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article 21c, issue 08


Regarding the amount of work which men can provide during their daily work, according to the different ways they use their strength.

Charles-Augustin de Coulomb
1781/1821, translation started 21 December 2014

Editors' notes: This is so far mostly a translation by Google. We plan a human translation and a proper representation of the equations, but have so far only achieved the first few and the last sections. Anybody who would like to help, please contact the editors.
The male gender "man" is used instead of "human" because Coulomb used the word "homme" exclusively and described types work presumably done exclusively by men.

1. The human body, a collection of flexible well-designed parts, can assume an infinite number of forms and positions. Viewed this way it is a marvelous machine which can preform highly complex tasks requiring nuances of force, speed and direction.
Although the strength of men is limited, it is sometimes preferable to the greater strength of animals. Working together, it is easy for groups of humans to compensate for the limited force of a single individual. Humans require less space when working than do animals and are easier to transport. And the unique attribute of human intelligence allows humans to save their strength and moderate their efforts in accordance with the task at hand.

2. Although the strength of men is limited, it is sometimes preferable to the strength of animals. Working together, several humans can be as strong as an individual animal. Humans require less space when working than do animals. Humans are easier to transport than animals. And the unique attribute of human intelligence allows humans to save their strength and moderate their efforts in accordance with the task at hand.

3. The effect of any work could be evaluated by measuring the resistance which a given weight must overcome, multiplied by the speed and the duration of the action; or, equivalently, the product of this resistance, multiplied by the distance that this resistance will have traveled in a given time, for we obviously see that it follows the same effect, that a movement of a ten kilogram (kg) weight through one meter (m) or 1 kg through 10 m, since ultimately the work is ten times the height of one meter per kilogram [force].

Regardless of the number of wheels or levers of which a machine is made, if a weight acts against another with a uniform motion, the force produced by the falling weight, multiplied by the vertical distance it travels, is in theory equal to the force of the rising weight multiplied by the height through which it rises; the latter quantity is the effect. Thus, in practice, the effect altered by friction, vibration, and all the inconveniences of the machines, is always less than the force produced by a weight multiplied by the distance through which it has traveled.

4. We have seen that the effect of a machine can always be measured as a weight multiplied by the height through which it is raised. Now, to compare the effect with the fatigue that men experience in producing this effect, it is necessary to determine the fatigue that meets a certain level of action. I call action the quantity resulting from the pressure exerted by a man multiplied by the speed and time of this action; a quantity, as one can see, which can be represented by a weight which is dropped from a certain height in a given time; and if, by producing this amount of action, a man experiences all the fatigue that he can endure every day without degrading his animal vigour, the amount of action measures the effect it can produce in a day, or, if you will, the weight that can reach a certain height in one day. So the whole question is reduced to inquire what is the way we must combine the different degrees of pressure, velocity and time, so that a man with given fatigue, can provide the greatest amount of action.

Daniel Bernoulli, who discussed the matter, having regard to the greater part of its elements, said that the fatigue affecting  men is always proportional to the quantity of action; within limits set by their natural forces the velocity, pressure and time can be varied at willand, provided that the product of these three quantities is a constant quantity, it always results in the same level of human fatigue.
He added that in any way man uses his strength, either by walking or by applying force to  either a crank or on the rope of a bell, or by lifting a sheep to surmount a step  along a path, it will produce the same degree of fatigue, the same amount of action, and therefore the same effect. It assesses the daily work of men in all kinds of jobs, to be equivalent to lifting a weight of 1728000 pounds one foot high, which amounts to 274,701 kg being lifted through one meter. (Academy Award, Volume VIII, page 7.)
Désaguilliers, and most authors who had a need  to calculate the work of  machines, and to evaluate human action, adopted nearly the same results: all authors cite these experiences; but I observe that most of the experiments they cite only lasted a few minutes, and that men can, for a few minutes, give an amount of action in which they would not survive an hour a day: so we cannot reach any conclusion from these studies.

5. Although, as will be told later, fatigue is not proportional to the amount of action, as assumed by the famous D. Bernoulli; however, whatever the formula which represents the fatigue, it must necessarily be a function of the pressure, the velocity of the point of pressure, and the duration of the labor. So there must be a formula in the combination of these three quantities, such that at equal strain, we have maximum action and therefore the greatest effect that men can produce in one day.
This combination is different, as we shall see later, according to the ways in which man uses his strength: hence there results the consequence that, as in any work we must strive to provide the greatest effect  which expresses the maximum amount of action in relation to fatigue, should be the main object of the research. This amount is even more interesting to determine, that, according to the theory of maxima and minimis when it is known, we can vary substantially the elements that compose it, that is to say, velocity, force  and duration, without substantially increasing fatigue.
The amount of work that men can provide when they encounter during a working day, a ramp or stairs with a load or without a load.

