Usor:Tchougreeff/QUOMODO sive HOW TO/Methodus Inveniendi parallel

A good example is the best teacher. Obviously, Euler's Latin by far supercedes 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 English translation was too much polished and made both "good English" and "well comprehensible" 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. 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. 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.

De motu projectorum in medio non resistente, per Methodum maximorum ac minimorum determinando	Concerning the motion of particles in a non-resistant medium, determined by a method of maxima and minima
1. Quoniam omnes naturae effectus sequuntur quandam maximi minimive legem; dubium est nullum, quin in lineis curvis, quas corpora projecta, si a viribus quibuscunque sollicitentur, describunt, quaepiam maximi minimive proprietas locum habeat. Quaenam autem sit ista proprietas, ex principiis metaphysicis a priori definire non tam facile videtur; cum autem has ipsa curvas, ope Methodi directae, determinare liceat; hinc, debita adhibita attentione, id ipsum, quod in istis curvis est maximum vel minimum, concludi poterit. Spectari autem potissimum debet effectus a viribus sollicitantibus oriundus; qui cum in motu corporis genito consistat, veritati consentaneum videtur hunc ipsum motum, seu potius aggregatum omnium motuum qui in corpore projecto insunt, minimum esse debere. Quae conclusio etsi non satis confirmata videatur, tamen, si eam cum veritate jam a priori nota consentire ostendero, tantum consequetur pondus, ut omnia dubia quae circa eam suboriri queant penitus evanescant. Quin-etiam cum ejus veritas fuerit evicta, facilius erit in intimas Naturae leges atque causas finales inquirere; hocque assertum firmissimis rationibus corroborare.	1. Since all natural phenomena follow certain maximum or minimum law, there is no doubt that in the curved lines, which thrown bodies acted upon by whatever forces draw, some maximum or minimum property would have place. What, however, property should it be does not seem to easy determine from a priori metaphysical principles. Yet if the very curves could be determined by a direct method, provided that sufficient care is taken, the property which in these curves is minimum or maximum would be determined. After considering the effects of external forces and the movements they generate, it seems most consistent with experience to assert that the integrated momentum (i.e., the sum of all momenta contained in the particle's movement) is the minimized quantity. This assertion if not seen sufficiently proven [at present]; however, if I shall show it to be connected with some truth known a priori, it will carry such weight as to utterly vanquish every conceivable doubt. If indeed it could be verified, it will be easier to investigate the deepest laws of Nature and their final causes, and also to firmer identify a rationale for this assertion.
2. Sit massa corporis projecti = $M$, ejusque, dum spatiolum = $ds$ emetitur, celeritas debita altitudini $v$; erit quantitas motus corporis in hoc loco = $Mv$; quae per ipsum spatiolum $ds$ multiplicata, dabit $Mvds$ motum corporis collectivum per spatiolum $ds$. Jam dico lineam a corpore descriptam ita fore comparatam, ut, inter omnes alias lineas iisdem terminis contentas, sit $\int Mvds$, seu, ob $M$ constans, $\int vds$ minimum. Quod si autem curva quaesita tanquam esset data spectetur, ex viribus sollicitantibus celeritas $v$ per quantitates ad curvam pertinentes definiri, ideoque ipsa curva per Methodum maximorum ac minimorum determinari potest. Ceterum haec expressio ex quantitate motus petita aeque ad vires vivas traduci poterit; posito enim tempusculo, quo elementum $ds$ percurritur, = $dt$; quia est $ds=vdt$, fiet $\int vds=\int v^{2}dt$; ita ut, in curva a corpore projecto descripta, summa omnium virium vivarum, quae singulis temporis momentis corporis insunt, sit minima. Quamobrem neque ii qui vires per ipsas celeritates, neque illi qui per celeritatum quadrata aestimari oportere statuunt, hic quicquam quo offendantur reperient.	2. Let the mass of a thrown body be equal to $M$, and let its speed, while being moved over an infinitesimal interval $ds$, be $v$. The amount of motion of the body at this place will be $Mv$ that is, when multiplied by the same interval $ds$, will give $Mvds$, the <amount of> motion of the body accumulated over the interval $ds$. Now I assert <as it will be shown the true> line (among all other lines connecting the same endpoints) drawn by the body would be <the line that makes> $\int Mvds$ or (since M is constant) $\int vds$ minimal. Since the sought curve is given by the acting forces the speed $v$ can be determined by the quantities pertinent to the curve <itself>. Similarly the same curve can be determined by the method of maxima and minima. Meanwhile, this expression derived from the amount of motion, equally could be reduced to the live forces. For, given an infinitesimal time $dt$ during which the element $ds$ is traversed, so that $ds=vdt$. Hence, there will be $\int vds=\int v^{2}dt$ so that in the curve drawn by a thrown body the sum of all live forces in all moments of time present in the body [the integral over time of its instantaneous kinetic energies] should be minimal. Accordingly, either those who favor momentum for mechanics calculations or those who favor live forces will not be offended by this minimum principle.
3. Primum igitur, si corpus a nullis prorsus viribus sollicitari ponamus, ejus quoque celeritas, ad quam hic solum attendo (directionem enim ipsa Methodus maximorum & minimorum complectetur), nullam patietur alterationem; eritque ideo $v$ quantitas constans, puta = $b$. Hinc corpus a nullis viribus sollicitatum, si utcunque projiciatur, ejusmodi describet lineam, in qua sit $\int dsb$ vel $\int ds=s$ minimum. Via ergo haec, inter omnes iisdem terminis contentas, ipsa erit minima; atque adeo recta; prorsus uti prima Mechanicae principia postulant. Hunc quidem casum non adeo hic affero, quo principium meum confirmari putem; quamcunque enim, loco celeritatis $v$, aliam assumsissem functionem ipsius $v$, eadem prodiisset via recta; verum a casibus simplicissimis incipiendo facilius ipsa consensus ratio intelligi poterit.	3. First, thus, if we assume a body affected by absolutely no external forces, its speed, to which I only pay attention (and its [motion's] direction is embraced by the same method of minima and maxima) suffers no change; so that quantity $v$ will be a constant set to $b$. This body affected by no forces, if anyhow been thrown, by the same way draws a line in which $\int bds$ or $\int ds=s$ would be minimal. Hence, the <true> path <among> of all those connecting the same endpoints has the same quantity minimal; and as well is a straight line, just as the fundamental principles of Mechanics require. I do not bring here this case as evidence for my principle as its confirmation, since the integral of any function of the constant speed $v=b$ would, upon minimization, produce a straight path: beginning with the simplest cases facilitates to understand the reasoning.
