Pacioli’s Summa proved to be one of the most consequential books of alltime.

In 1540 a Venetianprinter named Domenico Manzoni excerpted them, without attribution(Pacioli himself had acknowledged most, but not all, of his sources) butusefully adding hundreds of worked examples which illustrated Pacioli’spoints. Tellingly, Manzoni retitled the work Quaderno Doppio, ‘the doubleledger’. Selling even better than Maestro Luca’s original, it went throughsix or seven editions and prompted a wave of adaptations and translations.

of the six hundred pagesof the Summa, only twenty-seven covered bookkeeping.

Pacioli’s reader, in whose company he would spend most of thefollowing decade, was Leonardo da Vinci.

Paganino de Paganini

In short: Book ix of the Summawas the nearest thing to an MBA textbook that the fifteenth century had tooffer. And one of the first lessons that its aspirational readers digested wasthat every business needed at least four blank books – the memoriale, orday book, the giornale, or journal, the quaderno, or general ledger, and abook for correspondence – and maybe even a fifth, the squartofoglia, or

He supplemented the commercial arithmetic with instruction in goodpractice in letter-writing, record-keeping, filing – and even that staple of theworkplace notebook, the things-to-do list

And buried deep inside,Book ix of the Summa presents a concise and surprisingly readable coursein double-entry bookkeeping, spelling out exactly how a business should berun – and why the Florentine-Venetian system of double entry was the bestway to do it. ‘Without double entry, businessmen would not sleep easily atnight’, he writes. ‘Their minds would keep them awake with worry.’

An ambitious synthesis of all the mathematical knowledge he could find,Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalitais a baggy monster of a book. Six hundred and fifteen pages long, nearlyhalf a million words, full folio in size, closely printed on fine paper, itcomprehensively sums up the state of European mathematical knowledge,and was intended for a wide audience – Fra Luca wrote informally, inTuscan, not Latin, making it accessible to anyone with a basic education.The book combines a general treatise on theoretical and practical arithmetic– including the Liber Abaci of the then little-known Fibonacci, whichPacioli had discovered on a monastery bookshelf – with an introduction toalgebra, currency conversions, multiplication tables, weights and measuresof the Italian states, a summary of Euclidean geometry, and accounts ofArchimedes, Euclid and Piero della Francesca.

5 curiosidades sobre a história e o acervo do Museu do Louvre

ξi(t) =ξip+t Xi−t22ΓijkXjXk+O(‖tX‖3)

he canonical divergence D induces the metric g and the connections∇and∇∗. The same holds for the mean canonical divergence D∇mcd

if∇is integrable, then it is notgenerally true that X(q,p) =−gradqD∇mcd(p‖·)

mean canonical divergenceD∇mcd(p‖q):=12(D(p‖q) +D∗(q‖p))(64)which obviously satisfiesD(∇∗)mcd(p‖q) =D∇mcd(q‖p)

he energy of the geodesicγp,qas the symmetrized version of the canonical divergence:12(D(p‖q) +D(q‖p))=12∫10∥∥ ̇γp,q(t)∥∥2dt

D(p‖q) =∫10t∥∥ ̇γp,q(t)∥∥2dt(61)whereγp,qdenotes the geodesic from p to q.

D(p‖q) =∫10(1−t)∥∥ ̇γq,p(t)∥∥2dt

inverse exponential map atγq,p(t)satisfiesXt(q,p)= (1−t) ̇γq,p(t)

n-dimensional dual manifold(M,g,∇,∇∗). Consider a∇-geodesicγq,p:[0, 1]→Mconnectingqandp. We define a tangent vector fieldXt(p,q)along this geodesic:Xt(q,p):=X(γq,p(t),p)(52)Obviously,X0=X(q,p)(53)X1(q,p) =0(54)Definition 3.A canonical divergence from p to q is defined by the path integralD(p‖q) =∫10〈Xt(q,p), ̇γq,p(t)〉dt

∫10〈X(γ(t),p), ̇γ(t)〉dt=−∫10〈gradγ(t)Dp, ̇γ(t)〉dt=−∫10(dγ(t)Dp)( ̇γ(t))dt=−∫10d Dp◦γd t(t)dt=Dp(γ(0))−Dp(γ(1))=Dp(q)−Dp(p) =Dp(q) =D(p‖q)(13)In particular, we can apply this derivation to the geodesic connectingqandpeven when theintegrability ofXis not guaranteed and obtain the definition of a general canonical divergence

functionsDpsatisfying the condition of Equation (12) then they are uniqueup to a constant that can vary withp, and we can therefore assumeDp(p) =0

aRiemannian metricgonM. Given such a metric, we assumeintegrabilityofXand∇, respectively,in the sense that for allpthere exists a functionDpsatisfyingX(q,p) =−gradqDp

although being quite restrictive in general, thisproperty will be satisfied in our information-geometric context, wheregis given by the Fisher metricand∇is given by them- ande-connections and their convex combinations, theα-connections

pointspandq, one can interpret anyXwith expq(X) =pas a difference vectorXthattranslatesqtop

p−q=−gradqDp(9)Here, the gradient gradqis taken with respect to the canonical inner product onRn

fixed pointp∈M, we want to define a vector fieldq7→X(q,p), at least in a neighbourhood ofp, thatcorresponds to the difference vector field

manifold is dually flat, a canonical divergence was introduced by Amari and Nagaoka [2], which isa Bregman divergence

a divergence exists for any such manifold. However, it isnot unique and there are infinitely many divergences that give the same geometrical structure

find a divergenceDwhich generates a given geometrical structure(M,g,∇,∇∗)

the coefficientsDΓijk(p) =−∂i∂j∂′kD(ξp‖ξq)∣∣q=p(5)DΓ∗ijk(p) =−∂′i∂′j∂kD(ξp‖ξq)∣∣∣q=p(6)define a pair of dual affine connectionsD∇andD∇∗[1]. The duality of the connections holds withrespect to the Riemannian metricDgin terms of the following condition:X〈Y,Z〉=〈D∇XY,Z〉+〈Y,D∇∗XZ〉(7)for all vector fieldsX,YandZ, where the brackets〈·,·〉denote the inner product with respect toDg

he coefficients of the Riemannian metric can be written asDgij(p) =−∂i∂′jD(ξp‖ξq)∣∣∣q=p=∂′i∂′jD(ξp‖ξq)∣∣∣q=p

When a coordinate systemξ:p7→ξp= (ξ1p, . . . ,ξnp)∈Rnis given inM, we pose one condition that, for two nearby pointsξpandξq=ξp+∆ξ,Dis expanded asD(p‖q) =12Dgij(p)∆ξi∆ξj+O(‖∆ξ‖3)(2)and(Dgij(p))ijis a positive definite matrix.

A divergence functionD(p‖q)is a differentiable real-valued function of two pointspandqin amanifoldM. It satisfies the non-negativity conditionD(p‖q)≥0(1)with equality if and only ifp=q.

Distribuições uniformes

um modelo estatístico regular de dimensãon−1

Riemannian manifolds (Mm,g) and (Nn,h), and a functionψ:M→R,defining a Riemannian space ( ̃M, ̃g), where ̃M=M×Nand ̃g=g+e2ψh.