Novi Commentarii Academiae Scientiarum Petropolitanae

Original Titles
English Titles

133On the surface of scalene cones and of other conic bodies
134Theorems on divisors of numbers
135Various geometric demonstrations
136On the propagation of pulses in an elastic medium
137Examination of a design for propelling a ship by an internal principle of motion
138On the motion of the lunar nodes, and on the variation of their inclination to the ecliptic
139To what degree the motion of the earth is perturbed by the moon, more accurately investigated
166On the reduction of curved lines to the arcs of circles
167On the solution of a most difficult problem proposed by Fermat
188Method of integrating differential equations of higher degrees, further developed
189On the determination of series, or a new method for finding the general terms of series
190Consideration of certain series which are gifted with particular properties
191On the partitions of numbers
192Solution of a problem of geometry
193On the perturbation of the movement of planets from a figure not arising from a sphere
194On machines in general
195On the tautochrone motion of pendulums
196Improvement of the magic lantern and solar microscope
228On numbers which are the sum of two squares
229On the construction of the best gears
230Elements of the doctrine of solids
231Proof of some of the properties of solid bodies enclosed by planes
232On the movement of celestial bodies perturbed by any number of forces
241Proof of a theorem of Fermat that every prime number of the form 4n+1 is the sum of two squares
242Proof of a theorem of Fermat that every number whether whole or fraction is the sum of four or fewer squares
243Observations on the sums of divisors
244A demonstration of a theorem on the order observed in the sums of divisors
245On a method analogous to that of Diophantus in the analysis of the infinites
246A contribution to the calculations of sines
247On divergent series
248On the screw of Archimedes
249On finding the best shape for gear teeth
250A method for finding infinitely many isoperimetric curves predicated with a common property
251On the integration of the differential equations (m dx)/√(1-x4) = (n dy)/√(1-y4)
252Observations on the comparison of arcs of irrectifiable curves
253On indeterminate problems which appear to be quite determinate
254On the expression of integrals as factors
255General solution of certain Diophantine problems, which are ordinarily thought to admit only special solutions
256Example of the use of observation in pure mathematics
257On friction of rotating bodies
258Principles of the motion of fluids
259On the motion and reaction of water flowing through moving tubes
260Attempt at a theory of the friction of fluids
261Example of another new methods for comparing transcendental quantities; on the comparison of the arcs of ellipses
262Theorems about the remainders left by division by powers
263An example of a new method for the quadrature and rectification of curves and of comparing other quantities which are transcendentally related to each other
264Proof of a theorem and solution of a theorem proposed in the Acta Eruditorum of Leipzig
265On differential equations of the second degree
269On the integration of differential equations
270The solution of a problem about searching for three numbers, of which the sum and not only their product but the sum of their products two apiece, are square numbers
271Demonstration of a new method in the Theory of Arithmetic
272A reinforcement of some arithmetic theorems, supported by several demonstrations
273Consideration of formulas, of which the integral can be obtained by sections of arcs of cones
274Construction of the differentio-differential equation Ay du2 + (B+Cu)du dy + (D+Eu+Fuu)ddy = 0, assuming the element du to be constant
275Annotations to a certain passage of Descartes for finding the quadrature of the circle
276Clarifications into fluid resistance
277Principal theory of machines
278On the motion and grinding of lenses while they are polished upon curved plates
279On the resolution of formulas of squares of indeterminates by integral numbers
280On progressions of arcs of circles, of which the accompanying tangents proceed by a certain law
281A specimen of a singular algorithm
282On the resolution of equations of any order
283On very large prime numbers
284On the resolution of the equation dy + ayy dx = bxm dx
285Investigation of functions from a given condition of differentials
286On the vibrating motion of a flexible wire, weighted with however many small particles
287On the vibrating motion of unevenly thick strings
288Thoughts on constructing ramparts/walls
295On the reduction of integral formulas for the rectification of the ellipse and hyperbola
296The Elements of the Calculus of Variations
297Explanation of the analytical method in the maxima and minima
298On the notable advancement of the inverse method of tangents
299Explanation of the tautochrone in a resistant medium
300Demonstration of a Bernoullian theorem in which cycloids are ultimately produced from the evolution, continued to infinity, of any right-angled curve
301De motu corporis ad duo centra virium fixa attracti
