Euc lid's Elements and Ptolemy's Almagest. See Ms. Modena B.E. Lat145, f. 1-1V, P. G a r u t i [ed.], pp. 10 7-1 08 . Bianchini wa s aware of the value of the doctrineelucidated b y him on only few folios: "[...] opusculum [...] corpore quidem pusil lum , verumcogn itione ac viribus am plum ". Ms. M ode na B.E. Lat. 145, f.2, P. G a r u t i [ed.], .108.1 3 [Ms. M oden a B .E. Lat. 145, f.3. P. G a r u t i [ed.], p .l 10: "A c etiam linea cu iuslibetpedis dividatur in partes decem equales terminantes per lineas minoris longitudinis qu amsint linee terminantes pedes. Que divisiones untie vocitentur. Que untie etiam in partesdecem dividantur et signentur per lineas etiam minores vel per punctos. Que divisionesminuta vocenftur] [...] Et nota quod iste divisiones terminantur sem per d e decem in decem ,ut multiplicationes et divisiones per eas faciende, per doctrinam quam inferius docebo,facilius operantur". Follow s description of the pinn ule.

1 4 [Ms.Modena B.E. Lat. 145, f.l2v. P. G a r u t i [ed.], pp.125-126. In this chapterBianchini considers the measurement of the distance between two inaccessible objects.For this purpose the reduction of the length of the bars to only one foot is advised.Moreover, Bianchini recommends the construction of a device that permits to avoid thetranspo rtation of the wh ole instrum ent. Actu ally, it is a ma tter of a model of the foot m adein metal - "me nsura uniu s pedis de mtallo". The model, suplied with the decimal scale,permits to construct easilly an instrument in the field using all sort of objects accessiblethere such as "lanceas, pe rticas et similia", provided they remain pe rpend icular to the soil.

1 5 M s. Mod ena, B.E . Lat 145, f.4v- 5. P. G a r u t i [ed.], p .l 12. Elements 1,29; 1,3017 Elements VII,9. Cf. Th. L. H e a t h: The thirteen books of Euclid's Elements. Vol.

I I pp. 309 -3101 8 M s. Mo dena , B.E. Lat. 145, ff.5r-5v. P. G a r u t i [ed.], pp .11 3-1 14 .1 9 Ms. Modena, B.E. Lat 145, f.5v. P. G a r u t i [ed], p.l 14: "Nunc vero prefactas

conclusiones, que in quantitate continua demonstrate sunt, ad quantitatem discretamreducam , ut ad num erum propositum in numeris fiat conclusio". (I added the pu nctuationto the text).20 Ibidem.

2 1 1 give here my-ow n reading of the Compositio, M s. M odena, B.E. La t. 145, ff .6-6v :Iterum qu e catecus .d.g. est pes .0. untie .8. m inuta .3. et secunda .4., cui add ito catecu s .f.d.erique totus catecus .f.g. pedes .4. untie .0. minuta .8. secunda .8., qui erunt .88. parteshab ito respectu ad .834. partes altitudinis turris a pun cto .. Quia concluditur quod sicut sehabet .88. ad .834. et ita pedes [6>] .4. unc ie .0. minu ta .8. et secun da .8. ad totam turrim.

2 2 G. R s i s a: A chap ter in the history of the Renaissance mathema tics: negativenumb ers and the formulation of the law of signs (Ferrara, Italy ca.1450). In: "KwartalnikHistorii Nauk i i Te chn iki" ("Quarterly Journal of the History of Science and T echn ology ").Vol. 40 1995 (1) pp.3 -20 .

2 3 M s. Mo dena, B .E. La t. 145, f.6, P. G a r u t i [ed.], p.l 16.24 Ibidem.2 5 1 give here my own lecture of the Compositio, f.7r-7 v: Num era ergo quot figure sunt

in num erum mu ltiplicantem ultra pedes, id est .909., que sunt fractiones et trs numro. Ettotidem sunt in numro multiplicando, quia idem sunt. In utraque igitur erunt sex figure.


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Decim al positional fractions 3 1Et ideo resseca sex ultimas figuras de productis. Que relinquentur erunt quatuor, videlicetun tie .O.minuta .8. sec un da .2. tertia .2. qu arta .8. qu inta .1. (G a r u t i, p .l 16 re ad s ". 1.9 09 "instead of "id est .909.".

2 6 G a r u t i o p . c i t . p . 1152 7 1 d e m , p . l 16.2 8 These words are of George S a r t n, op. cit . p.172. They were intended by their

author as a possible just i f ica t ion of the procedures adopted by Pie tro Borghi (1484) ,Francesco Pel l izzat i (1492) and Chris toph Rudolff (1525) .

