Front. Mech. Eng. 2016, 11(1): 26 – 32 32 DOI 10.1007/s11465-016-0368-z
RESEARCH ARTICLE
Hidetaka KUROKI
How did Arc rchi him medes dis isc cover th the e la law w of buoyanc ncy y by experiment?
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2016
Archimedes and Vitruvius Vitruvius era, for for more Abstract After Archimedes than 2000 years, it has been believed that the displaced water measurement of golden crown is impossible, and at his Eureka mome moment, nt, Archimedes Archimedes discovered discovered the law of buoyancy (Proposition 7 of his principles) and proved the theft the ft of a gol goldsm dsmith ith by we weigh ighing ing the gol golden den crown crown in water. A pr prev evio ious us st stud udy y sh show owed ed th that at a sm smal alll am amou ount nt of displa dis placed ced wa water ter was abl ablee to be mea measur sured ed wi with th eno enough ugh accuracy by the introduced method. Archimedes measured the weight of displaced water. He did not �nd the law of buoyancy but rather speci�c gr grav avit ity y of th thin ings gs at th thee moment. After Af ter whi which, ch, Ar Archi chimed medes es con contin tinued ued to mea measur suree the speci�c gravity of various solids and � uids. Through these measur mea sureme ements nts,, he rea reache ched d the dis discov covery ery of the law of buoyancy directly by experiment. In this paper, the process to the discovery of Archimedes ’ principle (Proposition 5) is presented. princip nciple, le, buo buoyan yancy, cy, spe speci ci�c Keywords Archimedes’ pri grav gr avit ity, y, Eu Eure reka ka,, Vit itru ruvi vius us,, di disp spla lace ced d wate water, r, ba bala lanc nce, e, �oating body
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Intro In trodu ducti ction on
Archimedes ran through a street in Syracuse naked with much joy shouting “Eureka! (I found it) ” repeatedly. This story was reported by Vitruvius, an architect during the 1st century BC, in his book Ten Ten Books on Architecture [ [1 1,2]. Archimedes has been widely believed to be the one that discovered the law of buoyancy at that time. However, Vitruvius did not mention about the law of
Received July 19, 2015; accepted October 20, 2015 Hidetaka KUROKI ( ✉) Okazaki, Aichi-ken 444-0076, Japan E-mail:
[email protected]
buoyancy though he knew Archimedes’ great achie achievevements well, suggesting that Archimedes found the density of things at his Eureka moment. More than 1500 years later, Galileo Galilei Galilei wrote his �rst short treatise entitled “La Bilancetta” (The Little Balance) in 1586 [3 [3] and stated his disbelief in the story of Vitruvius. Vitruvius. Galil Ga lileo eo tho though ughtt tha thatt the me measu asurem rement ent of dis displa placed ced (or over �owed owed)) wate waterr volu volume me by a golde golden n crow crown n (may (maybe be the shape was a wreath) was impossible. Finally, Finally, he concluded thatt Arc tha Archim himede edess had found the law of buo buoyan yancy cy and measured the weight of those things in water to prove the theft of a goldsmith at the moment. Even in modern times, it is believed that the measurement of displaced water volume is impossible [4 [ 4]. Thee ra Th rati tion onal alee is as fo foll llow ows. s. Th Thee go gold lden en cr crow own n is supp su ppos osed ed to be 10 1000 00 g, an and d 30% of th thee go gold ld of th thee crown was stolen and replaced with silver. The volume difference between the golden crown and the same weight of a pure gold lump will be only 13 cm 3 if the opening diameter of the vessel is 20 cm; the difference of 13 cm 3 is only onl y 0.4 mm in wat water er hei height ght.. Suc Such h a sma small ll dif differ ferenc encee cannot be measured. Furthermore, The Furthermore, The New Encyclopaedia Britannica says “The story that he determined the proportion of gold and silver in a wreath made for Hieron by weighing it in water is probably true” [ [5 5]. Numerous books or writings about the Eureka story have been made through more than 2000 years worldwide. As a result, these conclusions were made, and there seem to be no comments about the process to the great discovery discovery.. A paper titled “What did Archimedes �nd at “Eureka” moment?” was was pr pres esen ente ted d at th thee Ar Arch chim imed edes es 20 2010 10 conferenc conf erencee in Syrac Syracuse use [6]. In the pap paper er,, the followi following ng items are pointed: 1) Th Thee di disp spla lace ced d wa wate terr vo volu lume me me meas asur urem emen entt by a golden gol den cro crown wn is pos possib sible le wi with th eno enough ugh acc accura uracy cy by the introduced method through the use of items that Archimedes could prepare. 