302On the motion of vibrations in drums
303Tentamen de sono campanarum
304Considerationes de motu corporum coelestium
322The use of discontinued functions in analysis
323Concerning the use of a new algorithm to solve the Pell problem
324Properties of triangles for which certain angles have a ratio between themselves
325Easy solutions to some difficult geometric problems
326Analytical observations
327On the rectilinear motion of three bodies mutually attracted to each other
328On the motion of a body attracted to two fixed centers of forces
329On the appearance of the heavens observed through transparent spherical segments
330Supplement on the shape of the teeth of wheels [gear teeth]
331On the motion of fluids arising from different degrees of heat
345On integrated equations of the type dx/√(A+Bx+Cx2+Dx3+Ex4) = dy/√(A+By+Cy2+Dy3+Ey4)
346De arcubus curvarum aeque amplis earumque comparatione
347More general development of formulas serving for the comparison of curves
348Methodus facilis motus corporum coelestium utcunque perturbatos ad rationem calculi astronomici revocandi
349Disquisition on the true law of refraction of rays of different colors
350On a new type of microscope composed of six lenses
351On a telescope made from four lenses and its perfection
368On a hyperbolic curve expressed by this equation y = 1*2*3*...*x
369How very large numbers are to be tested for whether they are prime or not
370A new criteria for acquiring the imaginary roots of equations
371Considerations on completing the theory of the motion of the moon and especially of its variation
372Note of certain precautions to be taken in the investigation of the inequalities by which the heavenly bodies are perturbed in their motion
373More accurate investigation of the phenomena which can be produced in the diurnal motion of the earth by celestial forces
374On the equilibrium and motion of bodies connected by flexible curves
375First section on the state of equilibrium of fluids
390(Considerations of orthogonal trajectories)
391On double integral formulas
392Development of a notable paradox concerning the equality of surfaces
393On the sum of series involving the Bernoulli numbers
394On the partition of numbers into a number of parts of a given type
395On finding however many mean proportionals without regard to extraction of roots
396Section two of Principles of Fluid Motion
397Exposition of methods, not only for determining the parallax of the sun from an observed transit of Venus across the sun, but also for finding longitudes of places on the earth from observations of eclipses of the sun, together with calculations and conclusions deduced therefrom
405A solution of a problem about two numbers which are searched for, of which their product, increased or decreased by their sum or difference, will be a square
406Observations about the roots of equations
407An algebraic problem that is notable for some quite extraordinary relations
408On rectifiable curves on spherical surfaces
409The third chapter on the linear motion of fluids
410Natural basic principles of the study of the state of equilibrium, and the motion, of bodies that are both perfectly flexible and elastic
411On the impact of bullets exploded against a plate
419On solids whose (entire) surface can be unfolded onto a plane
420A new and easy method for treating the calculus of variations
421Solution of a formula for the integral ∫ x f-1 dx (log x)m/n the integration being extended from the value x = 0 to x = 1
422The solution of a certain altogether remarkable geometrical problem
423Considerations on the measurement of circles
424Section four of the motion of air in tubes
425On the perturbation of the motion of the Earth due to an action arising from Venus
427An investigation of a certain Diophantine problem
428Observations about two biquadratics, of which the sum is able to be resolved into two other biquadratics
429On various types of integrals
430Observations on the differential equation y dy + My dx + N dx = 0
431Consideration of the differential equation (a+bx)ddz + (c+ex)(dxdz/x) + (f+gx)(zdx2/xx) = 0
432Analytical exercises
433An aside on trajectories, both orthogonal and oblique-angled
434On the collision of spinning bodies
435On the collision of hanging bodies, both oblique and spinning
436On the true tautochrone in fluids
437On the tautochrone in a thin medium, where the resistance is directly proportional to the speed
438Clarifications regarding the tautochrone
439A further inquiry into vibrating chords
440Inquiries into the Bernoulli solution, in Volume XVI of the New Commentaries, regarding the motion of strings composed of two parts of differing density
441On the vibratory motion of strings composed of an arbitrary number of parts of differing density
442On the vibratory motion of strings possessed of a density that varies arbitrarily
443On the vibratory motion of elastic laminates, where many new types of vibrations, not treated heretofore, are solved
444On the most rapid (descending) motion of weights along curves of a specified type
447The summation of the progressions
sin(φλ) + sin(2φλ) + sin(3φλ) + ... + sin(nφλ);
cos(φλ) + cos(2φλ) + cos(3φλ) + ... + cos(nφλ).
