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A good example is the best teacher. Obviously, Euler's Latin by far superceedssupercedes any one possibly produced by our natural Science oriented contemporaries. Having parallel texts is probably the best method to learn the new (old) way of expression. The problem we faced while preparing this piece was that the available [[wikisophia:Translation:Methodus_inveniendi/Additamentum_II|English translation]] was too much polished and made both "good English" and "well comprehenciblecomprehensible" for modern reader. This breaks with the "''philological''" approach to translation, that is "not to add and not to omit words of the original". We thus attempted to make it "worse", but closer to the way Euler expressed himself - more literal. This highlights the complexity of ideas Latin is able to express. Doing so we tried to omit (put in angular <> brackets according to philological tradition) words added by the English translator and to position the pieces of English phrases in the order Euler put the Latin ones, but without (too much) breaking English grammar.<ref>The Latin phrase in general has a "bracket" structure (Complement-)Subject(-Complement)-Object(-Complement)-(Adjunct-)Predicate, like German; and the agreement in case, number and gender permits to understand whether complement refers to Subject or Object. In English the phrase (in Indicative) follows a different pattern: Subject(-Complement)-(Adjunct-)Predicate-Object(-Complement), thus...</ref> Moreover, in several occasions the English translator omitted segments of Euler's text; these are, of course, restored.
Although, it is said that mathematics is an art of convenient notation, among mathematicians themselves two kinds can be found: ones who pay attention to notations and others who, like Jordan.<ref>It was said of Jordan's writings that if he had 4 things on the same footing (as ''a'', ''b'', ''c'', ''d'') they would appear as <math>a,M^{\prime}_3, \epsilon_2,\Pi_{1,2}^{\prime\prime}</math>.</ref> Euler apparently was closer to the second one: he did not hesitate to denote by the same letter ''t'' either time or radius and for him ''v'' apparently was a mnemonic for ''vis viva'' rather for ''velocitas'' (for the '''velocity''' he regularly uses ''celeritas''). In order to make the text more comprehensible for a natural Science student of today we replaced the original Euler's formulae by those given in the available English translation.
|13. From these cases the most perfect agreement of the principle here established with experience manifests <itself>; the doubt whether it would take place in more complicated cases can be excessive. For that reason, in more general sense this principle ought to be studied carefully so that it is not ascribed more that its nature permits. To explain that, all motions of thrown <bodies> must be divided into two kinds: one, in which the speed of a body is determined by the position only, so that at the same position <a body> always acquires the same speed; which happens if a body is attracted to one or several centers by forces which would be some<ref>whatever</ref> functions of the distances <of a body> from these centers<ref>In the first case, for example, the speed of a particle acted upon by one or more centers of force will be a function only of the distances to those center(s)</ref>. To another kind I ascribe those motions of thrown bodies, where the speed of a body is not determined by its sole position where it sticks; it becomes useful, either when the centers to which the body is attracted themselves had became moving, or the motion may be in a resistant medium. ''With this division it is to be noted to which of the previous kinds the motion'' of a body belongs, that is, if a body is attracted not only to a single, but to multiple fixed centers, by whatever forces, sum of its elementary motions should be minimal.
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|14. Hoc ipsum autem postulat indoles Propositionis: dum enim, inter datos terminos, ea quaeritur curva, in qua sit <math>\int ds \, v</math> minimum; eo ipso assumitur, celeritatem corporis in utroque termino eandem esse, quaecunque curva corporis viam constituat. Quotcunque autem fuerint centra virium fixa, celeritas corporis in quovis loco <math>M</math>, exprimitur functione determinata ambarum variabilium CP = <math>CP =y</math>, & <math>PM = <math>x</math>. Sit igitur <math>v^{2}</math> functio quaecunque ipsarum <math>x</math> & <math>y</math>, ita ut sit <math>dv^{2} = 2 F_{x} dx + 2 F_{y} dy</math>; atque videamus, an principium nostrum veram exhibiturum sit projectoriam corporis. Cum autem sit <math>dv^{2} = 2 F_{x} dx + 2 F_{y} dy</math>; corpus perinde movebitur, ac si sollicitetur in <math>M</math> a duabus viribus, altera <math>F_{x}</math> in directione abscissis <math>x</math> parallela, altera vero <math>F_{y}</math> in directione parallela applicatis <math>y</math>, ex quibus oritur vis tangentialis = <math>\frac{F_{x} dx + F_{y} dy}{ds}</math>, et vis normalis = <math>\frac{-F_{y} dx + F_{x} dy}{ds}</math>. Debet autem, ex natura motus liberi, esse <math>\frac{v^{2}}{R} = \frac{-F_{y} dx + F_{x} dy}{ds} = \frac{F_{x} - p F_{y}}{\sqrt{1 + p^{2}}}</math>; ad quam aequationem si Methodus maximorum ac minimorum deducat, erit utique principium nostrum veritati conforme.
|14. HereThis isby itself the essenceinnate character of ourthe proposition hypothesisrequires. That curve is sought between <two> given endpoints in which <the integral> <math>\int ds \, v</math> mustwould be minimal, where, by assumptionthat assuming, the speed <<math>v</math>> of the body at the endpoints isto be the same irrespective to what specific curve forms the path of the body. Even if there are many fixed centers of force, the speed of the body in the arbitrary pointplace <math>M </math> is expressed as a <well-defined > function of both variables CP = <math> CP =y</math> and PM =x <math> PM =x</math> (Figure 27). Thus,Would <the squared speed> <math>v^{2}</math> isbe also a function of the same <math>x</math> and <math>y</math>, so that there would be <math>dv^{2} = 2 F_{x} dx + 2 F_{y} dy</math>, from which we may see whether our principle will indeed provide the trajectory <ref>as [???]well uniquely used word ''projectorium''. The general sense is clear, but the precise translation is problematic.</ref> of the body. Since, meanwhile, <math>dv^{2} = 2 F_{x} dx + 2 F_{y} dy</math>, the body at a point <math>M </math> will move just as it is acted by two forces, one <math>F_{x}</math> in the direction parallel to the abscissa <math>x</math>, another truelytruly <math>F_{y}</math> in the direction parallel to applicata <math>y</math>, of which originate a tangential force <along the trajectory> <math>\frac{F_{x} dx + F_{y} dy}{ds}</math>, and a normal force <perpendicular to the trajectory> <math>\frac{-F_{y} dx + F_{x} dy}{ds}</math>. From the nature of the free motion there must be <math>\frac{v^{2}}{R} = \frac{-F_{y} dx + F_{x} dy}{ds} = \frac{F_{x} - p F_{y}}{\sqrt{1 + p^{2}}}</math><, where <math>R</math> is the local radius of curvature of the trajectory> . to Thus,which ourequation principleif of deriving mechanicsdeduced from athe principle of maxima and minima conformswould also conform with experience. |-
|15. Cum igitur, per hoc principium, debeat esse <math>\int dy \, v \sqrt{1 + p^{2}}</math> minimum, differentietur quantitas <math>v \sqrt{1 + p^{2}}</math>, atque, ob <math>dv^{2} = 2 F_{x} dx + 2 F_{y} dy</math>, orietur:
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