2) Ar Archi chime medes des co could uld no nott exp expre ress ss di displ splace aced d wa wate ter r volume numerically. There were no glass-made accurate
Hidetaka KUROKI. The process from the Eureka moment to the discovery of Archimedes’ principle
27
graduated measuring-cylinders in his era. Then, he 3 Method of displaced water volume measured the displaced water weights by a balance. (weight) measurement 3) He proved the theft of the goldsmith through this measurement. Then Vitruvius’ story should be recognized 3.1 Vessel for measurement of displaced water as the fact (the method of the story was not correct). The conclusion of the paper is that what Archimedes A vessel having a beak at one end is used, with an opening found at ‘Eureka’ moment was the speci�c gravity of diameter of 21 cm (Fig. 1). The diameter is large enough to things (SG). put the golden crown in. A thin triangle-shaped tongue is After this moment, he continued measuring the SG of attached under the beak (Fig. 2). A Roberval balance various solids and liquids. Through these measurements, (Murakami Kouki Co. ltd type: MS-1, Max. 1kg,) is used he reached the law of buoyancy well known as the to weigh the displaced water (Archimedes would use an Archimedes ’ principle. old shaped balance). In this paper, it is shown how he discovered the principle by experiment.
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Measurement of speci�c gravity
After the Eureka moment, Archimedes continued to measure the SG of various kinds of things. In this paper, the following objects are measured: 1) Solids heavier than water: A gold lump, a silver lump and a golden crown mixed with 25 % silver (GC25S) and a natural stone. The gold lump, silver lump and GC25S were replaced with the combination of glass balls having the similar volume. 2) Solids lighter than water: A beeswax lump which was widely used from ancient era for lighting or bonding things (e.g., as used by Icarus in Greek mythology to bond his wings), a wood block coated with beeswax to avoid liquid absorption, and a glass cup (many kinds of glass ware were used in the Archimedes era). 3) Liquids: Sea water drawn from the Japan Sea, 10 % salt water where 10 wt. % of salt is dissolved, and Italian olive oil (916 g/l, as derived from the bottle label) These solids and liquids could also be prepared by Archimedes with no dif �culties.
Fig. 2
Fig. 1
3.2
Vessel having a beak
Measuring method using over �ow � nish point (OFP)
A measuring method using OFP is as follows: 1) A cup is put under the tongue to receive water. 2) Water is added gradually to the vessel until over �ow begins (Fig. 2, left). 3) At �rst, water �ows out rapidly. The �ow decreases slowly, then turns to drops.
Water over �ow and stop at OFP
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Front. Mech. Eng. 2016, 11(1): 26 – 32
4) The water drop intervals become longer. Then, the �ow stops (Fig. 2, right). This is the basic measuring point (OFP). This point may be considered as the ‘very brim’ point that Vitruvius wrote in his book. 5) Next, the cup is changed to a measuring cup to receive the displaced water. 6) An object is put into the vessel. It should not be placed so roughly, but also not so quietly. 7) At �rst, water �ows out rapidly. The �ow decreases slowly and changes to water drops, then stops at OFP. 8) The water cup is taken off and its weight is measured by a balance. By using a counter weight equal to the empty cup, only the displaced water is easily weighed as shown in Fig. 3. 9) The measured object is taken out from the vessel and wiped off to dry for the next measurement.
