HERMANN'S

# PHORONOMIA

## Introduction.

This is a work which has never been translated into English, apart from a few sections ; it forms a bridge between Newton's Principia and Euler's Mechanics; indeed Euler and Hermann, distantly related, worked together at St. Petersburg for a few years, but long after the present work was composed, around 1712 and published in 1715. Some of Hermann's biographical details can be found in Wikipedia, so we will not linger over these here, except to say he was a student of James Bernoulli, was held in high regard by Leibniz, he had an understanding of physics from his work in Italy, and eventually worked at St. Petersburg, alongside Euler ; it is a difficult work to translate for a number of reasons; the more salient being the poor state of the diagrams, most of which are incomprehensible without a lot of work, there are numerous typographical errors in the text and diagrams : I mention here with gratitude the assistance provided to me by the library of the University of Ghent, where a helpful individual provided me with a complete set of all the figures in the Phoronomia, all 160 of which I have redrawn]. Hermann's method of reasoning lies somewhere between his two idols, Newton and Leibniz : this, of course, is what renders his work interesting ; it lets us know where matters stood, at least on the continent of Europe, at this time, at least for this individual. The first 3 chapters deal with statics in the first section of Book I, while the second section is devoted to dynamics, which comes as rather a shock, as his methods seem to be in advance of what was regarded as true at the time; in fact some of his equations can be considered as nothing less than the conservation of potential and kinetic energy: he arrived at these via Newton's second law set out with the acceleration  a = vdv/dx, which enables one to bypass the use of time in equations involving acceleration. Ch. VI of Book I, Section II is interesting in this regard; some of the ideas have come originally from Huygens.

Book II is concerned with Hydrodynamics, and historically it was the first work to tackle this science using Leibnizian Calculus, at least in part. Some of the basic ideas are wrong ; for example, the pressure in a static liquid does not depend on the total weight of the supported fluid, only on the depth; however, in a translation such as this, the idea is to get the thoughts of the author across, not to correct his work , unless of course it is a simple error. Section I is concerned with hydrostatics, while Section II is concerned with hydrodynamics, initially without friction of any kind. Some interesting results emerge in this section, essentially the existence of a form of the Bernoulli Principle for fluid flow, originating in the works of Castelli, Baliani, Torricelli,  Borelli, and used by Gulielmini to estimate the flow of water in rivers and channels. Book III considers the effects of resistance to the passage of a body, both parallel to the axis and at an angle to the axis. Section IV considers the effect of air resistance on bodies moving through air, under increasingly complicated situations, and where the resistance can vary in different ways; most of the working uses arcane geometric methods similar to those presented by Newton in his Principia. The appendices deal with some corrections and matters not handled in the text, as you can find out below.

References :

The work itself can be found in the Garden of Archimedes series on cd no. 1.

A complete listing of Hermann's works can be found  in A Catalog of the Works of Jacob Hermann (1678-1733), by Fritz Nagel, in HISTORIA MATHEMATICA 18 (1991), pp. 36-54.

The Latin text by Baliani mentioned in Book 1, Sect. 2, Ch. 1, in which the equations of accelerated motion are first considered, can be found at the e-rara website.

Reading the Principia, by Niccolo Guicciardini, contains reference to Hermann's Phoronomia, and provides another point of view regarding Prop. I of the Principia, where Newton established his area law, on p. 211 onwards, which Hermann derives independently in Ch. 2 of Section 2, Book I.

Jacob Hermann and the Diffusion of the Libnizian Calculus in Italy by Mazonne and Roero is a wonderful work in my opinion, as it contains a great deal of original material on the early days of the calculus; there is much mention of the Phoronomia, but no intimate details, as the work had never been translated until now as a whole. It was regarded as being hard to understand.

BOOK ONE: SECTION 1

Click here  : Ch's. I &2. Chapter 1 (§26−§34) is about levers and moments of forces, and shares a lot with Varignon's Mechanics, which had appeared a few years earlier. The conditions for equilibrium of a body under various forces is considered. Chapter 2 (§35−§83) considers forces of a continuous nature acting on a body of a generalized shape ; at present the description is general, but the body is regarded as quite rigid, and might represent the hull of a ship in the sea, the surface of a sail blowing in the wind, etc.

Click here   : Chapter 3 (§84−§113). This is an interesting chapter, as the curves formed by a flexible string or wire under various loading situations are investigated ; this is done geometrically at first, and finally Leibniz's calculus is used to find the equations in analytic form. The curves formed from extended two dimensional sheets such as sails are also investigated.

BOOK ONE : SECTION 2.