6. When you go up the stairs of our homes if we do not have to rise beyond 20-30 meters, we can climb at a rate of 14 meters per minute. To calculate, from this experiment, the amount of action provided by a man in this kind of work for a minute, he cares multiply the weight of the man by the height at which it stood. The average weight of a worker can be assumed 70 kilograms; thus, the amount of action that provides for a minute, to measure 70 kg multiplied by 14 meters, or, equivalently, 980 kg higher than one meter in height. Assuming that a man can support this work four hours a day, the amount of daily work would measure a weight of 235,200 kg higher than one meter in height. But the assumption of four hours of actual work per day absolutely hypothetical; when should rise to 15 or 20 meters in height, it can provide this level of action, and even a much larger; but we must rise beyond 30-40 meters, one feels forced to reduce speed and to slow its movement. I have often seen men to ride, without charge, 150 meters high, with a staircase carved into the rock, but quite convenient, and I found they used 20 minutes to rise to this height; I wanted to commit to eighteen times up the stairs in the day; what is required, according to my calculation, six hours of actual work. As I wanted and that I should not, according to the object I intended, give them the price of a day, not wanting to engage in forced labor, I have not been able to determine a walk that seemed to them as tiring as ridiculous. I was beginning to despair of me to get the measure of the amount of action that men can provide this kind of work, when I remembered that our friend Mr. Borda was corrected by very precise geometric operations the offending measures that we had before him the height of the peak of Teneriffe. Here is what he communicated to me, and that is confirmed by a report signed by all those who cooperated in his work. The one climbs the peak of Tenerife in two days; the first day at 2993 meters: the first day can be on horseback; but on the second day is at 857 meters from rising, as well as the hands with feet, on stone and slag which roll under the feet and lead you with every step; it is even necessary to climb the last hundred meters, support with ropes. After visiting the summit of the peak, it goes down to sleep at the station the day before. We can not, from this detail, we used to evaluate the daily work of men, the path in the first day. Mr. Borda traveled the first day on horseback, and all the officers of his ship; but there were eight men who accompanied him on foot; three guides, two men wearing compasses, barometers and thermometers; it estimates the load of each of these men to 7-8 kilograms; Both men led horses loaded; and the eighth was a traveler, son of Mr. Lalouette, doctor of Paris. When the men arrived on foot were they are still back down fifty meters to collect firewood in order to be able to light a fire; which proves not qu'ls were not exhausted with fatigue. The meters were installed in 2923 by eight men from nine in the morning until half past five; whereupon there was a break of three quarters of an hour for dinner; thus, there were only seven three-quarter hours of actual work. Note that most of these men were sailors unaccustomed to forced matches.

7. If we assume that men consumed walk, riding at this height, the entire amount of action they can provide in a day, it will be necessary to have this amount, multiply their weight, we evaluated 70 kg, per 2923 meters, height at which they are mounted on the first day, which gives a high equivalent to 204,610 kg in a quantity meter; but it should be noted that the very irregular ramp parcouraieut they were much more tired men as if they were connected by a convenient staircase; that the ramp had over 20,000 meters horizontal length, instead a convenient staircase, which would have risen to 2923 meters would have had wide steps 8 to 9000 meters, which necessarily unnecessarily consume part of action. But like riding a ramp or stairs there is a combination of horizontal movement and vertical movement, which could be open to debate, I will simply assume that men who ride a convenient staircase, they quelqu'habitués are in this kind of work, can not rise to the height of 2923 meters, as experience gives to men who climbed the peak of Teneriffe, on an uneven ramp, and where their feet n ' were not placed conveniently; whereby, as we have already found a quantity of action that can be evaluated in round numbers to 205 kg a mile high. Although, based on all observations in this article widespread, it is likely that this amount of 205 kg high action one kilometer is too weak to express the amount of daily work that can provide a man accustomed to this kind of work and amount freely convenient staircase, with no charge, however this action is so much greater than any that can provide the same man in any daily work, acting with his arms, or by other means that j 'prefer to run the risk of being a little below the true value of the kind of work I want to determine here, than risk the pass.

8. We have to evaluate 205 kg a kilometer high, the daily amount of action of men climbing a convenient staircase without being charged with any burden; must now try to compare this with the amount of action that men can provide when they get a burden. I often ride firewood to 12 meters in height; I've never been able to make up, by the same man, over six lanes in one day: he always told me that it would be impossible to continue such work for several days. This man was a force slightly above average strength; I paid at the rate of one franc per channel. So I can watch six channels of wood as the greatest burden that men can reach 12 meters in one day. So I have only to compare the amount of action that provides a man walking up stairs without charge, with that of a man who rises in the day such a burden. The way wood moderately weighed 734 kg; the man rose in eleven trips; in the first ten travel lanes in the past twelve; it amounted to 66.7 kg each trip; 68 can be assumed, because of the weight of the hooks. Add to this the weight of the load body of man, which we suppose to 70 kg, we will, for the amount of work provided in each trip, 138 kg to 12 meters high; and as the carrier was in the sixty-six day trips, it will be for the amount of action provided in the day, the three numbers 138, 66 and 12 multiplied together, or, equivalently, 109 kg a mile high. We saw in the preceding article, a man who is not responsible for any burden may, in its day, raising 205 kg to one kilometer; thus the action of the daily quantity men naturally ascend stairs, is that of the man in charge of 68 kg, 188 is as 100; report that, according to the above comments, I think too low; we depart somewhat from the truth assuming that two men riding under such a load, can provide the same amount of work without a single load. This result, which I believe have assessed too low, compared the amount of work provided by men climbing stairs freely, with that of the man in charge, contrary to the assertion of D. Bernoulli, and almost all authors who followed, who say that as long as the loads do not exceed the strength of animals, the amount of daily action will always be a constant quantity. I asked di fl Erens men who built my wood, what the greatest work of its kind that they could provide in one day was. Whoever went to the highest of his comrades told me he once mounted seventeen tracks of wood in a day, at a first stage, which he estimated the height to five meters; he was then able to work two days without. If we submit to calculate the work of this man, we find, according to his answer, he had to make 187 trips; the amount of aetion he provided is equivalent to a weight of 129 kg to a high kilometer. Although this amount of action to respond to a daily tiredness a very strong man can barely support, it is however the amount of action of man walking up stairs with fatigue certainly much less than in the ratio of 129 205, or nearly as 10 is to 16.

9. In the calculation, I have not given the amount of action that men consume down the stairs; but, as in the raid that they hardly traveled 1800 meters, and that, according to their own admission, it does not appear that it is much more tiring than walking down on a flat ground, where a man in a strong day's work, travels at least 50,000 feet, fatigue due to the descent can not be evaluated beyond the twenty-fifth part of the daily work; and can be even more neglected, the amount of daily work of the man climbing the wood is probably too high, relative to that of the man who rises freely and without load.