448A new infinite series that expresses the perimeter of an ellipse, and which converges very rapidly
449Demonstrations about the residues resulting from the division of powers by prime numbers
450Nova ratio quantitates irrationales proxime exprimendi
451A solution of the problem of finding a triangle, in which the lines from each angle bisecting the opposite sides are rational
452The resolution of the equation Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0 by rational and integral numbers
453Eminent properties of series within which the general term is contained as x = (1/2)(a+b/√k)(p+qk)n + (1/2)(a-b/√k)(p-qk)n
454On the resolution of irrationals by continued fractions, where a certain minor new and singular type is set forth
455A determination, sought from the first principles of mechanics, of the oscillatory motion that was treated in the preceding article
456On the pressure of a weight on the plane against which it leans
457De harmoniae veris principiis per speculum musicum repraesentatis
458A new method of reducing the motions of the principal planets to astronomical tables
459A disquisition concerning three-fold objective lens, which either create no distortion or which are capable of removing a given distortion arising from the other lens
460On the application of objective lenses to all types of telescopes
462On the value of the integral formula ∫ (zm-1 ± zn-m-1)/(1 ± zn) dz in the case in which after integration it is put z = 1
463On the value of the integral formula ∫ (zλ-ω ± zλ+ω)/(1 ± z)(dz/z)(lz)μ casu quo post integrationem ponitur z = 1
464A new method of determining integral qualities
465A demonstration of a theorem of Newton on the expansion of the powers of a binomial by cases, in which the exponents are not integral numbers
466A singular Diophantine problem
467On the table of prime numbers continued up to one million and beyond, in which at once all the non-prime numbers are expressed by their smallest divisors
468On the minimal oscillations of a pendulum laden with a given weight
469On the oscillatory movement of two plates suspended from a balance
470Explicatio motus oscillatorii mirabilis in libra maiore observati
471De motu turbinatorio chordarum musicarum; ubi simul universa theoria aequilibrii quam motus corporum flexibilium simulque etiam elasticorum breviter explicatur
472Commentatio hypothetica de periculo a nimia cometae approponquatione metuendo
474A solution of several Diophantine problems
475Analytical speculations
476Observations about a new and singular type of progression
477Meditations about a singular type of series
478General formulas for the translation of arbitrary rigid bodies
479A new method for generating the motion of a rigid body
480Regula facilis pro diiudicanda firmitate pontis aliusve corporis similis excognita firmitate moduli
481De gemina methodo tam aequilibrium quam motum corporum flexibilium determinandi et utriusque egregio consensu
482De pressione funium tensorum in corpora subiecta eorumque motu a frictione impedito. Ubi praesertim methodus traditur
483De traiectu citissimo stellae per duos circulos almucantarath datos pro qualibet elevatione poli.
484De circulo maximo fixo in coelo constituendo, ad quem orbitae planetarum et cometarum referantur.