Click here   : for Chapter 1 (§114−§152). This is a most interesting chapter. Hermann clearly had an excellent understanding both of Newton's work, as well as Leibniz's calculus. Here he sets the foundations for future chapters on dynamics, were he demonstrates in a rather antiquated notation a number of results which have survived the test of time, and amount to the conservation of energy, although of course his equations do not make this connection; nevertheless the equations are there. He even gets ½ mv2 for the work done by gravity for a falling body: thus future workers would have done well to have considered his work in more detail; his approach is essentially modern, yet who has even heard of him today?

Click here   : for Chapter 2 (§153−§169). This is another most interesting chapter, packed with Hermann's way of finding the curve traced out by a mobile body obeying a general law of gravitation. As usual, a lot of information is packed into the diagrams, esp. Fig. 37 : which shows together diagrams for the force, velocity, displacement, changes in the tangents, etc. for a body moving relative to a fixed attracting body. Eventually, a general law is produced for the orbit of such a body; later, he specializes with algebraic curves of various kinds as examples. I have not worked through the work in the Acta Erud. referred to, regarding the dissection of angle, so this is taken on trust at present; if this is of particular concern to anyone, I can probably translate these pages : finding diagrams is always a problem, however. Google is a great disappointment in this regard : why go to the trouble of scanning a book if you miss out fold-out diagrams? As somebody said, they are more interested in quantity than quality: for it negates the whole effort – rather like selling you a car without the wheels, which can't be had for love nor money……

Click here   : for Chapter 3 (§171−§186). I have become quite attached to this chapter, and I am sorry to have to leave it; in it you will find much information about gravitational isotones in general, before Hermann delves into the cycloid and epicycloid by his own mainly geometrical methods, which he relates finally to calculus : essentially he had found the work energy relations involving potential and kinetic energy; however, because these physical concepts were unknown at the time,  their mathematical representations remain just that, and no attempt was made to put things onto a physical footing. Do not however, expect an 'easy read', though I have tried to lighten the load via notes.

Click here   : for Chapter 4 (§187−§196). This chapter is taken from the Principia mainly as Hermann indicates, about the motion of apses, with a diversion into Varignon, who handled the question of apses from the Cartesian/Huygens viewpoint, though that point is not made clear, and unfortunately centrifugal forces are introduced. Interesting from the early calculus point of view.

Click here   : for Chapter 5 (§197−§211). This chapter is only of academic interest, I think ; a compound pendulum is immersed in a fluid, the parts of the pendulum of equal volume are made from materials with different densities to each other and to the fluid, and the equivalent simple pendulum is found; no account is taken of fluid resistance, etc. Hermann shows that in these circumstances the centre of oscillation and percussion of the pendulum are different, which of course is noteworthy.

Click here   : for Chapter 6 (§212−§237). This chapter is full of interesting material, (but not all of which is immediately accessible to the modern reader, as almost everything is handled in terms of ratios), for example : Whatever the masses, in an elastic collision in one dimension between a moving body and one at rest in the lab. ref. frame, the latter moves off with twice the speed of the C. of M., which remains at a constant speed throughout. Hermann had been looking at one of Huygens' posthumous works, de Motu Corporum ex Percussione, [The motions of colliding bodies] and presented solutions to the propositions, which are presented without proof.

BOOK TWO : SECTION 1.

Click here   : for Chapter 1 (§238−§262). This is the introductory chapter, where definitions etc. are set up, and some theorems are developed relating pressure to depth of fluid, density, etc.

Click here   : for Chapter 2 (§263−§289). Here Hermann sets out his thoughts on the pressure exerted on the walls of a chamber, which appear to be surprisingly accurate in parts ; however, he seems to have been led astray in his analysis of deformable chambers; not all, I must confess, I have understood completely according to his line of thinking, especially his last geometric proposition; and of course one cannot take moments in liquids as if they were solids under stress, etc.

Click here   : for Chapter 3 (§290−§302). This chapter is concerned with the equilibrium of immersed or bodies floating in a fluid. Unfortunately, there is no mention of metacentric height, so that the presentation is rather limited, and occasionally quite wrong.

Click here   : for Chapter 4 (§303−§311). This chapter is concerned with the deformation of flexible vessels, such as the shape adopted by a sail filled with water, which is found to agree with the calculations of the Bernoulli brothers.

Click here   : for Chapter 5 (§312−§324). This chapter is concerned with barometers mainly, and is largely descriptive. Refinements are made to barometers to make them more sensitive to atmospheric changes.

Click here   : for Chapter 6 (§325−§338). This chapter is concerned with the construction of vacuum pumps mainly, with some early views on the nature of pressure. I have added some pages from Boyle's work describing the construction of his pump. Quite interesting.

Click here   : for Chapter 7 (§339−§353). This chapter is concerned with showing the proportionality between pressure and density of air; it follows Newton's approach as cited in the text.

Click here   : for Chapter 8 (§354−§383). This chapter is concerned with showing the proportionality between pressure and density of air, in which different models are examined. This is a continuation of the last chapter, and has some interesting applications to measuring heights of mountains, used by Cassini in his cartography of France.

