Latin Authors

"The remarkable principle of James Bernoulli consists exactly of this... namely, that the mean given by a series of trials falls near the number sought within limits so much the more narrow as the trials are more multiplied. All the properties which result from his learned researches constitute one of the most honourable monuments to his memory. But Bernoulli established his calculations on the hypothesis that the number sought was fixed and determined. ... It may happen that this quantity will experience small variations... But the principle of Bernoulli is still applicable to this case and has been demonstrated by M. Poisson by means of analysis. ...In the case before us the experiments should generally be very numerous: it is for this reason that M. Poisson has designated the extension of Bernoulli's principle as the law of great numbers."

"In 1690... Jacob Bernoulli brought up the problem of the catenary in a memoir... in the '...Huygens' solution represents the past... a complex, though skillful, geometrical method. Leibniz, using his new [infinitesimal calculus] reaches a correct analytical formula...y/a = (b^\frac{x}{a} + b^\frac{-x}{a})/2 where a is [a] segment... and b... corresponds to... e... Johann Bernoulli ...supplied two correct constructions ...presents valid statistical arguments and... new and important... equations of equilibrium in differential form. ...In 1697-1698, Jacob Bernoulli was the first to derive the general equations that not only solved the problem, but also permitted the treatment of the more general theme of the equilibrium of a flexible rope, subject to any distribution of tangential (f_t) and normal (f_n) forces. Bernoulli's equations are...\frac{dT}{ds} + f_t = 0, \qquad \frac{T}{r} + f_n= 0where T is the tension, s the curvilinear abscissa, and r the radius of curvature."

"Elastic Curve is the name that James Bernoulli gave to the curve which is formed by an elastic blade, fixed horizontally by one of its extremities in a vertical plane, and loaded at the other extremity with a weight, which by its gravity bends the blade into a curve... This problem is resolved by James Bernoulli in the "Memoirs of the Acad. of Sciences for 1703;" and other solutions have been given by some of the most celebrated mathematicians of Europe..."

"The term "induction" had been used by John Wallis in 1656, in his Arithmetica infinitorum; he used the induction known to natural science. In 1686 Jacob Bernoulli criticised him for using a process which was not binding logically and then advanced in place of it the proof from n to n + 1. This is one of the several origins of the process of mathematical induction."

"The art of measuring, as precisely as possible, probabilities of things, with the goal that we would be able always to choose or follow in our judgments and actions that course, which will have been determined to be better, more satisfactory, safer or more advantageous."

"[P]robability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers..."

"Eadem mutata resurgo [Changed and yet the same, I rise again]"

"The tract in which Leibnitz deals with series appeared late in the seventeenth century and was among the first on the subject. ...the question of their convergence or divergence ...was in those days more or less ignored. ...It was not until the publication of Jacques Bernoulli's work on infinite series in 1713 that a clearer insight into the problem was gained. ...Bernoulli's work directed attention towards the necessity of establishing criteria of convergence. The evanescence of the general term, i.e., of the generating sequence, is certainly a necessary condition, but this is generally insufficient. Sufficient conditions have been established by d'Alembert and Maclauren, Cauchy, Abel, and many others. ...to recognized whether a series converges or diverges is even today rather difficult in some cases."

"[T]he writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"Notwithstanding the broad foundation for mechanics laid by Newton in his Principia, and notwithstanding the indefatigable labors of Clairaut, d'Alembert, the Bernoullis, and Euler, there was near the end of the eighteenth century no comprehensive treatise on the science. Its leading principles and methods were fairly well known, but scattered through many works, and presented from divers points of view. It remained for Lagrange to unite them into one harmonious system. Mechanics had not yet freed itself from the restrictions of geometry, though progress since Newton's time had been constantly toward analytical... methods. The emancipation came with Lagrange's Mécanique Analytique published one hundred and one years after the Principia."

"[H]e was soon seconded by two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of it that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. ...I am speaking of the two brothers James and John Bernoulli."

"Quos omnes libros conscripserunt homines otiosi, male feriati, imperiti, vitiis ac spurcitiae dediti, in queis miror quid delectet nisi tam nobis flagitia blandirentur. Eruditio non est exspectanda ab hominibus qui ne umbram quidem eruditionis viderant. Iam cum narrant, quae potest esse delectatio in rebus quas tam aperte et stulte confingunt?"

"In corpore ipso quid forma est? nempe cuticula bene colorata?"

"We may therefore reckon Arnobius among those, who hold that the souls of beasts are rational. It is from him, without doubt, that Lactantius learned to put no other difference between them and man, but religion."