4. Progredior ergo ad casum gravitatis uniformis, seu quo corpus projectum ubique, secundum directiones ad horizontem normales, deorsum sollicitetur a vi constante acceleratrice = $g$. Sit $AM$ curva, quam corpus in hac hypothesi describit, sumatur recta verticalis AP pro axe, ac ponatur abscissa $AP$ = $x$, applicata $PM=y$, & elementum curvae $Mm=ds$; erit ergo, ex natura sollicitationis, $dv^{2}=2gdy$, & $v^{2}=v_{0}^{2}+2gy$. Hinc curva ita erit comparata, ut in ea sit $\int ds{\sqrt {a+gx}}$ minimum. Ponatur $p\equiv {\frac {dx}{dy}}$, ut sit $ds=dy{\sqrt {1+p^{2}}}$, atque minimum esse debet $\int dy{\sqrt {v_{0}^{2}+2gy}}\,{\sqrt {1+p^{2}}}$; quae expressio cum formula generali $\int Zdx$ comparata dat $Z={\sqrt {v_{0}^{2}+2gy}}\,{\sqrt {1+p^{2}}}$ ; quare, cum positum sit $dZ=Mdx+Ndy+Pdp$, erit $M=0$ & $P={\frac {p{\sqrt {v_{0}^{2}+2gy}}}{\sqrt {1+p^{2}}}}$. Quia ergo valor differentialis est $Mdy=dP$; ob $M=0$, fiet praesenti casu $dP=0$, & $P={\sqrt {C}}$. Habebitur ergo ${\sqrt {C}}={\frac {p{\sqrt {v_{0}^{2}+2gy}}}{\sqrt {1+p^{2}}}}={\frac {dx{\sqrt {v_{0}^{2}+2gy}}}{ds}}$; unde sit $Cdx^{2}+Cdy^{2}=dx^{2}\left(v_{0}^{2}+2gy\right)$, & $dx={\frac {dy{\sqrt {C}}}{\sqrt {v_{0}^{2}-C+2gy}}}$; quae integrata dat $x={\frac {1}{g}}{\sqrt {C\left(v_{0}^{2}-C+2gy\right)}}$.	4. Thus, I proceed to the case of uniform gravity or, one where a body thrown along the direction normal to the horizon is acted upon by a downwards force of constant acceleration $g$. Let $AM$ be the curve, which the body draws under this hypothesis (Figure 26), taking the vertical strait line $AP$ for an axis and assuming abscissa $AP=x$, applicata $PM=y$, and the element of the curve $Mm=ds$, thus, by the nature of the action <of the force>, there will be $dv^{2}=2gdy$ & $v^{2}=v_{0}^{2}+2gy$. The <sought> curve will so be set that $\int ds{\sqrt {v_{0}^{2}+2gy}}$ would be minimal. Defining $p\equiv {\frac {dx}{dy}}$, so that <there would be> $ds=dy{\sqrt {1+p^{2}}}$; also $\int dy{\sqrt {v_{0}^{2}+2gy}}\,{\sqrt {1+p^{2}}}$ must be minimal. This expression compared with the general formula $\int Zdy$, yields $Z={\sqrt {v_{0}^{2}+2gy}}\,{\sqrt {1+p^{2}}}$ and, therefore, assuming <to be> $dZ\equiv Mdx+Ndy+Pdp$, there will be $M=0$ and $P={\frac {p{\sqrt {v_{0}^{2}+2gy}}}{\sqrt {1+p^{2}}}}$. For that reason the differential [value] obey $Mdy=dP$ and, since $M=0$ in the present case there will become <as well>, $dP=0$ and $P={\sqrt {C}}$ <, where $C$ is a constant>. Hence, we shall have <the differential equation> ${\sqrt {C}}={\frac {p{\sqrt {v_{0}^{2}+2gy}}}{\sqrt {1+p^{2}}}}={\frac {dx{\sqrt {v_{0}^{2}+2gy}}}{ds}}$, where from it should be $Cdx^{2}+Cdy^{2}=dx^{2}\left(v_{0}^{2}+2gy\right)$ and <separated> $dx={\frac {dy{\sqrt {C}}}{\sqrt {v_{0}^{2}-C+2gy}}}$, which when integrated gives <the trajectory solution> $x={\frac {1}{g}}{\sqrt {C\left(v_{0}^{2}-C+2gy\right)}}$.
Figure 26
5. Manifestum quidem est hanc aequationem esse pro Parabola. At ejus consensum cum veritate attentius considerasse juvabit. Primum ergo patet tangentem hujus curvae esse horizontalem, seu $dy=0$ ; ubi est $v_{0}^{2}-C+gy=0$. Cum igitur principium abscissarum $A$ ab arbitrio nostro pendeat, sumatur id in hoc ipso loco, fietque $C=v_{0}^{2}$; tum vero ipse axis per hoc punctum curvae summum transeat, ita ut, posito $y=0$, fiat simul $x=0$. His consideratis, aequatio pro curva erit haec $x=v_{0}{\sqrt {\frac {2y}{g}}}$; quam non solum patet esse pro Parabola; sed etiam, cum celeritas in puncto $A$ sit $v_{0}$, altitudo $CA$, ex qua corpus labendo ab eadem vi $g$ sollicitatum eam ipsam acquirit celeritatem, qua in puncto A movetur, erit = ${\frac {v_{0}^{2}}{2g}}$; hoc est, quartae parametri parti aequatur; prorsus uti ex doctrina motus projectorum per Methodum directam intelligitur.	5. Obviously to everybody, this equation is to be one for a parabola. Its agreement with the reality should be carefully considered. First, the <initial> tangent to the curve is seen to horizontal, or, $dy=0$ <at the point> where $v_{0}^{2}-C+gy=0$. Since the origin $A$ of the abscissa <will> depend on our decision, it is set at the same place, so that same point, where $C=v_{0}^{2}$ (thus, truly, the same axis transects the highest point of the curve, so that setting $y=0$ would give immediately $x=0$). So considered <in this coordinate frame>, the equation for the curve thus will be $x=v_{0}{\sqrt {\frac {2y}{g}}}$, which not only shows to be <itself> one for a parabola, but even more, with the speed in point $A$ being $v_{0}$, the height $CA$ from which the falling body acted upon by the same force $g$ acquires the same speed will be ${\frac {v_{0}^{2}}{2g}}$ (by this equals to ${\frac {1}{4}}$ of the parameter <of the parabola>), entirely as derived from learning on motions of thrown <bodies> by the direct method.