BOOK TWO : SECTION 2.

Click here   : for Chapter 9 (§384−§406). This chapter is concerned with the flow of water from vessels, and a model is developed which is a fore-runner of what is now called the Bernoulli Principle; however, although formulas are obtained by which the flow from a simple hole or from a larger section in the wall of a vessel maintained at a fixed level with water can be calculated, these do not involve the conservation of energy as such, which lay in the future, but which amount to the same thing. The arguments depend on the final velocity of the water out flowing being calculated from the time for a weight to fall without friction through the same height; clearly the water does not perform this motion, but does so more slowly, yet conserving energy, if no frictional forces are present. The filaments introduced would seem to be analogous to streamlines. I have not gone to the trouble of transposing the arguments into the terms of modern calculus, which means it may be  hard  to follow at times, but it seems to be better to leave matters as presented.

Click here   : for Chapter 10 (§407−§420). This chapter is a continuation of the previous one, but establishes the use of the final theorems therein; it is an attempt to provide solutions to real problems involving hydrodynamics, such as gauging the amount of water flowing though a sluice gate into a channel, etc., as well as investigating the speed of water in an inclined channel, etc.

BOOK TWO : SECTION 3.

Click here   : for Chapter 11 (§421−§433). Hermann is now ready to consider forces exerted between fluids in contact other than pressures induced by their weight, and the interactions between fluids and solid bodies, whichever is moving. He presents some theorems that may seem naïve, and one is left wondering whether he considers the collisions to be elastic or inelastic and no explanation is given of what the fluid does after the collision : he considers it merely to 'slip away' ; no attempt is made to consider the viscosity or resistance at this stage; however, he produces a formula for what he considers the most efficient angle to extract motion from a moving fluid, such as windmill sail in the wind, which is interesting.

Click here   : for Chapter 12a (§434−§462 for whole chapter). I have split the present chapter into two parts, as the going is rather heavy, and a lot of new material is presented. The work deals with the ratio of the resistances experienced by a body moving along its axis where the shape is some given curve, or a curve to be found, where the vertex leads or the base of the curve leads. The resistance is taken proportional to the square of the speed.  A lot of the material can be found in Elements of Hydrostatics published by Miles Bland, a Cambridge lecturer, in 1827, and available from Google with its diagrams : without any reference to Hermann at all! Here the advances in analysis and algebra are evident, and a good historical comparison can be made. The reader may wish to consult this book to aid understanding, where I have not provided notes.

Click here   : for Chapter 12b. Hermann now considers uniform motion of a fluid at an angle to the main axis of a body, or conversely the resistance of the same body not moving along its main axis, but at some angle, through a fluid at rest. The analysis considers the impressed force to be separated into horizontal and vertical components at the incremental level, then summed or integrated to find the equivalent vertical and horizontal forces, acting on an arbitrary convex curve. A number of examples are then detailed for well-known curves, in which the necessary calculations are performed. The texts I have used have poor printing, so that there may remain the occasional misprint, letters are occasionally lacking in the diagrams, which I have inserted.

Click here   : for Chapter 13 (§463−§476). Hermann now considers the shape of a sail in the wind; he relies on the analysis of Johan Bernoulli to produce his own version based on the catenary, but he elaborates greatly on the derivation of the results both geometrically as well as analytically. There is a basic problem trying to put differential elements on a diagram, as squares are no longer squares, a point on a tangent line may lie inside the circle, etc. Use is made of the equilateral hyperbola rather than the equiangular hyperbola in working out the areas of sectors of the hyperbola in terms of a logarithmic curve, the inverse of which was the catenary curve. I have provided notes for the first proposition mainly; there is a lot of interesting mathematics in this chapter, not all of which I have time to investigate at present ……

BOOK TWO : SECTION 4.

Click here   : for Chapter 14 (§477−§494) . Hermann now sets out some properties of  his logarithmic curve that he is going to find useful in the following chapters, in which he intends to find the motion of bodies in mediums with the resistance proportional to the speed, the square of the speed, etc.  I have indicated that this will probably be incorrect, as such motions depend on the exponential function, not yet developed by Euler ; however, it will be of some interest to see what transpires; Hermann's logarithmic curve seems to be elusive at present, as it cannot be the graph of simple logarithms, as he seems to be indicating. Perhaps some knowing person can enlighten me ……

Click here   : For Huygens thoughts on the logarithmic curve. Well, nobody enlightened me but I have worked it out myself, or rather Huygens did, in a manner of speaking, in a note entered at the end of his Treatise on Optics, which you can look at here. The curve in question is the exponential curve, to some base a in general, or e, in which case the subtangent has unit length always.