10. In this work it is an interesting observation concerning the effectiveness of the work. When a man mounts a burden, he formed his own weight with the burden; and as each journey, it drops to empty, there is a useful purpose in the amount of action that provides the transportation burden. But it is clear from the foregoing that as the burden increases, the total amount of daily share decreases; so that it would be void if a man was responsible for 150 kilograms, weight under which he could hardly move; On the other hand, if he rode without burden, though for the amount of action in daily is the maximum of all quantities of action it can provide for its daily work, the burden is zero, the effect would also be useful. Between these two limits of action, there must be for the weight of the load, such a value, that provide useful daily work, with a maximum effect; it is interesting to determine this value.
To succeed in an exact way, there should be a ormule which should represent the amount of daily work that men can provide different loads, but in practice, it can be satisfied with approximate formula; and simplest, provided it gives a continuous decrease as the load increases, and it is consistent with the weight limit that serve the maximum and minimum action, she understands more a intermediate value provided by the expérieuce, will almost certainly less greater than the differences resulting from two experiments at different days mistakes. It will be easy, by complying with this observation, to determine the load that gives maximum effectiveness.

11. When a man mounted freely a staircase, we have seen that in neglecting the fractions, which there is no need to take into account in a search of this kind, its quantity of daily action has been represented by 205 kilograms pupils at a kilometer; but that when it is carrying a load of 68 kilograms, its quantity of daily action has been represented by 109 kilograms pupils to a kilometer. Thus, in striking out this second number from the first, we will find that a burden of 68 kilograms has decreased the quantity of action that a man provides when it mounted freely a staircase, of 96 kilograms pupils to a kilometer.
It seems now that we can assume, without large error, in a question of the kind which we occupied, that the quantities of action lost are proportional to the loads; and for when, if we appoint P any kind of load, we will have the quantity of action that this load is lost, by doing 68 : 96 :: P: the quantity of action lost, which is therefore equal to 96/68 P = 1.41 P, or 1.41 kilometer multiplied by P.
As well, such as the quantity of action that the man provides in amount freely a staircase is 205 kilograms pupils to a kilometer, we will have, for the quantity of daily actions that it can provide under the load P, the formula 205 - 1.4 P. Here 205 Represents 205 kilograms pupils to a kilometer, and 1.41 represents one kilometer 41 hundredths, height or is high the weight P.
If h is assumed the height at which the man responsible for the weight P can be elevated by his daily labor, Ph will be the useful effect of the work, and (70 P)h will be the total quantity of action provided by the man, whose gravity is 70 kilograms, that it high at the same time that the weight P. Thus, we have the equality
(70 P)h = 205 - 1.41 P;
of or is the result, for the useful effect,
Ph = (205 - 1.41 ) / (70 P) P.
Doing so 205 = a, 1.41 = b, 70 = Q, we have
Ph = (a-bp)P/Q P ,
quantity in which, for having the Maximum of Ph, it must do vary P, and the differecre of the quantity that represents Ph equal to zero; the result will be, for the value of P, P
=Q[ (1 a/Bq) ^1/2 - 1] .
By substituting the numerical values of a, b, Q, we can find P = 0.754 x Q = 53 kilograms.


12. If, in the formula Ph = (205- 1.41 P)P/ (70P), which represents the useful action, I substituted in the place of P, 53 kilograms, i have Ph = 56 kilograms pupils to a kilometer. As well, this kind of work or the men rise of burdens, and then come back down to take a new load, does not provide in useful work that 56 kilograms pupils to a kilometer, while the rights amount freely provides a quantity of daily action which has to measure 205 kilograms pupils to a kilometer. The result is that this kind of work done unnecessarily consume almost three-quarters of the action of men, and costs therefore four times more that a job or, after having mounted a staircase with no load, they would fall back by any means, in causing and raising a weight of a gravity roughly equal to the weight of their body. As well, this kind of work, although very in use especially in the cities, should never be used in the workshops which require of the celerity, the economy and a continuous job.
To verify if the assumption we made of the decrease in the quantity of action proportional to the loads , can give large errors in the practice, we need to see whether the amount of action that the man can provide in a day determined according to the formula (205 - 1.41 P), will give, to the point where it becomes 0 (because the man is in charge of the largest weight that it can bear), an approximate quantity of that provided by the expericnce. Thus doing 205 - 1.41 P = 0, we will have P = 145 kilograms, weight actually the largest that a man of a medium strength can bring to a very small distance.
As well it seems, according to this result, that the formula that we have drawn from the experience to determine the maximum of the useful effect that can provide the men in a staircase under any kind of load, meets at the same time the two limits, that is to say, to the maximum of action total rights amount freely and without charge, to the minimum of action when the man is in charge of a weight so enormous that it can no longer move, and in a quantity of intermediate 68 kilograms provided by the experience, regular load of men who rise of burdens.