6. Sollicitetur, ut ante, corpus ubique verticaliter deorsum, at ipsa vis sollicitans non sit constans, sed pendeat utcunque ab altitudine CP. Scilicet posita abscissa $CP$ = $y$, sit vis qua corpus in $M$ deorsum nititur = $F_{y}(y)$ functioni cuicunque ipsius $y$. Si ergo vocetur applicata $PM$ = ${\textstyle x}$, elementum arcus Mm = $ds$, & $dy=pdx$; erit $dv^{2}=2F_{y}\,dy$ & $v^{2}=v_{0}^{2}+\int 2F_{y}\,dy$; unde minimum esse debet haec expressio $\int dy{\sqrt {v_{0}^{2}+\int 2F_{y}\,dy}}\,{\sqrt {1+p^{2}}}$, ex qua pro curva descripta AM obtinebitur haec aequatio ergo ${\sqrt {C}}={\frac {p{\sqrt {v_{0}^{2}+\int 2F_{y}\,dy}}}{\sqrt {1+p^{2}}}}$ & $p={\frac {\sqrt {C}}{\sqrt {v_{0}^{2}-C+\int 2F_{y}\,dy}}}$; seu $x=\int {\frac {dy{\sqrt {C}}}{\sqrt {v_{0}^{2}-C+\int 2F_{y}\,dy}}}$. Tangens ergo curvae erit horizontalis ubi $\int 2F_{y}\,dy=C-v_{0}^{2}$. Haec vero eadem aequatio trajectoriae corporis per Methodum directam reperitur.	6. Let as previously the body be acted upon downwards vertically by the same acting force, not constant, but eventually dependent somehow on the height $CP$ (Figure 27). Obviously, if $y$ is set to be abscissa along CP, the downwards vertical force $F_{y}(y)$ by which the body in <point > $M$ is pressed would be a function of that same $y$. If consequently the applicata $PM$ is denoted as $x$, the element of the arc <represent the infinitesimal distance>$Mm=ds$ and $dy=pdx$, there will be $dv^{2}=2F_{y}\,dy$ and $v^{2}=v_{0}^{2}+\int 2F_{y}\,dy$. Hence, this expression $\int dy{\sqrt {v_{0}^{2}+\int 2F_{y}\,dy}}\,{\sqrt {1+p^{2}}}$ must be minimal, where from for the described curve $AM$ this equation ${\sqrt {C}}={\frac {p{\sqrt {v_{0}^{2}+\int 2F_{y}\,dy}}}{\sqrt {1+p^{2}}}}$ & $p={\frac {\sqrt {C}}{\sqrt {v_{0}^{2}-C+\int 2F_{y}\,dy}}}$will be obtained , or $x=\int {\frac {dy{\sqrt {C}}}{\sqrt {v_{0}^{2}-C+\int 2F_{y}\,dy}}}$. The tangent to the curve will be horizontal whenever $\int 2F_{y}\,dy=C-v_{0}^{2}$. Namely, this equation for the body's trajectory is recovered by direct method.
Figure 27
7. Sollicitetur nunc corpus in $M$ a duabus viribus, altera horizontali = $F_{x}$ secundum directionem MP, altera verticali = $F_{y}$ secundum directionem $MQ$ (Fig. 27). Sit autem $F_{x}(x)$ functio quaecunque rectae verticalis $MQ=CP=y$ & $F_{x}(x)$ functio quaecunque applicatae $PM=y$. Positis ergo ut ante ${\textstyle dx=pdy}$, erit $dv^{2}=-2F_{x}dx-2F_{y}dy$, fietque $v^{2}=v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy$; unde minimum esse debet haec formula $\int dy{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$. Differentietur ${\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$, atque prodibit ${\frac {-F_{x}dx{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}-{\frac {F_{y}dy{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}+{\frac {pdp{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\sqrt {1+p^{2}}}}$ Hinc posito $M={\frac {-F_{x}{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}$, & $P={\frac {p{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\sqrt {1+p^{2}}}}$; erit pro curva quaesita haec aequatio $0=M-{\frac {dP}{dy}}$, seu $Mdy=dP$. Hinc ergo sit ${\frac {-F_{x}dy{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}={\frac {dp{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\left(1+p^{2}\right){\sqrt {1+p^{2}}}}}-{\frac {pF_{y}dy-pF_{x}dx}{{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}}$ seu ${\frac {dp{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\left(1+p^{2}\right){\sqrt {1+p^{2}}}}}={\frac {F_{y}dx-F_{x}dy}{{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}}$ ideoque ${\frac {dp}{1+p^{2}}}={\frac {F_{y}dx-F_{x}dy}{v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$ Hanc aequationem veritati esse consentaneam patebit, si loco $v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy$ ponatur $v^{2}$, erit enim ${\frac {v^{2}dp}{\left(1+p^{2}\right)^{3/2}}}={\frac {F_{y}dx-F_{x}dy}{\sqrt {1+p^{2}}}}$ At est radius osculi $R\equiv -{\frac {\left(1+p^{2}\right)^{3/2}dy}{dp}}$, quo introducto est ${\frac {v^{2}}{R}}={\frac {F_{x}dy-F_{y}dx}{ds}}$; ubi est ${\frac {v^{2}}{R}}$ vis corporis centrifuga, & ${\frac {F_{x}dy-F_{y}dx}{ds}}$ exprimit vim normalem ex viribus sollicitantibus ortam; quarum virium aequalitas utique in omni motu projectorum locum habet.	7. Now let the body in $M$ be acted upon by two <external> forces, one horizontal <denoted as> $F_{x}$ in the direction MP another vertical $F_{y}$ in the direction $MQ$ (Figure 27). Let $F_{y}(y)$ be meanwhile a function of the vertical strait line $MQ=CP=y$ and $F_{x}(x)$ be a function of the applicata <horizontal distance> $PM=x$. Setting as before ${\textstyle dx=pdy}$ , there will be $dv^{2}=-2F_{x}dx-2F_{y}dy$, and as well there will become $v^{2}=v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy$. Hence, this expression: $\int dy{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$ must be minimal. Differentiating <the integrand> ${\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$ as well will yield: ${\frac {-F_{x}dx{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}-{\frac {F_{y}dy{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}+{\frac {pdp{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\sqrt {1+p^{2}}}}$ Setting $M={\frac {-F_{x}{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}$, and $P={\frac {p{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\sqrt {1+p^{2}}}}$, for the sought curve there will be this equation: $0=M-{\frac {dP}{dy}}$ or $Mdy=dP$. Thus, there would be ${\frac {-F_{x}dy{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}={\frac {dp{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\left(1+p^{2}\right){\sqrt {1+p^{2}}}}}-{\frac {pF_{y}dy-pF_{x}dx}{{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}}$ or ${\frac {dp{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}{\left(1+p^{2}\right){\sqrt {1+p^{2}}}}}={\frac {F_{y}dx-F_{x}dy}{{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}}}$ which is, ${\frac {dp}{1+p^{2}}}={\frac {F_{y}dx-F_{x}dy}{v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$ This equation looks out to agree with reality, <since,> if instead of $v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy$, $v^{2}$ is inserted, there will be ${\frac {v^{2}dp}{\left(1+p^{2}\right)^{3/2}}}={\frac {F_{y}dx-F_{x}dy}{\sqrt {1+p^{2}}}}$ The <instantaneous> radius <of curvature of the trajectory> $R\equiv -{\frac {\left(1+p^{2}\right)^{3/2}dy}{dp}}$, while being inserted yields ${\frac {v^{2}}{R}}={\frac {F_{x}dy-F_{y}dx}{ds}}$, where, ${\frac {v^{2}}{R}}$ is the centrifugal force, and ${\frac {F_{x}dy-F_{y}dx}{ds}}$ expresses the normal force <perpendicular to the trajectory> coming from the acting forces. The equality of these forces has place in motion of all thrown bodies.
8. Aequatio autem inventa ${\frac {dp}{1+p^{2}}}={\frac {F_{y}dx-F_{x}dy}{v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$ ita generaliter est integrabilis, si multiplicetur per ${\frac {p\left(v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy\right)}{1+p^{2}}}$; fiet enim ${\frac {pdp\left(v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy\right)}{\left(1+p^{2}\right)^{2}}}+{\frac {F_{x}dx-p^{2}F_{y}dy}{1+p^{2}}}=0$ quae integrata dat ${\frac {\int 2F_{x}dx-p^{2}\int 2F_{y}dy-v_{0}^{2}}{1+p^{2}}}=C$ seu $\int 2F_{x}dx-p^{2}\int 2F_{y}dy=v_{0}^{2}+C+Cp^{2}$ unde $p={\sqrt {\frac {B+\int 2F_{x}dx}{C+\int 2F_{y}dy}}}$, posito $B$ pro $-v_{0}^{2}-C$. Cum ergo sit $p={\frac {dx}{dy}}$ , erit $\int {\frac {dx}{\sqrt {B+\int 2F_{x}dx}}}=\int {\frac {dy}{\sqrt {C+\int 2F_{y}dy}}}$ aequatio pro curva quaesita, in qua variabiles $x$ & $y$ sunt a se invicem separatae. Vel si constantes $B$ & $C$ in negativas convertantur, erit $\int {\frac {dx}{\sqrt {B-\int 2F_{x}dx}}}=\int {\frac {dy}{\sqrt {v_{0}^{2}-\int 2F_{y}dy}}}$ Ex quibus etsi curvae constructio facilis habetur, tamen aequationes algebraicae, quoties quidem in ipsis continentur, non tam facile eruuntur. Sint $F_{x}$ & $F_{y}$ functiones similes & quidem potestates ipsarum $x$ & $y$, ita ut sit $\int {\frac {dx}{\sqrt {b^{n}-x^{n}}}}=\int {\frac {dy}{\sqrt {a^{n}-y^{n}}}}$ quae aequatio, si $n=1$, praebet Parabolam; sin $n=2$, Ellipsin centrum in $C$ habentem: etsi hoc casu utraque integratio quadraturam Circuli requirit. Verisimile ergo videtur etiam aliis casibus, quibus neutra integratio succedit, curvas algebraicas satisfacere; quarum autem inveniendarum Methodus adhuc desideratur.	8. However, the found equation ${\frac {dp}{1+p^{2}}}={\frac {F_{y}dx-F_{x}dy}{v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy}}$ is generally so integrable: if multiplied by ${\frac {p\left(v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy\right)}{1+p^{2}}}$ it will become ${\frac {pdp\left(v_{0}^{2}-\int 2F_{x}dx-\int 2F_{y}dy\right)}{\left(1+p^{2}\right)^{2}}}+{\frac {F_{x}dx-p^{2}F_{y}dy}{1+p^{2}}}=0$ which <when> integrated gives ${\frac {\int 2F_{x}dx-p^{2}\int 2F_{y}dy-v_{0}^{2}}{1+p^{2}}}=C$ or $\int 2F_{x}dx-p^{2}\int 2F_{y}dy=v_{0}^{2}+C+Cp^{2}$ where from, $p={\sqrt {\frac {B+\int 2F_{x}dx}{C+\int 2F_{y}dy}}}$, <where> $B$ is set $-v_{0}^{2}-C$. Then, since $p={\frac {dx}{dy}}$, there will be $\int {\frac {dx}{\sqrt {B+\int 2F_{x}dx}}}=\int {\frac {dy}{\sqrt {C+\int 2F_{y}dy}}}$ - the equation for the sought curve in which the variables $x$ and $y$ are separated from each other. Or, if the constants $B$ and $C$ are changed into their negatives, there will be $\int {\frac {dx}{\sqrt {B-\int 2F_{x}dx}}}=\int {\frac {dy}{\sqrt {v_{0}^{2}-\int 2F_{y}dy}}}$ Where from, even though, the construction of the curve turns easy <in principle>, it may be not that easy to express it as algebraic equations. <For example,> let $F_{x}$ and $F_{y}$ be similar and <proportional to/of> the same power of $x$ and $y$, so that it would be $\int {\frac {dx}{\sqrt {b^{n}-x^{n}}}}=\int {\frac {dy}{\sqrt {a^{n}-y^{n}}}}$ which equation yields a parabola if $n=1$, whereas, if $n=2$, <it describes> an ellipse having its center in $C$. (Even in this case, each integration requires cyclic quadratures). It is verisimilar, that in other cases, neither integration succeeds <in producing> curves <described by> algebraic <functions>. Still, a method for finding these trajectories is desirable.