4 Speci�c gravity measurement of various things
4.2.1
Floating objects are sunk forcibly by using a 3-needledevice and the displaced water is measured. The results are shown in Table 2. 4.2.2
4.1
4.3
Liquids
The displaced liquid weight by each object is divided by the displaced water weight that is measured in former measurements. Here, these solid objects are used only for volumetric bodies. Sea water
The SG of sea water is shown in Table 4, which is determined as 1.02.
Solid objects heavier than water 4.3.2
The weight of a measured object is divided by the displaced water weight. The SG values in Table 1 is clari�ed by Archimedes by this experiment for the �rst time in the world.
10% salt water
The SG of 10% salt water is determined as 1.08 (it is found as lighter than 1.10) as shown in Table 5. 4.3.3
4.2
Measurement at naturally �oating condition
The �oating objects are measured at a naturally �oating condition (Table 3). The weights of the �oating objects are found just the same as displaced water weights.
4.3.1
The results in each table are the average of 10 measurements.
Measurement at forcibly sunk condition
Olive oil
Solid objects lighter than water
The liquid characteristics of olive oil are much different from water, such as viscosity, surface tension, SG, etc.
Three �oating objects are tested (Fig. 4).
Fig. 3
Water weight measurement
Hidetaka KUROKI. The process from the Eureka moment to the discovery of Archimedes’ principle
Table 1 Speci�c gravity measurement of the chosen heavier solid objects Object
Weight/g
Displaced water/g Speci�c gravity
Gold lump
1000.0
51.4
19.5
Silver lump
1000.0
95.3
10.5
GC25S
1000.0
65.3
15.3
610.1
227.2
2.7
Natural stone
Therefore, its measuring condition is also different from water. The tongue attached under the beak of the vessel needs to be a large triangle shape (its width at the end of the beak is 20 mm and its length is 40 mm) and a time of more than 20 min is needed to �nish dropping. Figure 5 shows the dropped olive oil weight curves after the changing point of continuous � ow to drops. The results show the different curves. Figure 6 shows the curves on 2 to 20 min (cleared to 0 g at 2 min), which show very similar curves. At 20 min, their difference is only 1.3 g. Then the 2 min after changing point is chosen as the end point for the olive oil �ow.
Fig. 4
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In the Archimedes era, a clepsydra is used for time keeping. It is a vessel having a hole at the bottom. And it sinks in water for an arranged time length. An example of clepsydra is shown in Fig. 7. This sinks around 2 min after immersed in water. Archimedes might use a clepsydra like this (in this paper, a countdown timer is used). The results of Table 6 are the average of 5 measurement data. The SG of olive oil is determined as 0.91. As the beeswax is 0.95 heavier than olive oil, it sinks. From these results, the SG of various solids and liquids are clearly presented. However, Archimedes did not mention anything regarding this in his writings. The physical properties, such as SG, may not be worthwhile for him. 4.3.4
Notice of principle of � oatation by Archimedes
In item 4.2.2, the weights of �oating objects are the same as the weights of displaced water. “Maybe, this is the principle of �oatation” Archimedes noticed. “Any solid lighter than water will, if placed in water, be so far immersed that the weight of the solid will be equal to the
Floating objects: Glass cup, beeswax, and wood block
Table 2 Speci�c gravity measurement at forcibly sunk condition
Table 3
Object
Object
Weight/g
Displaced water/g
Speci�c gravity
Beeswax
261.9
275.5
0.95
Wood block
132.9
303.7
0.44
Glass cup
210.0
–
–
Displaced water measurement of � oating objects Weight/g
Displaced water/g
Difference/g
Beeswax
261.9
261.8
– 0.1
Wood block
132.9
133.1
+ 0.8
Glass cup
210.0
210.0
0.0
Table 4 Speci�c gravity measurement of sea water Object
Displaced sea water/g Displaced water/g Speci�c gravity
Gold lump
52.5
51.4
1.02
Silver lump
97.5
95.3
1.02
GC25S
65.9
65.3
1.01
Beeswax*
282.1
261.8
1.02
Wood block*
309.3
303.7
1.02
*: Beeswax and wood block are forcibly sunk by 3-needle-device.