13. Returning to the consideration of maximum effectiveness. We have now found that for a man to provide this, it is necessary that each trip it covers only 53 kg; However, we saw that it was responsible, in our experience, 68 kg each trip. This difference between the calculation results and experience deserves that we seek to develop the causes.
The first thing to determine is the difference resulting to the effectiveness of the work, the substitution of a weight of 68 kg instead of 53 kg weight given by the formula represents the amount of useful action.
According to the art experience. 8, the worker has sixty-six trips. On each trip he went up to 12 meters high a burden of 68 kilograms; which gives, for the amount of useful action 12.66. 68 = 53.86 kilogramnes a kilometer high. We found previous article, when the load is 53 kg, the practical effect is a maximum, and its value is 56 kg higher one kilometer; amount that was little more than a twenty-sixth majority as that provided by the man charged with 68 kg.
It is understood, according to this comparison, that workers who perform oes kinds of work can have no idea of ​​such a small difference, while their interest to be associated by their peers for profit businesses, go to very strong; besides, what they should do illusion is that they reduce the number of trips by increasing each particular load.
If one wants to be convinced of the truth of these reasons, we have only to ask the stronger this kind of workers, those who boast to mount a load of wood to 12 meters in height in seven to eight travel, they can climb the six paths in forty-eight trips; they all confess that this is not possible, and that when this work is to last a considerable part of the day, it is necessary to decrease expenses, increase the proportion of trips; qu'autremeut we would soon be exhausted with fatigue.
Comparing the amount of action that men can provide when traveling in a horizontal way, with a load or no load.
14. When men hang traveling several days without charge, they can travel easily in their day 50 kilometers. If I assume the average weight of 70 kg, as I did in the preceding articles, the amount of action they provide will be represented by 70 kg multiplied by 50 kilometers, or, equivalently, by 3500 kg transported one kilometer.
Order now compare the amount of daily work that man can provide when traveling without load, with the amount of action that provides the same man when traveling with a burden, this is how I'm me taken.
I proposed to different porters carry furniture for housing in another, at a distance of 2 km, being in charge, on every trip, weighing 58 Kilogramm .: they all me said that all they could do was six trips in the day, and it would be impossible for them soutinssent for two days on such a work. None of them wanted to undertake at least 12 to 15 tenths per trip.
If we determine our calculation on these data, we will find, joining the weight of the man, who is 70 kg, with the load, which is 58 kilogratmnes, the weight carried 2 kilometers each trip is 128 kg. Thus, for the amount of work provided in six trips, multiply 128 kg per 12 km, amount equivalent to 1,536 kg transported one kilometer.
But to have the total amount of daily work, add to this first quantity fatigue resulting from 12 km that men travel in returning for a new load. As here they have no burden, and that men, in a day, can travel 50 kilometers, they consume in this return to approximately the fourth part of their daily work; and 1556 kg carried one kilometer, which represent part of their work when loaded, are three-quarters of the working day. So the work or the amount of work that men can provide in a day, with a load of 58 kg, can be estimated at an amount equivalent to 2,048 kg transported one kilometer.
Hence it follows that the amount of daily acuon that men can provide when walking freely, is that they can provide when loaded with 58 kg, as 3500 is 2048, approaching as 7 4.
15. I then interviewed several hawkers, to know what was the largest weight they carried in their travels, and what path length they could go in a day with this weight. The average result of the response of those who seemed to me the strongest, that was loaded with 44 kg, all the way they could do that day was 18 to 20 kilometers.
To calculate the total amount of action provided from the response hawkers, add the weight of the man, who is 70 kilograms, his dependents, who is 44; which will give a mass of 114 kg transported in the day 19 kilometers, or, what amounts to the meme, 2166 kg transported one kilometer. We found in the preceding article, according to the request of porters for the amount of daily action 2048 kg transported one kilometer amount slightly less than that provided by eté we work hawkers; but it should be noted that the charge of the porters was greater than that of hawkers; which, according to the results of the experiment necessarily lose part of the action. The agreement is between these two results shows us that we do not move away us a lot of truth, if we assume that a load of 58 kg, men, traveling in a horizontal way, can provide for their daily work an equivalent amount of action è weight 2000 kg transported one kilometer.
Here I am approaching an outcome that provided by porters, because I almost always found that hawkers accuse load a little stronger than they are, that their days are also very irregular, they can not have an imperfect idea of ​​the amount of their daily work.
16. We have, from the experiments above, determine what must eter man's burden at equal tired he can produce the greatest effectiveness. This effect is measured by the load transported multiplied by the distance at which it is transported; because here, as in the previous question, the amount of work required by human body transport is absolutely pure loss for the effectiveness of the work.
Begin by determining the amount of action that the burden is lost; throughout the rest we will follow the method we have explained, in the preceding articles, a man walking up stairs.
So we find first that when men travel freely and without charge, they can travel 50 kilometers; for when they provide in their daily work a quantity of actiou equivalent to a weight of 3,500 kg transported one kilometer.
Secondly we find that when men are responsible for 58 kg, they provide for their daily work equivalcnte a quantity of Action to weight 2000 kg transported one kilometer. Thus the amount of daily action lost a load of 58 kg is equivalent to a weight of 1,500 kg transported one kilometer.
 transported one kilometer.
If now we assume, as we have seen above that it is psnible to do it in a search of this kind, that action losses are proportional to the charges; naming the load P and x the amount of action that lost this charge, we will have 1500: x :: 58: P, where
 x = 1500P / 58 = 25.86 P.
 Thus, the amount of daily action can provide a man under the load P is equal to the amount of work it can provide no load, reduced by the amount of work lost due to the load P; which gives, for the amount of daily action 3500 to 25.86 P, where 3500 represents 3,500 kilograms multiplied by a kilometer and is 25.86 miles.

17. If we look from this formula what is the more weight a man can carry, or, equivalently, that under which it ceases to act will require the amount of action 3500- 25,86P = 0; which gives P = 135.4 kg, quantity which is actually almost that a hommee of average strength can wear for a very short time. This quantitée that gives the limit of human action in this kind of work, which was provided by the assumption of the amount of action we lost proportional to the load, is a proof that this oertaine supposilion n was unable to make us commit to significant errors.

18. It is now necessary to determine the load under which the man who carries burdens can provide maximum effectiveness. Suppose that under the load P, the man in his daily work, traverses the space, the amount of daily action by Q = 70 kg, which is the weight of his body, will (P + Q) l; amount that has to be equal to (P 3500 to 25.85), which represents the same amount of action when the man is responsible for weight P; thus one has
 (P + Q) l = (3500 to 25.86 P);
 of or pulling
 Pl = (3500-25,86P) / P + Q.
 This quantity multiplied Pl represents the load pa space it has traveled, and therefore e'effet useful work. This is the amount that must be differentiated by variable P and the difference 0, for the greatest effectiveness. I suppose if 3500 = a, b = 25.86, it will result from the difference in the amount equal to 0, the same formula as the X1 article; P = Q [(1 + a / bQ) ^ 1 / 2-1]; in which equality, if we substitute the numbers, we will
 P = 0.72 x Q = 50.4 kg.