9. Urgeatur corpus $M$ perpetuo versus punctum fixum secundum directionem $MC$, vi quae sit ut functio quaecunque distantiae $MC$. Positis ut ante $CP=y$, $PM=x$, & $dx=pdy$; sit $CM={\sqrt {x^{2}+y^{2}}}=r$, atque sit $F_{r}$ ea functio ipsius $r$, quae exprimit vim centripetam. Resolvatur haec vis in laterales secundum $MQ$ & $MP$, erit vis trahens secundum MQ = ${\frac {F_{r}y}{r}}$; & et vis secundum MP = ${\frac {F_{r}x}{r}}$; ex quibus oritur acceleratio $dv^{2}=-{\frac {2F_{r}xdx}{r}}-{\frac {2F_{r}ydy}{r}}=-2F_{r}dr$, ob $xdx+ydy=rdr$; unde sit $v^{2}=v_{0}^{2}-\int 2F_{r}\,dr$. Quamobrem minimum esse debet haec expressio $\int dy{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}$. Jam, secundum Regulae praeceptum, differentietur quantitas, ${\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}$, prodibitque $-{\frac {F_{r}dr{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}+{\frac {pdp{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}}}}$ Ob $dr={\frac {xdx+ydy}{r}}$, erit ergo $M={\frac {-F_{r}x{\sqrt {1+p^{2}}}}{r{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}}$ & $P={\frac {p{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}}}}$; ex quibus efficitur aequatio pro curva $Mdy=dP$, quae praebet, ${\frac {-F_{r}xdy{\sqrt {1+p^{2}}}}{r{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}}={\frac {dp{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\left(1+p^{2}\right){\sqrt {1+p^{2}}}}}-{\frac {pF_{r}dr}{{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}}$ haecque reducta abibit in istam, ${\frac {F_{r}\left(ydx-xdy\right)}{r\left(v_{0}^{2}-\int 2F_{r}\,dr\right)}}={\frac {dp}{1+p^{2}}}$	9. <Now> let the body $M$ be acted upon by a force that always directs to a fixed point along $MC$ and is a function <only> of the distance $MC$ (Figure 27). Setting, as before, $CP=y$, $PM=x$, and $dx=pdy$; there would be $CM={\sqrt {x^{2}+y^{2}}}=r$ and $F_{r}$ would be a function of that same $r$ which expresses the centripetal force. Let this force be resolved into components along <the lines> $MQ$ and $MP$; then the dragging force along $MQ$ is ${\frac {F_{r}y}{r}}$, and the force along $MP$ equals ${\frac {F_{r}x}{r}}$, of those an acceleration $dv^{2}=-{\frac {2F_{r}xdx}{r}}-{\frac {2F_{r}ydy}{r}}=-2F_{r}dr$ rises (because $xdx+ydy=rdr$), from which there would be $v^{2}=v_{0}^{2}-\int 2F_{r}\,dr$. For this reason, the expression which must be minimal is $\int dy{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}$. Now, following the previously accepted rule, the quantity ${\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}$, being differentiated, will yield $-{\frac {F_{r}dr{\sqrt {1+p^{2}}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}+{\frac {pdp{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}}}}$ Since $dr={\frac {xdx+ydy}{r}}$, there will be $M={\frac {-F_{r}x{\sqrt {1+p^{2}}}}{r{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}}$ and $P={\frac {p{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}}}}$; where from the equation for curve $Mdy=dP$ appears, which yields ${\frac {-F_{r}xdy{\sqrt {1+p^{2}}}}{r{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}}={\frac {dp{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\left(1+p^{2}\right){\sqrt {1+p^{2}}}}}-{\frac {pF_{r}dr}{{\sqrt {1+p^{2}}}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}}$ which simplified will go to the ${\frac {F_{r}\left(ydx-xdy\right)}{r\left(v_{0}^{2}-\int 2F_{r}\,dr\right)}}={\frac {dp}{1+p^{2}}}$
10. Quamvis haec aequatio quatuor contineat litteras diversas, tamen debita dexteritate integrari potest. Cum enim sit $xdx+ydy=rdr=pxdy+ydy$, erit $dy={\frac {rdr}{y+px}}$ & $dx={\frac {prdr}{y+px}}$, qui valores in aequatione substituti dabunt ${\frac {\left(py-x\right)F_{r}dr}{\left(y+px\right)\left(v_{0}^{2}-\int 2F_{r}\,dr\right)}}={\frac {dp}{1+p^{2}}}$ seu ${\frac {F_{r}dr}{v_{0}^{2}-\int 2F_{r}\,dr}}={\frac {dp\left(px+y\right)}{\left(1+p^{2}\right)\left(py-x\right)}}$ Harum expressionum utraque per logarithmos est integrabilis, est enim $\int {\frac {F_{r}dr}{v_{0}^{2}-\int 2F_{r}\,dr}}=-{\frac {1}{2}}\log \left(v_{0}^{2}-\int 2F_{r}\,dr\right)$, & $\int {\frac {dp\left(px+y\right)}{\left(1+p^{2}\right)\left(py-x\right)}}$ resolvitur in $\int {\frac {ydp}{py-x}}-\int {\frac {pdp}{1+p^{2}}}=\log {\frac {py-x}{\sqrt {1+p^{2}}}}$; ita ut sit ${\frac {C}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}={\frac {py-x}{\sqrt {1+p^{2}}}}$; qua aequatione declaratur, celeritatem corporis in $M$, quae est $v={\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}$, esse reciproce ut perpendiculum ex $C$ in tangentem demissum; quae est proprietas palmaria horum motuum.	