Grayna RosiskaD Z I E SI T N E U A M K I P O Z Y C Y J N E .

I C H Z A S T O S O W A N I E W M I E R N I C T W I E .(F E R R AR A, 1 4 4 2 )

Artyku przedstawia wyniki kontynuacj i bada prowadzonych przez autork w la tachosiemdziesitych i dotyczcych pierwszych na Zachodzie tabl ic dziesitnych funkcj it rygono me trycznyc h (przyp is 1) . W tabl icach tych jedn ak, wyliczon ych przez G iova nnie -go Bianchiniego okoo poow y XV wieku , wartoci l iczbow e nie byy ujte ja ko uam ki(przyp isy 2 ,4 ) .

W obec nej publikacj i , autorka posu gujc s i ma ter ia em rkopim ienny m z X V wiek uoraz wydanymi a le n ie opracowanymi dotd rdami, ukazuje pen koncepcj uamkwdziesitnych, opracow an przez Bianchiniego w celu uproszc zenia obl icze w ynik ajcychz pom iarw doko nywan ych przy zas tosowaniu w yna lez ionego i op isanego przez B ianch i -niego przyrzdu mierniczego (przypisy 5 , 6) . Poza opisem konstrukcj i i skalowaniainstrumentu, zachowanym w Bibl ioteca Estense w Modenie (sygn. Lat . 145) i opubliko-wa nym , bez komentarza , w 1992 roku (przypis 5), autorka odw ou je s i take do p rzyg o-towanej przez s iebie do druku edycj i krytycznej De arithmetica Bianchiniego, t rakta tu wktrym wy janion a jes t idea uam kw dziesitnych. Bianchini uywa naw et przy te j okazj ite rminu fractio decenaria (w przeciw iestw ie do Stevina, k try tego terminu nie uyw a) ,n ie posuw a s i jedn ak w koncepcj i u am kw dziesitnych dale j , n i to uczyn i Stevin wLa disme, dziele opub likow anym w 1585 roku (przypisy 3 ,4 ) .

W traktacie Bianchiniego o instrumencie mierniczym podane s , podobnie jak tozostao dokonane przez Stevina blisko 150 lat pniej, reguy dziaali na uamkachdziesitnych. Bianchini idzie dale j n i Stevin w tym sensie , e wprowadza oznaczenie(punkt) odzie la jcy cz cakowit l iczby od je j czci u amkowej. Na systematycznewprowadzenie podobnego oznaczenia t rzeba byo czeka a do poowy XVIII wieku.

Trakta t Giovanniego Bianchiniego Compositio instrumenti powsta w roku 1442(datacja ta zostaa ustalona przez autork na podstawie analizy rda oraz krytykiwewntrznej tekstu) i zosta of iarowany, w tyme roku, Leonel lo d 'Est z Ferrary jakohomagium.


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32 G. RosiskaWyoona w nim doktryna uamkw dziesitnych najprawdopodobniej nie miaa w

przyszoci w pyw u na prace Stevina i jego nastpcw. Miaoby zatem miejsce ponow neodkrycie" uamkw dziesitnych o goszone wraz z La disme.

Seria pub likacji na temat dokon a B ianchiniego w dziedzinie astronom ii i matem atyki(w tym pub likacje w KH NiT), ukazu je w nowym wietle matema tyk wosk XV wieku,zwaszcza matematyk zwizan ze rodowiskami uniwersyteckimi. W przeciwiestwiebow iem do uniwersytetw, rodowiska m atematyczne kupieckich scuole d'abbaco zostayju w duej mierze opracowane.

*W krgu wpyww Bianchiniego ksztaci si w Ferrarze zwizany z Kopernikiem

Dom enico M aria Nov ara. Zanim jednak mg Kopernik zetkn si z dzieem Bianchinie-go w e Woszech, w ykorzysta on, we wpisach do swego studenckiego notatnika uczynio-nych w czasie krakowskich studiw, Bianchiniego tablice astronomiczne ruchu planet,znane od okoo poowy XV wieku w Uniwersytecie Krakowskim. (KHNiT 1984, R. 29,pp.637-644). Zna take wyliczone przez Bianchiniego dziesitne tablice funkcji trygo-nometrycznych. (KHNiT 1981 R. 26 s. 567-577).

*

Miejscem, w ktrym powstawao to studium, bya Bibl ioteka przy Ist i tuto eMuseo di Storia del ia Scienza we Florencji oraz Bibl ioteka PAN, Bibl iotekaInstytutu Histori i Nauki PAN i Bibl ioteka Instytutu Matematyki PAN w Warsza-wie . Kompetencja i yczl iwo Bibl iotekarzy bya mi nieocenion pomoc.

rosinska - bianchini decimaltables from ms. modena bibl.estense, lat145

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