19. In the kind of work we present here the calculation, there is a special case which almost always occurs in transportation which are in cities; This is when healthy men of expenses or backpack, or on stretchers, return empty each trip to find a new load. It is necessary to determine what kind of work which is the load under which a man can provide the greatest effectiveness. If l = 50 km, length of the path that a man can travel in a day when it is responsible for any burden, always assuming Q = 70 kg, the weight of his body, Ql will be the amount of action it can provide in the day when it carries no weight; but if it runs without load space x, smaller than l, Qx will be only a part of his daily work. If we divide this portion of work per Ql, which is the work that it can provide in the day, Wx / or Ql (x / l) is the portion of a day's work without load, including cst unit all; since becoming the x, x / l will be equal to unity.
But as the man travels here lc meme path x loaded and unloaded, and when the man is responsible for weight P, we found the amount of work it can provide in their daily work, equal to 3500 - 25.86 P; since the portion of the action under this load is represented by P (P + Q) x, the ratio of this amount with the amount of daily share represents the portion of the daily work he has provided in this load. Thus, we have, for that portion of work, (P + Q) x / 3500-25,86P; and as the sum of the work of the man in charge, and labor granny man walking freely, must equal the working day, we will have
x / L + (P + Q) x / 3500-25,86P = 1.
But because Ql = 3500, which is the amount resulting from the weight of the Q multiplied by the man the way he can travel in a day when it is responsible for any burden, let h = 25.86 kilometers; the above equation becomes Px = P (QL2-h / P) / 2QL + P (lh) or Px expresses the portion of action which is equal to the useful effect that man can supply in a day work.
We must differentiate the Px value by variable P, and suppose the difference 0.
For simplicity, I QL2 a = b = hl = 2QL c, f = lh; and Px = aP-BP2 / c + fP. Differentiating the second member, the difference equaled 0, we have, by ordering the formula, ca-2bcP-BFP ^ 2 = 0; hence resulting P = c / f [(1 + fa / bc) ^ 1 / 2-1].
In presenting the numbers instead of letters, we have
f = l - h = 24.14.
a = Ql² = 70.50 ^ 2 = 175000,
b = hl = 25,86.50,
2QL c = = = 2.70.50 7000.
These substituted values, we will draw P = 61.25 kg.
This burden is very nearly that worn men of average strength when they are forced to make several trips in a day at great distances; so it should not remain in doubt about the accuracy of the elements of which this result is deducted.
20. If we want to have, according to this value of P = 61.25 kg, the amount of useful work that men provide this kind of work must be replaced to 61.25 instead of P in the formula-bP aP ^ 2 / c + fP, which represents Px, and we will find, according to this substitution, Px = 692.4 kg transported one kilometer, which represents the largest amount of useful action or effect a man can give in his day.
Substituting in the formula, instead of P, 58 kg, weight we first assumed the man in charge, we would find, for the amount of useful work, Px = 691 kg transported one kilometer.
If we assume P equal to 65 kg, we would find Px = 690 kg transported one kilometer; and we see an increase or reduction in charge from April to May kilgrammes produces only insensitive to differences in the maximum useful effect.
If we wanted to compare the amount of action that man provides walking freely with the amount of useful effect it can produce this kind of work, we would find a man walking without burden that can produce a quanlité Action represented by 3,500 kg transported one kilometer, while lelffet useful measure is to 692.4 kg transported one kilometer, these two quantities are to each other as 505 to 100, approaching very like 5 is 1; that is to say, in this kind of work usefully employed the amount of action is the fifth part of it that can be provided in his day a man walking without burden.
21. Quantities of action provide men climbing stairs, are not the same kind as those of men who freely walk on level ground, because in the first case, they are forced, at every step, raising the center of gravity to the height of a step, while Hummes traveling a horizontal path give their body a velocity parallel to the ground; this speed is not destroyed by their gravity, so they have to produce every step alternative transportation legs and little considerable rise in their center of gravity, which rises falls ct at each step by an oscillatory movement of 2 or 3 millimeters; which depends mainly on the art that men acquire when they travel often, to raise their very low center of gravity, and support approximately parallel to the ground on which they walk.
But though these two types of action are not da same nature, it is nonetheless interesting to try to compare, for equal fatigue, the height where a man can raise his center of gravity, with the way that it can travel on level ground. The results of calculations and experiments above will provide Celtic comparison.
When men go up stairs without any burden, amount of daily work is measured by 205 kg higher one kilometer; they traverse a horizontal path, quantity of daily work is measured by 3,500 kg transported one kilometer. These two quantities are roughly between them as 1 to 17.
The usual height you a stair can be assumed to 135mm, its width being about three times its height. And seventeen times 135 millimeters or 2295 millimeters, represent the length of the horizontal path that a man can travel with the same degree of fatigue when walking up 135 millimeters. But as ordinary horizontal steps of a man is 65o millimeters, it follows that man has the same degree of fatigue by mounting a march of 135 millimeters, that by three and a half steps on a horizontal path.
Quankte of action that men can provide in their daily work when carrying burdens on brouetts.
22. The kind of work that we will submit the calculation is in use in all civilian and military jobs that require ground transportation. The Marshal Vauban, which of all engineers is perhaps the one that has done the most performing this type of work, has left us in a printed statement in the Science Belidor Engineers, the results of several experiments after which one can try to calculate the amount of work that men can provide daily in this kind of work. Here's what Vauban said; I reduce the measures he used to our current measures.