10. Although this equation contains four individual characters <variables ($x,y,p$ and $r$)>, with a due skill it can be integrated. Since $xdx+ydy=rdr=pxdy+ydy$; there will be, $dy={\frac {rdr}{y+px}}$ and $dx={\frac {prdr}{y+px}}$. These values substituted into the equation will give ${\frac {\left(py-x\right)F_{r}dr}{\left(y+px\right)\left(v_{0}^{2}-\int 2F_{r}\,dr\right)}}={\frac {dp}{1+p^{2}}}$ or ${\frac {F_{r}dr}{v_{0}^{2}-\int 2F_{r}\,dr}}={\frac {dp\left(px+y\right)}{\left(1+p^{2}\right)\left(py-x\right)}}$ Either of two expressions is integrable with logarithms, like $\int {\frac {F_{r}dr}{v_{0}^{2}-\int 2F_{r}\,dr}}=-{\frac {1}{2}}\log \left(v_{0}^{2}-\int 2F_{r}\,dr\right)$; <expanding> $\int {\frac {dp\left(px+y\right)}{\left(1+p^{2}\right)\left(py-x\right)}}$ <in partial fractions,> resolves into $\int {\frac {ydp}{py-x}}-\int {\frac {pdp}{1+p^{2}}}=\log {\frac {py-x}{\sqrt {1+p^{2}}}}$, so that, there would be ${\frac {C}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}={\frac {py-x}{\sqrt {1+p^{2}}}}$. By this equation the particle's speed in $M$ which is $v={\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}$ is stated to be inversely proportional to the perpendicular drawn from $C$ to the tangent <to the trajectory>, an excellent property of these motions.
11. Hoc vero idem Problema commodius resolvi potest ipsam rectam $CM$ pro altera variabili assumendo. Verum Methodus supra tradita non postulat, ut ambae sint coordinatae orthogonales, dummodo sint ejusmodi binae quantitates quibus determinatis simul curvae punctum determinetur. Hanc ob causam, non liceret distantiam $CM$ cum perpendiculo ex $C$ in tangentem demisso pro binis illis variabilibus accipere; quoniam etiamsi detur & distantia a centro & perpendiculum in tangentem, hinc tamen locus puncti curvae non definitur. Nihil autem impedit, quo-minus distantia $CM$, & arcus circuli $BP$ centro $C$ descripti, in locum duarum variabilium substituantur; quia dato arcu $BP$, & distantia $CM$ curvae punctum $M$ aeque determinatur ac per coordinatas orthogonales. Hac ergo annotatione usus Methodi multo latius extenditur, quam alioquin videri queat.	11. This same problem may be solved more conveniently by assuming the same strait line $CM$ as a second variable <(Figure 28)>. Really, the method described above does not require that the coordinates should be orthogonal to each other, merely that the pair of quantities when defined suffice to determine the point of the curve. For this reason, it would be not allowed to accept the distance $CM$ and the perpendicular to the tangent line set from $C$ as two variables, since even if both the distance from the center and the perpendicular to the strait line would be given so position of a point on the curve is not defined. Nothing, however, prevents <us> from taking the distance $CM$ and the arc $BP$ of a <unit> circle drawn around the center $C$ as our coordinates; since, as arc $BP$ and distance $CM$ are given the point $M$ is equally determined, exactly, just as by orthogonal coordinates. By this remark the usage of our Method is extended more broadly than might be imagined otherwise.
Figure 28
12. Sit igitur distantia corporis a centro $MC=r$, & vis qua corpus ad centrum $C$ sollicitatur sit = $F_{r}(r)$ functioni cuicunque ipsius $r$. Centro $C$, radio pro lubitu assumpto $BC=c$, describatur circulus, cujus arcus $BP$ teneat locum alterius variabilis $\theta$, ita ut sit $Pp=d\theta =pdr$. Ex vi autem sollicitante est $dv^{2}=-2F_{r}\,dr$, unde $v^{2}=v_{0}^{2}-\int 2F_{r}\,dr$. Centro $C$, radio $CM=r$ describatur arculus $Mn$, erit $mn=dr$; & $CP:Pp=CM:Mn$, unde sit $Mn=prdr$, & elementum spatii $Mm=ds=dr{\sqrt {1+p^{2}r^{2}}}$. Quamobrem minimum esse debet haec formula $\int dr{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}{\sqrt {1+p^{2}r^{2}}}$, ex qua oritur valor differentialis ${\frac {d}{dr}}\left[{\frac {pr^{2}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}r^{2}}}}\right]$, qui, per Regulam, nihilo aequalis positus, praebebit hanc aequationem: ${\sqrt {C}}={\frac {pr^{2}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}r^{2}}}}$, seu $C+Cp^{2}r^{2}=\left(v_{0}^{2}-\int 2F_{r}\,dr\right)p^{2}r^{4}$, ex qua sit $p={\frac {\sqrt {C}}{\sqrt {\left(v_{0}^{2}-\int 2F_{r}\,dr\right)r^{4}-Cr^{2}}}}={\frac {\sqrt {C}}{r{\sqrt {\left(v_{0}^{2}-\int 2F_{r}\,dr\right)r^{2}-C}}}}$ seu $d\theta ={\frac {dr{\sqrt {C}}}{r{\sqrt {\left(v_{0}^{2}-\int 2F_{r}\,dr\right)r^{2}-C}}}}$ quae eadem aequatio etiam per Methodum directam invenitur.	