"I should wish, however, to know what this reason is, through which we are more excellent than all the tribes of animals. Is it because we have made for ourselves houses, by which we can avoid the cold of winter and heat of summer? What! do not the other animals show forethought in this respect? … But if nature, which gave them life, had chosen to give to them also hands to help them, they too would, without doubt, raise lofty buildings and strike out new works of art. Yet, even in those things which they make with beaks and claws, we see that there are many appearances of reason and wisdom which we men are unable to copy, however much we ponder them, although we have hands to serve us dexterously in every kind of work."

"Will you, laying aside all partiality, consider in the silence of your thoughts that we [humans] are creatures either quite like the rest [of animals], or separated by no great difference? For what is there to show that we do not resemble them? or what excellence is in us, such that we scorn to be ranked as creatures? Their bodies are built up on bones, and bound closely together by sinews; and our bodies are in like manner built up on bones, and bound closely together by sinews."

"But if men either knew themselves thoroughly, or had the slightest knowledge of God, they would never claim as their own a divine and immortal nature; nor would they think themselves something great because they have made for themselves gridirons, basins, and bowls, because they have made under-shirts, outer-shirts, cloaks, plaids, robes of state, knives, cuirasses and swords, mattocks, hatchets, ploughs. Never, I say, carried away by pride and arrogance, would they believe themselves to be deities of the first rank, and fellows of the highest in his exaltation, because they had devised the arts of grammar, music, oratory, and geometry. For we do not see what is so wonderful in these arts, that because of their discovery the soul should be believed to be above the sun as well as all the stars, to surpass both in grandeur and essence the whole universe, of which these are parts."

"Alius quidam veterum poetarum, cuius nomen mihi nunc memoriae non est, Veritatem Temporis filiam esse dixit."

"Non exprimis, aliquis inquit, Ciceronem. Quid tum? Non enim sum Cicero; me tamen, ut opinor, exprimo."

"To understand , Boccaccio, Petrarch, and Politian are just as valuable as and Sismondi."

"Ludimus effigiem belli, simulataque veris Praelia, buxo acies fictas, et ludicra regna, Ut gemini inter se reges albusque, nigerque Pro laude oppositi certent bicoloribus armis."

"But see! each Muse, in Leo's golden days, Starts from her trance, and trims her withered bays; Rome's ancient genius, o'er its ruins spread, Shakes off the dust, and rears his reverend head. Then sculpture and her sister-arts revive; Stones leaped to form, and rocks began to live; With sweeter notes each rising temple rung; A Raphael painted, and a Vida sung! Immortal Vida! on whose honoured brow The poet's bays and critic's ivy grow: Cremona now shall ever boast thy name, As next in place to Mantua, next in fame!"

"Tunc longe sale saxa sonant, tunc et freta ventis Incipiunt agitata tumescere: littore fluctus Illidunt rauco."

"Haud satis est illis utcunque claudere versum, Et res verborum propria vi reddere claras; Omnia sed numeris vocum concordibus aptant, Atque sono, quaecunque canunt, imitantur."

"Gratantes plausu excipient: tua gloria coelo Succedet, nomenque tuum sinus ultimus orbis Audiet, ac nullo diffusum abolebitur aevo."

"Ipse viam tantum potui docuisse repertam Aonas ad montes, longeque ostendere Musas Plaudentes celsae choreas in vertice rupis."

"Principio quoniam magni commercia coeli Numina concessere homini, cui carmina curae, Ipse Deum genitor divinam noluit artem Omnibus expositam vulgo, immeritisque patere: Atque ideo, turbam quo longe arceret inertem, Angustam esse viam voluit, paucisque licere."

"Jam vero cum rem propones, nomine nunquam Prodere conveniet manifesto: semper opertis Indiciis, longe et verborum ambage petita Significant, umbraque obducunt: inde tamen, ceu Sublustri e nebula, rerum tralucet imago Clarius, et certis datur omnia cernere signis. Hinc si dura mihi passus dicendus Ulysses, Non ilium vero memorabo nomine, sed qui Et mores hominum multorum vidit et urbes Naufragus, eversae post saeva incendia Trojae, Addam alia, angustis complectens omnia dictis."

"Primus at ille labor versu tenuisse legentem Suspensum, incertumque dia qui denique rerum Eventus maneant."

"Saepe etiam memorandum inter ludicra memento, Permiscere aliquid breviter, mortalia corda Quod moveat, tangens humanae commoda vitae, Qodque olim jubeant natos meminisse parentes."

"Praeterea haud lateat te nil conarier artem, Naturam nisi ut assimulet, propiusque sequatur. Hanc unam vates sibi proposuere magistram: Quicquid agunt, hujus semper vestigia servant."

"Nec dubitem versus hirsuti saepe poetae Suspensus lustrare, et vestigare legendo, Sicubi se quaedam forte inter commoda versu Dicta meo ostendant, quae mox melioribus ipse Auspiciis proprios possim mihi vertere in usus, Detersa prorsus prisca rubigine scabra."