Click here   : for Chapter 15 (§495−§521) . Hermann now sets out to establish the formulas for the acceleration, velocity, and distance gone, for a body projected downwards, upwards, at an angle, and at an angle to an inclined plane where the air resistance is proportional to the speed ; in which he appears to be successful in establishing the requisite formulas.  He also demonstrates that his method produces the same results as Huygens, Varigon, and Newton obtained by different methods, so there is little doubt that he is correct. However, explanations are lacking, the same exponential decay curve is used for each kind of motion, and there is some confusion about what the various lines represent. Part of the trouble is the lack of analytical formulas to supplement the mainly geometric style of proof. It is now clearer to me that Hermann spend considerable ingenuity is establishing geometrical arguments and finally taking what he called moments, to effect the differential equations and their integrations which we now assert at once from conservation of mechanical energy, work done by friction, etc. In a sense, he was 100 years ahead of his time, as these principles were unknown at the time. The natural logarithm curve forms the backbone of these calculations, and that equates an arithmetic progression of the times to the log of a geometric progression of the speeds, while the inverse logarithmic curve [i.e. exponential] turns a geometric progression of the speeds into the anti-log of an arithmetic progression of the time, as Huygens had surmised. I am at present making small adjustments to these latter chapters by way of notes, etc.

Click here   : for Chapter 16 (§522−§539) . Hermann now sets out to establish the formulas for the acceleration, velocity, and displacement for situations similar to the above, but in which the air resistance varies as the square of the speed.

Click here   : for Chapter 17 (§540−§561) . Hermann now sets out to establish the formulas for the acceleration, velocity, and displacement for situations similar to the above, but in which the air resistance varies as the square of the speed but also includes a term proportional to the speed to account for the viscosity of the air. The propositions are presented geometrically, and are hard to understand at first; it is probably a good idea to have a look at the next chapter while reading this chapter, as the motivation for the geometrical procedures adopted can be understood in terms of presenting logarithmic integrals that can be evaluated. There is a problem with the viscous term, as the two complementary physical situations presented in the text do not have the same resistive forces acting ; we know for example, that if a ball is thrown upwards, it takes longer to come down than to go up; I am not sure if Hermann appreciated this fact in his discussions.

Click here   : for Chapter 18 (§562−§580) . Hermann now sets out to establish the formulas for the acceleration, velocity, and displacement for situations similar to the above analytically; this is a great help in understanding the propositions presented above. One wonders why he went to the bother of presenting geometrical proofs, which in the end have to be performed analytically in any case. The case where the density of the air is varied is also considered here.

Click here   : for Chapter 19 (§581−§601) . Hermann now applies his theory do motions on special curves, and he concentrates on the cycloid, with the air resistance proportional to the square of the speed. I have indicated in a note the pre-energy vs work done ideas used by Hermann in his analysis, which were not appreciated at the time.

Click here   : for Chapter 20 (§602−§623) . A body is projected along a given curve with air resistance proportional to the square of the speed, while it is attracted by masses located at some points on another curve. The forces at any point on the former curve are resolved and integrated to describe the motion. However, the analysis is performed geometrically, and  a number of limiting cases are examined, and in which generally the rectangular hyperbola is used to change ratios of quantities into areas and vice versa, essentially the logarithms of the ratios, although logs are not referred to in the text. As in a lot of Hermann's work, no particular problem is actually solved numerically: instead, curves are sketched generally and the analysis is geometrical. In other words, what you see is what you get, as geometry is essentially a visual form of mathematics, devoid of actual numbers.

Click here  : for Chapter 21 (§624−§629). Here the air resistance of the wind on a sail, taken proportional to the square of the relative speed of the wind to the sail, and with the water resistance removed in a similar manner, is taken as the  accelerating force generating essentially the kinetic energy of the ship.

Click here   : for Chapter 22-24 (§624−§629). In Ch. 22 a number of results are established for the shape of the earth, similar to those developed by Newton and Huygens. Ch. 23 enlarges on Newton’s method for finding the speed of sound, and employs some of the geometry associated with simple harmonic motion. Ch. 24 gives the first indication of the kinetic theory, with the pressure depending on the mean speed squared of the particles and the density of the gas.

Click here   : for the Appendix to Book I. A number of issues are investigated here in more detail, especially Kepler's Laws from an analytical point of view, and the calculation of the position of the centre of oscillation of an extended body, a forerunner of the moment of inertia analysis of Euler. Unfortunately, I have not the time to spend doing a thorough analysis of these matters.

Click here   : for the rest of the Appendix. A number of items are added here finally. An analysis of Hermann's interpolation schemes; some notes on the rate of flow of a liquid from the bottom of a vessel, erroneous of course, and the correction of  Prop. XXV, Book I, where an incorrect sign had crept into the calculation.

Ian Bruce.  16th June, 2016; latest revision. Copyright : I reserve the right to publish this translated work in book form. You are not given permission to sell all or any part of this translation as an e-book. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. See note on the index page.