12. Let, thus, the distance of the body from the center be $MC=r$, and let the force by which the body is acted towards the center $C$ be equal to $F_{r}(r)$ some function of the same $r$. Assuming $C$ to be center of attraction and drawing a circle of radius $BC=c$ whose arc-length $BP$ would take place of the second variable $\theta$; so that $Pp=d\theta =pdr$ <(Figure 28)>. From the acting force there is $dv^{2}=-2F_{r}\,dr$, from which $v^{2}=v_{0}^{2}-\int 2F_{r}\,dr$. An <infinitesimal circular> arc $Mn$ of radius $CM=r$ is drawn about the center $C$, then $mn$ <represents the infinitesimal change in radius> $=dr$ and <from the geometrical similarity> $CP:Pp=CM:Mn$ where from would be $Mn=prdr$, and the infinitesimal distance $Mm=ds=dr{\sqrt {1+p^{2}r^{2}}}$. For this reason, this quantity $\int dr{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}{\sqrt {1+p^{2}r^{2}}}$ must be minimal, where from originates the differential ${\frac {d}{dr}}\left[{\frac {pr^{2}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}r^{2}}}}\right]$, which when, following our Rule, is set equal to zero, will yield this equation ${\sqrt {C}}={\frac {pr^{2}{\sqrt {v_{0}^{2}-\int 2F_{r}\,dr}}}{\sqrt {1+p^{2}r^{2}}}}$, or rather $C+Cp^{2}r^{2}=\left(v_{0}^{2}-\int 2F_{r}\,dr\right)p^{2}r^{4}$, from which there would be $p={\frac {\sqrt {C}}{\sqrt {\left(v_{0}^{2}-\int 2F_{r}\,dr\right)r^{4}-Cr^{2}}}}={\frac {\sqrt {C}}{r{\sqrt {\left(v_{0}^{2}-\int 2F_{r}\,dr\right)r^{2}-C}}}}$ or $d\theta ={\frac {dr{\sqrt {C}}}{r{\sqrt {\left(v_{0}^{2}-\int 2F_{r}\,dr\right)r^{2}-C}}}}$ This same equation may also be derived using the direct method.
13. Ex his igitur casibus perfectissimus consensus principii hic stabiliti cum veritate elucet: utrum autem iste consensus in casibus magis complicatis locum quoque sit habiturus, dubium superesse potest. Quamobrem quam late pateat istud principium diligentius erit investigandum, quo plus ipsi non tribuatur quam ejus natura permittit. Ad hoc explicandum, omnis motus projectorum in duo genera distribui debet; quorum altero celeritas corporis, quam in quavis loco habet, a solo situ pendet; ita ut, si ad eundem situm revertatur, eandem quoque sit recuperaturum celeritatem; quod evenit, si corpus vel ad unum vel ad plura centra fixa trahatur viribus, quae sint ut functiones quaecunque distantiarum ab his centris. Ad alterum genus refero eos, projectorum motus, quibus celeritas corporis per solum locum in quo haeret non determinatur; id quod usu venit, vel si centra illa ad quae corpus sollicitatur fuerint mobilia, vel si motus fiat in medio resistente. Hac facta divisione; notandum est, quoties motus corporis ad prius genus pertineat, hoc est, si corpus non solum ad unum sed ad quotcunque centra fixa sollicitetur viribus quibuscunque, toties in motu hoc summam omnium motuum elementarium fore minimam.	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 functions of the distances <of a body> from these centers. 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.
14. Hoc ipsum autem postulat indoles Propositionis: dum enim, inter datos terminos, ea quaeritur curva, in qua sit $\int ds\,v$ 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 $M$, exprimitur functione determinata ambarum variabilium $CP=y$ & $PM=x$. Sit igitur $v^{2}$ functio quaecunque ipsarum $x$ & $y$, ita ut sit $dv^{2}=2F_{x}dx+2F_{y}dy$; atque videamus, an principium nostrum veram exhibiturum sit projectoriam corporis. Cum autem sit $dv^{2}=2F_{x}dx+2F_{y}dy$; corpus perinde movebitur, ac si sollicitetur in $M$ a duabus viribus, altera $F_{x}$ in directione abscissis $x$ parallela, altera vero $F_{y}$ in directione parallela applicatis $y$, ex quibus oritur vis tangentialis = ${\frac {F_{x}dx+F_{y}dy}{ds}}$, et vis normalis = ${\frac {-F_{y}dx+F_{x}dy}{ds}}$. Debet autem, ex natura motus liberi, esse ${\frac {v^{2}}{R}}={\frac {-F_{y}dx+F_{x}dy}{ds}}={\frac {F_{x}-pF_{y}}{\sqrt {1+p^{2}}}}$; ad quam aequationem si Methodus maximorum ac minimorum deducat, erit utique principium nostrum veritati conforme.	14. This by itself the innate character of the proposition requires. That curve is sought between <two> given endpoints in which <the integral> $\int ds\,v$ would be minimal, by that assuming, the speed of the body at the endpoints to 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 place $M$ is expressed as a defined function of both variables $CP=y$ and x $PM=x$ (Figure 27). Would <the squared speed> $v^{2}$ be also a function of the same $x$ and $y$, so that there would be $dv^{2}=2F_{x}dx+2F_{y}dy$, from which we may see whether our principle will indeed provide the trajectory of the body. Since, meanwhile, $dv^{2}=2F_{x}dx+2F_{y}dy$, the body at a point $M$ will move just as it is acted by two forces, one $F_{x}$ in the direction parallel to the abscissa $x$, another truly $F_{y}$ in the direction parallel to applicata $y$, of which originate a tangential force <along the trajectory> ${\frac {F_{x}dx+F_{y}dy}{ds}}$, and a normal force <perpendicular to the trajectory> ${\frac {-F_{y}dx+F_{x}dy}{ds}}$. From the nature of the free motion there must be ${\frac {v^{2}}{R}}={\frac {-F_{y}dx+F_{x}dy}{ds}}={\frac {F_{x}-pF_{y}}{\sqrt {1+p^{2}}}}$<, where $R$ is the local radius of curvature of the trajectory> to which equation if deduced from the principle of maxima and minima would also conform with experience.