"Idcirco si quando ducum referenda virumque Nomina dura nimis dictu, atque asperrima cultu, Illa aliqui, nunc addentes, nunc inde putantes Pauca minutatim, levant, ac mollia reddunt."

"When will a man guilty of any crime or sin accept with a tranquil mind that his wickedness is a product of his own will, not of necessity, and allow what he now strives to attribute to nature to be ascribed to his own free choice? It affords endless comfort to transgressors of the divine law if they are able to believe that their failure to do something is due to inability rather than disinclination, since they understand from their natural wisdom that no one can be judged for failing to do the impossible and that what is justifiable on grounds of impossibility is either a small sin or none at all."

"If you depart from evil but fail to do good, you transgress the law, which is fulfilled not simply by abominating evil deeds but also by performing good works."

"Under the plea that it is impossible not to sin, they are given a false sense of security in sinning ... Anyone who hears that it is not possible for him to be without sin will not even try to be what he judges to be impossible, and the man who does not try to be without sin must perforce sin all the time, and all the more boldly because he enjoys the false security of believing that it is impossible for him not to sin ... But if he were to hear that he is able not to sin, then he would have exerted himself to fulfil what he now knows to be possible when he is striving to fulfil it, to achieve his purpose for the most part, even if not entirely."

"The best incentive for the mind consists in teaching it that it is possible to do anything which one really wants to do."

"We can never enter upon the path to virtue unless we have hope as our guide and companion."

"We must now take precautions to prevent you from being embarrassed by something in which the ignorant majority is at fault for lack of proper consideration, and so from supposing with them, that man has not been created truly good simply because he is able to do evil. ... If you reconsider this matter carefully and force your mind to apply a more acute understanding to it, it will be revealed to you that man's status is better and higher for the very reason for which it is thought to be inferior: it is on this choice between two ways, on this freedom to choose either alternative, that the glory of the rational mind is based, it is in this that the whole honor of our nature consists, it is from this that its dignity is derived."

"Nothing impossible has been commanded by the God of justice and majesty. ... Why do we indulge in pointless evasions, advancing the frailty of our own nature as an objection to the one who commands us? No one knows better the true measure of our strength than he who has given it to us nor does anyone understand better how much we are able to do than he who has given us this very capacity of ours to be able; nor has he who is just wished to command anything impossible or he who is good intended to condemn a man for doing what he could not avoid doing."

"Yet we do not defend the good of nature to such an extent that we claim that it cannot do evil, since we undoubtedly declare also that it is capable of good and evil; we merely try to protect it from an unjust charge, so that we may not seem to be forced to do evil through a fault of our nature, when, in fact, we do neither good nor evil without the exercise of our will and always have the freedom to do one of the two, being always able to do either."

"Whenever I have to speak on the subject of moral instruction and the conduct of a holy life, it is my practice first to demonstrate the power and quality of human nature."

"Love of wealth is insatiable, desire for honour knows no fulfilment; possessions destined to meet with a speedy end are sought endlessly. But divine wisdom, heavenly riches, immortal honours we neglect in our indifference and sloth, and, as for spiritual riches, either we do not touch them at all or, if we get a slight taste of them, we at once suppose that we have had enough. The divine Wisdom invites us to its feasts in quite different terms: Those who eat me, she says, will hunger for more, and those who drink me will thirst for more (Sir.24.21). No one can have enough of such feasts or ever suffers from squeamishness because he has had too much: the more he drinks from that source the greater will be each man's capacity and eagerness for more."

"Do you consider a man to be a Christian by whose bread no hungry man is ever filled?"

"Let no man judge himself to be a Christian, unless he is one who both follows the teaching of Christ and imitates his example."

"He is a Christian"

"Their faith alone will not profit them, because they have not done works of righteousness."

"Unless a man has despised worldly things, he shall not receive those which are divine."

"Whenever I have to speak on the subject of moral instruction and conduct of a holy life, it is my practice first to demonstrate the power and quality of human nature and to show what it is capable of achieving, and then to go on to encourage the mind of my listener to consider the idea of different kinds of virtues, in case it may be of little or no profit to him to be summoned to pursue ends which he has perhaps assumed hitherto to be beyond his reach; for we can never end upon the path of virtue unless we have hope as our guide and compassion."

"It was because God wished to bestow on the rational creature the gift of doing good of his own free will and the capacity to exercise free choice, by implanting in man the possibility of choosing either alternative. ... He could not claim to possess the good of his own volition, unless he was the kind of creature that could also have possessed evil. Our most excellent creator wished us to be able to do either but actually to do only one, that is, good, which he also commanded, giving us the capacity to do evil only so that we might do His will by exercising our own. That being so, this very capacity to do evil is also good – good, I say, because it makes the good part better by making it voluntary and independent, not bound by necessity but free to decide for itself."