"A man in his daily work, can carry in a wheelbarrow 14.79 cubic meters of earth to 29.226 meters apart; causes that land mass five hundred trips: thus he traveled 14.613 kilometers loaded and unloaded as much on bringing the wheelbarrow. "
We must join these data Vauban some other remarks. When the wheelbarrow loaded, men, seizing the arms of the wheelbarrow 15 decimeters roughly away from the axle, support a portion of the load and some of the weight of the wheelbarrow; the rest of the weight is carried by the point of the terrain on which raises the wheel.
I found, supporting the loaded wheelbarrow, using a spring balance, to the point where men take arms, that they support some of the weight is 18 to 20 kg; when the wheelbarrow is empty, they relate only 5-6 kilograms.
I still found that when the wheelbarrow is loaded, the arm being supported by ropes attached to a very high point, the force required to push the wheelbarrow on dry land and united is 2 to 3 kg. This latter force is largely dependent on small ledges that the wheel experiences in the field: it varies according to the address of the worker, who can not always take control of the movement of his wheelbarrow.
23. To determine, based on the experience in this kind of work, the amount of useful work that men provide, we noted that the average load wheelbarrows, in a workshop composed of powerful men, is about 70 kilograms; the weight of barrows, which varies widely, moderately 30 kilograms.
But as the effectiveness is measured by the amount of land transported multiplied by their courses; since men are rolling the wheelbarrow loaded at 14.61 km away, the daily effectiveness will be measured by the product of the two numbers 70 and 14.61 multiplied by one another, which gives an amount equivalent to 1022.7 kg transported one kilometer.
But we found, Art. 21, when a man carrying backpack burdens, the maximum of the effectiveness of his work was to measure a weight of 692.4 kg transported one kilometer; thus the effectiveness provided by a man who carries burdens on a wheelbarrow, the effectiveness is the same man when carrying the same burdens on his back as 1022.7: 692.4: 148: 100; so that on dry land, united and horizontal, 100 men with wheelbarrows will, in almost exactly the same amount of work as 150 men with hoods.
The amount of action that can provide our stud sounding, movement that executed when they raise sheep to beat and push piles.
24. In the action of men who raise sheep and hang down on the head of piles useful action they provide is determined by the weight they raise the height to which they rise, and the number of shots they can give in the day. Here is what is practiced very often, because there were many varieties in the distribution of weight, based on the strength of men.
Ordinary sheep weighing 350 to 450 kg. A rope that passes over a pulley supports one side the sheep; at the other end of the rope are attached dilférens cords men grasp with the hands.
When the sheep is about the pilot, men hold the cord at about the height of their hats; then fall leaving the upper part of their body, making an effort on the cord, they amount to about 11 sheep decimetres; it beats about twenty strokes per minute, and sixty to eighty strokes on, after which the men sit so long they worked. Despite that rest, we have to face the most often by the hour.
Following this work, taking into account the different rest, I have never seen workers withstand more than three hours of actual work during the day; the remaining time is used to the different rest we just mentioned, to place and move the bell, to redress the piles, etc. When men are very vigorous, it usually makes the doorbell a number of men such, that each student 19 kg weight of the sheep.
Based on these data, the amount of daily work in this kind of work will be measured by the product of the three numbers, 11 decimeters, 19 kg, and the number of shots beaten in three hours of actual work at the rate of twenty strokes per minute; which gives an equivalent amount à75,2 kilograms higher è one kilometer.
If we compare this amount of action with a product free man when climbing stairs, quantity that we found, by experience, equal to 205 kg a kilometer high, we will see that in the bell worker provides only a little over a third of the action it would produce in the latter case, and so it would be easy, by force of men in the most advantageous manner, to ensure that one man produced almost as much effect as three how they are used in the bell.
25. The calculation by which we come to determine the daily action of men beating the pilots, gives an amount too great, when compared with similar work followed for several months on è Monnaie de Paris, where men struck coin pieces with a sheep. Here is what constituted the working day.
The sheep weighed 38 kg; it was operated by two men, who were therefore an effort each 19 kilograms. The sheep was high, has every stroke, 4 dm tall; being beaten in the day 5200 pieces, where it is the same, it was the sheep 5200 times.
If, for the amount of action, we take the product of the three numbers, 19 kg, 4 decimeter and 5200, we found that the amount of daily work was represented by a weight of 39.5 kilograms high one kilometer; amount which is not that half of 75.2 kg, we found the quantity of action of men who beat pilings, and is only the fifth part of the daily amount of action provides a free man when climbing stairs.
But note that the same men worked currency for fifteen consecutive months; rather than beating of piles and men go to another kind of work when tired, which happens bientùt.
However, it seems to me likely that strong men, employees of eutreprise, could have provided in the work of the currency, a greater amount of action than that resulting from the above calculation. The person who was responsible for conducting the workshop there told me that a very strong man, had undertaken to conduct itself a sheep, but he was forced to give it up after a few hours.
I believe that this man could have worked several days, if, instead of raising alone weighs 38 kg at 4 decimetres, he would have made an effort only 19 kilograms; that his hand had traveled 8 decimeters instead of 4, and, by some means, the sheep had just been raised from 4 decimetres, as it was by the action of the two men; which produced a fall which, from experience, suffaisait to the footprint of the pieces. By combining the strength and resistance, it is likely that this very vigorous man would have supplemented the two men who beat the currency, since in his daily work he would have provided that the amount of action meme that men who beat pilots may provide for a few days.