Table 5 Speci�c gravity measurement of 10% salt water Object
Displaced 10% salt water/g
Displaced water/g
Speci�c gravity
Gold lump
55.3
51.4
1.08
Silver lump
101.6
95.3
1.07
Natural stone
244.5
227.2
1.08
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Front. Mech. Eng. 2016, 11(1): 26 – 32
Fig. 5
Olive oil weight curve after � ow changing
Fig. 6
Olive oil weight curve 2 to 20 min Table 6 Speci�c gravity measurement of olive oil Object
Displaced olive oil/g Displaced water/g Speci�c gravity
Gold lump
45.7
51.4
0.89
Silver lump
86.4
95.3
0.91
Natural stone
207.0
227.2
0.91
Beeswax*
251.6
261.8
0.91
Wood block**
276.3
303.7
0.91
*: Beeswax sinks in olive oil; **: Wood Block is forcibly sunk by 3-needledevice.
4.4.1
Sea water
weight of water displaced”. However, only water is measured so far. Other liquids are tested for generalization.
The displaced liquid weight is measured in sea water being SG = 1.02. As shown in Table 7, difference is only – 0.5 to 0.4 g. Displaced sea water weight is also equal to object weight.
4.4
4.4.2
Fig. 7
Clepsydra, as 2 min timer
Floating object measurement in various liquids
The displaced liquid weights by �oating objects are measured in various liquids having different SG.
10% salt water
Displaced liquid weight is measured in 10 % salt water being SG = 1.08. As shown in Table 8, difference is at most
Hidetaka KUROKI. The process from the Eureka moment to the discovery of Archimedes’ principle
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Displaced liquid measurement by � oating object in sea water
Table 7 Object
Weight/g
Displaced sea water/g
Difference/g
Beeswax
261.9
262.3
0.4
Wood block
132.9
132.9
0.0
Glass cup
210.0
209.5
– 0.5
Displaced liquid measurement by � oating object of 10% salt
Table 8 water Object
Weight/g
Displaced 10% salt water/g
Difference/g
Beeswax
261.9
263.1
1.2
Wood block
132.9
134.4
1.5
Glass cup
210.0
211.2
1.2
1.5 g. Displaced 10 % salt water weight is also equal to object weight. 4.4.3
Olive oil
Fig. 8
Volume difference of displaced liquid by glass cup
The displaced liquid weight is measured in olive oil being SG = 0.91. As shown in Table 9, difference is only up to – 1.1 g. Displaced olive oil weight is also equal to object weight. Displaced liquid measurement by � oating object olive oil
Table 9 Object
Weight/g
Displaced olive oil/g
Beeswax
261.9
(Sink)
—
Wood block
132.9
131.8
– 1.1
Glass cup
210.0
209.8
– 0.2
4.4.4
Difference/g
Volume of displaced liquid
The displaced liquid weighted by �oating objects are recognized to have the same weights in all liquids. However, their volumes are signi�cantly different. An example is shown in Fig. 8. Reading volume by the scale of a 250 cm3 mess-cylinder is not accurate. Volume measurement is not easy even in nowadays. 4.5
Fig. 9 Displaced liquid (olive oil) is balancing with the � oating object (glass cup) �uid
displaced” [7]. The results obtained through above experiments in this paper are showing just the same as these words.