15. Cum igitur, per hoc principium, debeat esse $\int dy\,v{\sqrt {1+p^{2}}}$ minimum, differentietur quantitas $v{\sqrt {1+p^{2}}}$, atque, ob $dv^{2}=2F_{x}dx+2F_{y}dy$, orietur: ${\frac {F_{x}dx{\sqrt {1+p^{2}}}}{v}}+{\frac {F_{y}dy{\sqrt {1+p^{2}}}}{v}}+{\frac {pdpv}{\sqrt {1+p^{2}}}}$ ex quo obtinetur pro curva quaesita sequens aequatio, secundum praecepta tradita, ${\frac {F_{x}dy{\sqrt {1+p^{2}}}}{v}}=d\left[{\frac {pv}{\sqrt {1+p^{2}}}}\right]={\frac {dpv}{\left(1+p^{2}\right)^{3/2}}}+{\frac {pF_{x}dx+pF_{y}dy}{v{\sqrt {1+p^{2}}}}}$ seu ${\frac {-dpv}{\left(1+p^{2}\right)^{3/2}}}={\frac {F_{y}pdy-F_{x}dy}{v{\sqrt {1+p^{2}}}}}$ At est radius osculi in M = ${\frac {-\left(1+p^{2}\right)dy{\sqrt {1+p^{2}}}}{dp}}$; qui si ponatur = $r$, erit ${\frac {v^{2}}{R}}={\frac {F_{y}p-F_{x}}{\sqrt {1+p^{2}}}}$; omnio uti per Methodum directam invenitur. Dummodo ergo vires sollicitantes ita fuerint comparatae, ut eae reduci queant ad duas vires $F_{x}$ & $F_{y}$, secundum directiones coordinatis $x$ & $y$ parallelas sollicitantes, quae sint ut functiones quaecunque harum variabilium $x$ & $y$, tum semper in curva descripta erit motus corporis per omnia elementa collectus minimus.	15. Since by this principle $\int dy\,v{\sqrt {1+p^{2}}}$ should be minimal; let us differentiate the quantity $v{\sqrt {1+p^{2}}}$ so that $dv^{2}=2F_{x}dx+2F_{y}dy$, appears: ${\frac {F_{x}dx{\sqrt {1+p^{2}}}}{v}}+{\frac {F_{y}dy{\sqrt {1+p^{2}}}}{v}}+{\frac {pdpv}{\sqrt {1+p^{2}}}}$ from which for the sought curve a chain of equalities appears, according to previous prescriptions ${\frac {F_{x}dy{\sqrt {1+p^{2}}}}{v}}=d\left[{\frac {pv}{\sqrt {1+p^{2}}}}\right]={\frac {dpv}{\left(1+p^{2}\right)^{3/2}}}+{\frac {pF_{x}dx+pF_{y}dy}{v{\sqrt {1+p^{2}}}}}$ or ${\frac {-dpv}{\left(1+p^{2}\right)^{3/2}}}={\frac {F_{y}pdy-F_{x}dy}{v{\sqrt {1+p^{2}}}}}$. Since the radius of curvature <of the trajectory> at M equals ${\frac {-\left(1+p^{2}\right)dy{\sqrt {1+p^{2}}}}{dp}}$, which, if set to be $R$, will be ${\frac {v^{2}}{R}}={\frac {F_{y}p-F_{x}}{\sqrt {1+p^{2}}}}$, just as derived by the direct method. Thus, provided the acting forces were such that they can be resolved into two forces $F_{x}$ and $F_{y}$ acting along the directions parallel to the <axes of> coordinates $x$ and $y$ and being functions of these variables $x$ and $y$, then in a drawn curve the <amount of> motion accumulated over all elements of the path will always be the minimum.
16. Tam late ergo hoc principium patet, ut solus motus a resistentia medii perturbatus excipiendus videatur; cujus quidem exceptionis ratio facile perspicitur, propterea quod hoc casu corpus per varias vias ad eundem locum perveniens non eandem acquirit celeritatem. Quamobrem, sublata omni resistentia in motu corporum projectorum, perpetuo haec constans proprietas locum habebit, ut summa omnium motuum elementarium sit minima. Neque vero haec proprietas in motu unius corporis tantum cernetur, sed etiam in motu plurium corporum conjunctim; quae quomodocunque in se invicem agant, tamen semper summa omnium motuum est minima. Quod, cum hujusmodi motus difficulter ad calculum revocentur, facilius ex primis principiis intelligitur, quam ex consensu calculi secundum utramque Methodum instituti. Quoniam enim corpora, ob inertiam, omni status mutationi reluctantur; viribus sollicitantibus tam parum obtemperabunt, quam fieri potest, siquidem sint libera; ex quo efficitur, ut, in motu genito, effectus a viribus ortus minor esse debeat, quam si ullo alio modo corpus vel corpora fuissent promota. Cujus ratiocinii vis, etiamsi nondum satis perspiciatur; tamen, quia cum veritate congruit, non dubio quin, ope principiorum sanioris Metaphysicae, ad majorem evidentiam evehi queat; quod negotium aliis, qui Metaphysicam profitentur, relinquo.	16. Therefore, this principle is broadly applicable, so that only motion in a resistant medium is to be seen as an exception. The reason for this exception is easily seen, since in this case the body arriving at the same place by different ways will not acquire the same velocity. For that reason, neglecting any resistance to the motion of the thrown body, that permanent property will always take place, that the sum of all elementary motions should be a minimum. Moreover, this property is clearly seen not only for the motion of separate bodies, but also for collective motions of multiple bodies; no matter how they act upon each other, any way the sum of all motions is always minimal. Although, a motion may be more difficult to regain by calculation, using <our new> method; however, it is easier to grasp from first principles, than from agreement between calculations by whatsoever customary method. Because of their inertia, bodies are reluctant to change their state, and will obey applied forces as little as it will be possible, as if they would be free; from which follows that in the produced motion the effect coming from forces should be smaller than if the body or bodies would be promoted by whatever other way. <As for> the force of the <presented> reasoning, although, for the time, being not enough perceived, nevertheless, since it agrees with experiment, I do not doubt that by principles of a complete Metaphysics it would be possible to reach stronger proofs. But I leave this task to others, who profess Metaphysics.