26. Here again an experience that has some relation to the work of the bell. I did, for two days in a row, drawing water from a well that was 37 meters deep. One drew by a double bucket; I paid the man for 25 cents by ten buckets. He set up the first day, 125 buckets; the second, 119. The average effort, measured with a load cell was 16 kilograms. I will here 120 buckets for the amount of water that could raise in a day: thus, for the daily amount of action, multiply the three numbers together, 16 kg, 37 meters and 120; giving to the effect or the amount of daily share amounted to 71 kg one kilometer amount roughly the same as the one we found for the amount of daily action of men who beat pilots.
Men acting on the cranks.

27. I could not get me or make my own direct experiences to determine what kind of action; what follows is the result of quite a number of comments on the machines that are used in épuisemens. But in these machines, the resistance that men experience is very difficult to assess. In rosaries, for example, the shock pallet and hedgehogs, frictions of various parties, the loss of water by the play of the machine, while varying according to the state of the machine. These quantities are not the same in the machine in motion and the machine that you want to leave the state of rest. Besides, here it is very difficult to put men to the company, if you want to experience in completing some barrels, which lasts five or six minutes; men, to then provide an amount of action that often doubles announces daily produces effective. I would have been able to get more approximate results if, in the time I was taking this kind of work, I had substituted a crank winch with two buckets with rosaries exhaustion. There same appearance as this means much in use in the country, would have provided me with more advantageous than other machines; because there are few circumstances in which the two buckets, and a winch UUE crank, not be preferable to all machines to exhaustion.
It is estimated, in most mechanical works, a man exerts pressure on the crank handle, 12 or 13 kilograms. I do not believe that in a continuous work, this pressure can be estimated in excess of 7 kg. The crank handle travels usually a circle of circumference 23 decimeters, and there are about 30 revolutions per minute. But in examining several hours workers, we see that, when exercising a pressure of 7 kg, they do little as 20 to 22 revolutions per minute. Finally we evaluate the daily working time to ten hours per day; and in the great work, we apply the workers acting on the cranks, at most eight, on which they slow their movement or rest well enough so that it is possible to estimate that to six hours of actual working time, at 20 rpm.
In calculating the amount of action from these observations together seven kilograms multiply 23 decimeters, 20, and 360; which gives, for the amount of daily share amounted to 116 kg one kilometer. Based on these results, if you wanted to compare the different amounts of action provided by the men who freely ascend a staircase, with that of men acting on the crank and the bell, one would find that the amounts of Action provided by the Memec man in ccs different kinds of work are these as numbers 205, 116, 75; quantities that are almost as numbers 8, 5, 3 reports probably provide sufficient precision in practice; because in a matter of this kind, it is useless to seek an accuracy which variety, which is between the forces of different workers makes it impossible determination.
The practice, moreover, seems to have decided that the cranks are preferable to the bell; because almost all machines used in public works for épuisemens are in play per the cranks.
The amount of work that men consume in their daily work when they plow the earth with a spade.
28. There is a great variety in the results of cc kind of work, depending on the nature of the land and the seasons and the time when the granny preceding plowing were made, which left take the earth more or less sag, and plant roots that cover its surface, more or less extent and strength, that the calculations that follow should be viewed only as a particular example which should serve to throw some light on work that is similar to it.
The farmer that I used, and plowed on 8000 square meters of land, was strong, intelligent, and used to working with a spade. The land was very strong and produced excellent wheat: they were in this state average moisture and dryness that is best for plowing; but they were very sunken.
The laborer was paid per square meter, so that in a good day he could win 2 francs and 50 centimes. Here is what seemed to result from the experience, according to the average quantities rather difficult to assess.
The farmer dug his spade 25 centimeters and each sod it rose a moderate 6 kg weight of soil, he was the center of gravity, it is returning to a height that was very variable, but I ' I thought, taking an average measure, to evaluate 4 decimetres. The land, though very heavy, is ameublissait quite easily, and it was only after five or six spade he hit several times with his sharp to break up clods and unite the plow: he gave about Twenty spade strokes per minute. The first efffort to push the spade was an average of 20 kg: When the spade was down a few inches, the strength to continue to press it was little more than 12 kilograms.
In fine weather, this man was plowing a surface l8l square meters; and the mass of earth moved by plowing was 45.25 cubic meters. The cubic meter of soil weighed 1898 kg.
From these data it follows that since the land was high, to overthrow 4 decimetres, if one wants to have the first part of the equivalent amount of action in daily work, multiply together the numbers 1898 kg, weight one cubic meter, 45.25, number of cubic meters, and 4 decimetres height at which the center of gravity of each shovel of dirt is high the farmer; the product of these three quantities is equivalent to a weight of 34.3 kg higher one kilometer. But note that in addition to the weight of the earth, man, every spade, raises the weight of the tail, which can evaluate to 1.7 kg, roughly a quarter of the weight of the land that the spade return; and can be approximated by the amount of action rcprésenter consommnée to raise the land, 45 kg high one kilometer.
We must now look for the amount of work required to drive the spade, with each stroke, at a depth of 25 centimeters. The experience has given us a continued resistance of 12 kilogrannnes, which can be 15, because of the first effort that is at least 20 kilograms; and calculating, based on the weight of the land, the amount of spade blows in the day, at 6 kilograms per spade, we find that the farmer gives in the day 14316 shots tail. Therefore, for this second part of the action, multiply the three numbers together, 15 kg, pressure man exerts on the spade, 14316, number of strokes of spade, and 25 centimnètres, pressing the spade to each shot; the product of these three quantities equivalent to a weight of 55.6 kg higher one kilometer.
Add together the two amounts of action; we have for the total action of the day, 96.6 kg higher one kilometer.
It would be difficult to determine the amount of action that man uses to break up clods and spread the earth. According to the way our laborer was this, I do not think you can evaluate it much beyond the twentieth part of the daily work. Thus we may not be far from the true value of daily work, by estimating 100 kg a kilometer high.
In the work of the farmer, the two ways must be observed to use force. In the first, man, pressing the foot and the body on the spade sinks into the earth; it does not seem that this portion of the work can occur in the daily work much more tired than when a man walking up stairs.
In the other part of the work, the men raised by the effort of their arms, the land along the spade; and they probably tired as much as when they act on the doorbell. We'll see if, based on the calculations, we can admettte (admit?) These assumptions.
In the daily work of men climbing stairs, they can raise 205 kg one kilometer; but the portion of the daily work that responds to the depression of the tail, was found to be 53.6 kg higher one kilometer. Thus, assuming that these two types of work are of the same nature, the portion of the working day provided that the plowman shall sticking his spade, will be equal to 53.6 / 205 = 0.261 part of daily work.
 We must now add to this initial amount of action that the man who lifts the earth, assuming that at equal strain consumes the same amount of action at the bell: we found the three experiments different values; namely, for men who beat 75 kg high piles one kilometer; for the man who draws water from a well, 72 kg high one kilometer; for work followed for fifteen months at the Mint, 40 kg high one kilometer. Taking an average amount of these three values, we find, for the work of a day, 62.3 kg higher one kilometer. But we have seen in this article that the amount of action used to raise and turn the earth with the spade, was 45 kg higher one kilometer; and the portion of the daily work of the laborer would, for this part of the action, represented by 43 / 62.3 = 0.69 hundredths of daily work. Join these two portions of the work of the laborer, and we will, for his work for the day, 0.26 + 0.69 = 0.95, or, equivalently, 95/100 working day.
Thus, assuming that the man who sinks the spade no more tired than a man walking up stairs, and the man who reports the land with spade tired as a man used to the bell, we find, according to this comparison, one twentieth loss amount of action that can be overlooked in Nature Research those that are the subject of these experiences.
Before finishing this article; I must again warn that the result above is the measure of the work of an excellent farmer, used to working in the strongest land of the department of Eure-et-Loir. This is the art of plowing as all other arts which men consume all their daily forces; the ability always to employ all the action effectively. In plowing, for example, the distribution of human activities should vary according to the state, the nature of the land, and even the season when the work is done; but a good worker uses all parts of its work in a useful way, while a poor laborer, although very strong, drop each sod the upper part of his body more than is necessary to in darken the spade, and not being clever to return the land, he often rises more than necessary, and thus consumes wasteful part of its action; from which results that equal fatigue, giving a lower to the ground, he plows a lesser extent.
In this art, as in all others, when the observer wants to obtain the necessary evidence to establish the calculation of human action, follow a good worker paid by the piece; but at the same time, not in fl uence his momentary work, so do not let him know he is observed.