Discovery of the law of buoyancy 4.6
It can be said that the weight of a �oating object is the same as the displaced liquid weight in all liquids having different SGs. By using a balance, when the displaced liquid cup is placed on one dish and a �oating object is placed on another dish, they will be in balance as seen in Fig. 9. After these experiments, Archimedes reached the discovery of the law of buoyancy. His book On Floating Bodies Book 1 Proposition 5 says, “Any solid lighter than a �uid will, if placed in the �uid, be so far immersed that the weight of the solid will be equal to the weight of the
Discovery of Proposition 7
After discovery of Proposition 5, Archimedes continued measuring weights of heavier solids by placing them in liquid and hanging them on a thin string. He con �rmed that the decreased weight from its weight in the air is the same as displaced liquids. On Floating Bodies Book 1 Proposition 7 says, “ A solid heavier than a �uid will, if placed in it, descend to the bottom of the � uid, and the solid will, when weighed in the �uid, be lighter than its true weight by the weight of the �uid displaced”.
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Front. Mech. Eng. 2016, 11(1): 26 – 32
5 Flow from Eureka moment to the discovery Even nowadays, the process by which Archimedes discovered the law of buoyancy has not been made clear. In the previous and current paper [ 6], the process from the Eureka moment to the discovery is shown. The �ow is summarized as follows: 1) In a bath, Archimedes suddenly got the solution of King Hieron’s problem and ran naked on a Syracuse street, shouting “Eureka”. 2) He measured a small amount of the displaced water of the crown by using a certain method. 3) There were no glass-made measuring cylinders in his era. Thus, he measured the weight of the displaced water. He determined the purity of the golden crown and proved the theft of the goldsmith. 4) At the Eureka moment, Archimedes found the SG of things. 5) After this moment, he continued to determine the SG of many kinds of solids and liquids. 6) During these measurements, he noticed that the displaced water weights are the same as �oating bodies. 7) By conducting experiments on other liquids having different SGs, he reached the discovery of the law of buoyancy (Proposition 5).
The process to the discovery has not been explained until now. However, in this paper, the process from the Eureka moment to the discovery of Proposition 5 is clari �ed in this paper. Afterwards, Archimedes measured the weight of heavier solids in liquids and found Proposition 7. This is a very accurate method even in nowadays. It is used for investigating the purity of noble metal accessories. Archimedes ’ interest was mainly in the mathematical principle. He also made many great developments on the �eld of mechanics. His achievements are recognized as the important foundations for modern mechanics. This paper clearly showed the process on how Archimedes discovered the law of buoyancy, one of his great achievements, through an experiment.
References 1. Pollio M V. Ten Books on Architecture (trans. Morgan M H). Cambridge: Harvard University Press, 2010 (Original work published 1st century BC) 2. Pollio M V. Kenchikusho (in Japanese, trans. Morita Keiichi). Tokyo: Toukai University Press, 1979 (Original work published 1st century BC) 3. Galileo Galilei. The Little Balance (in Japanese, trans. Toshiyuki Fujita). Skaino-Meicho, 1975, 21: 36 – 41 (Original work published
6
Conclusions
1586) 4. Archimedes Homepage by Prof. Chris Rorres of Drexel University. http://www.math.nyu.edu/~crorres/Archimedes
After the Archimedes and Vitruvius era, for more than 2000 years, it has been thought that the displaced water is so little that the measurement of a golden crown is impossible. Thus, at the Eureka moment, Archimedes found the law of buoyancy and proved the theft by weighing it in water. This relates to Proposition 7 of Archimedes ’ the principle. Proposition 5 is the most famous but seems to be considered as a theoretical clause.
5. Encyclopaedia Britannica Inc. The New Encyclopaedia Britannica. 15th ed. Encyclopaedia Britannica, 2007 6. Kuroki H. What did Archimedes
�nd
at “Eureka” moment? In:
Paipetis S A, Ceccarelli M, eds. The Genius of Archimedes — 23 Centuries of In�uence on Mathematics, Science and Engineering. Amsterdam: Springer, 2010, 265 – 276 7. Heath T L. The Works of Archimedes. Cambridge: Cambridge University Press, 1897, 253 – 260