29. In all of the foregoing, I have sought to determine, from experience, what is the amount of daily actions that men can provide under any load, and I assumed that by this natural instinct to all men, they take under a given load speed which saves more forces. The remarks that follow will show that this assumption could not cause large errors in the results. It seems the same, according to the practice, than men in their work can, with equal fatigue, produce the same amount of daily action, we vary considerably speed, and cutting off their work by small rest intervals.
I will, for example, men who, according to Article 8 consumed all their daily work to climb the wood to 12 feet high. In this experiment. Each load 68 kg had risen to 12 meters high in just over a minute, just took 1.1 minutes. Thus, as in his daily working man rose 66 loads, consuming most of its daily action in 1:12 minutes. But the distribution of its action was cut by intervals of rest, or at least a little tiring work; such, for example, than to load its hooks log log, ot these intervals were much longer than those which he had borne on the back, because he climbed the six-lane wood in about half past six: ensure that the time to present the work being half past six, the effective time of fatigue was only 1:12 minutes and six hours and detnie were cut into 66 parts; each part into two others, one of 1.1 minutes, when man was under load, and the other 4.8 minutes, where the man came down the stairs, loaded its hooks and little tired .dropoff window
It seems that this way of cutting into small intervals of action and rest the work of men who carry great burdens, is the one that is best for the animal economy, and that the men walking with préfèreut speed for a few moments, and rest completely for a few more moments, a drive the same race in a time equal to two intervals with a slower speed, but continues.
This is what we see every day: for the men who carry loads of 60 to 70 kg on level ground, walk almost as fast as those who are not charged; but, as long as the race is long, they cut several rest intervals.
Moreover, regardless of how to divide the intervals between them, which probably varies for every man, according to his physique, it seems, as I have already said, that in the works for men to consume all their daily action, we must require of them, within twenty-four hours seven to eight hours of work, cut or not by small rest intervals. I talk about work when men consume in a violent exercise all their daily work; because there are many kinds of work, especially in the part of the arts, of such a nature that men, working ten to twelve hours a day, often consume only a little part of the considerable amount of action they can provide in the day.

30. I was mainly engaged in this thesis in determining how much greater or lesser a load decreases the amount of work that a man is able to do in a day. The experiences which have formed the basis for this determination were taken from the most natural movements common to all men, such as walking horizontally or climbing stairs. It seemed to me to give the evident result, that a man climbing stairs freely and without a load does almost double the work than if the same man is responsible for a load of 68 kg, which is about the average burden of men who carry wood upstairs in houses. But, as the only useful work done is that of carrying the load, the effective work of the man climbing stairs with loads is only one quarter of the total amount of work done during the day by a man who climbs stairs unloaded. Therefore if a man climbed a staircase unloaded and if by dropping back down by some means could raise a weight equal to his own weight, this would produce about as much work as four men carrying the same weight on their backs. This observation seems of utmost importance to engineers engaged in building human-powered machines to be always used at maximum efficiency.
I then sought to compare the total amount of work that men can provide when mounting stairs unloaded, with that which they produce by working a ram or crank handle, etc., and I found that a man who climbs up stairs freely can produce at least twice as much more work than by other means. The experiences that formed the basis for the assessment of the amount of work of the ram and the crank were always done in large workshops.  I ask those who wish to repeat the measurements, that if they do not have time to measure the results during several days of continuous work, to observe the workers at different times during the day, but without letting them know that they are being observed. One cannot be too aware of the risk of a wrong calculation of either the rate or the effective time of work, after a few minutes of observation.
The results of all of the foregoing give quantities considerably smaller than those used by most authors for calculating human-powered machines; but these are almost all based on experiments lasting only a few minutes and performed by selected men. They then, from these experiments, extrapolate the calculations, assuming seven to eight hours of actual work. But a man can, in almost all kinds of work, provide for a few minutes twice or even three times his average power and consume all his daily work in two or three hours. This is what we saw in the preceding article, where men who deliver wood consume all their daily work during the time when they are loaded, and this time is no more than one and a half hours per day.
The selection of the men also greatly affects the assessment of their average strength. I observed during ten years the transport of soil by troops, and paid, as it was said, by volume. I did the measurement every fortnight and almost always found that the working groups of grenadiers earned a third more than those of other companies, and often double that of the weaker groups. If I had determined the average strength of all individuals who formed the grenadiers' group, I would have found a third more than the average strength of other groups. It is true, and a necessary remark, that in this kind of work which mainly consists of transporting soil, not a single weak man could be found in the group of grenadiers, and that only two or three bad workers in each of the other groups completely slowed down the work.
Finally, the average power varies with the provided food, but even more with the climate. With the troops, I carried out major works in Martinique; the thermometer rarely below 20 degrees.  I did the same kind of work with the troops in France, and I can assure that below 14 degrees of latitude, where men are almost always flooded with their sweat, they are not capable of half the daily work which they can provide in our climates.

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