An Enquiry into the Measure of the Roman Foot; By Matthew Raper, Esq; F. R. S.

LXIX. An Enquiry into the Measure of the Roman Foot; by Matthew Raper, Esq.; F. R. S. Read May 15, 22, and June 5, 1760. THE methods that have hitherto been taken to discover the measure of the Roman foot, will, upon examination, be found so unsatisfactory, that it is no wonder the learned are not yet agreed in that point. The antient foot-rules now remaining; the representations of the foot in sculpture; and the measure of it, derived from the congius, differ so much among themselves, and from each other, as to be insufficient evidences separately: and the great disagreement of the foot from the congius, with the rest, has not hitherto been satisfactorily accounted for. The foot-rules found in old ruins at Rome, are of various lengths; and the age of none of them being certainly known, no precise measure can be determined from them, otherwise than by taking a mean from such as appear to be most perfect. But though this may have been the foot in use at some time or other, yet as these rules are probably of different ages, both the greatest and the least of them may have answered to the standards of their times. For though we have no account of any alteration ever made in the standard of the Roman foot, yet the wear of a standard measure by use, and the making new to replace the old ones, must, in all probability, create a difference; especially, as the Romans had not those inducements to so precise an accuracy accuracy in these matters, as the later discoveries in natural philosophy (particularly the invention of the pendulum) have introduced among the moderns. Add to this, that the different state of Europe, which has, for some ages, been divided into many considerable kingdoms and sovereign states, independent on each other, equally civilized, and carrying on a more constant and regular mercantile commerce with each other, than was known to the Romans, must necessarily introduce more frequent enquiries into the weights and measures of the different states, and a more careful examination of the respective standards of each, than the Romans could have any occasion for: and use in these matters, is the parent of accuracy. We can arrive at no greater certainty from the marbles, than from the foot-rules. These, indeed, do not differ so widely from each other, as the rules; which seems to be the reason why most of the writers on this subject have given them the preference: but of the four that are extant, no two agree in the same measure; nor is the age of any one of them known: and as they were intended for representations only, and not for use, their accuracy may reasonably be doubted. Festus, Frontinus, and Rhemnius Fannius, say, the side of the quadrantal (which contained 8 congi) was a Roman foot (1). A standard congius of Vespasian (1) Quadrantal vocabant antiqui quam ex Graeco ἀμφορί dicunt, quod vas pedis quadrati octo et quadraginta capit textarios. Festus in v. Quadrantal. Pes quadratus concavus capit amphora trimodia Vespasian is still in being, and has been measured by several learned men; but the foot derived from it exceeds those on the marbles, and the greatest number of the foot-rules, so much, that Mr. Greaves could find no better way of accounting for so great a difference, than by supposing what Festus and Fannius say (for he does not quote Frontinus), to be a vulgar error (2): whereas the name of this standard shews its figure to have been a cube (3); which adds probability to their testimony, that its side was intended for the measure of the foot. The measures of public roads in the itineraries, can be of little use in this enquiry; for they omit fractions, and we do not know whether the distances of the towns are reckoned from the market-places or from the gates; but a difference of half a mile in sixty, is equivalent to the tenth part of an inch in the foot: therefore, no exact measure is to be expected from thence, even though the modern mensurations of Cassini, Riccioli, and others, were more unexceptionable than they really are. trimodia [read, amphoram trimodiam]. Frontinus, in Expositio Formarum. Though the passage is corrupt, there can be no doubt of the meaning. Pes longo spatio latoque notetur in anglo, Angulus ut par sit quem claudit linea triplex; Quattuor ex quadratis medium cingatur inane, Amphora fit cubus: quem ne violare liceret, Sacraeere Jovi, Tarpeio in monte, Quirites. Rh. Fannius, de Pond. et Menf. (2) See Dr. Birch's edition of Greaves's Works, p. 228. (3) Quae illi κύβος, nos quadrantalia dicimus. A. Gellius, l. i. The distances between the antient mile-stones are not liable to these objections; and if a sufficient number of such as stand nearest to Rome were carefully measured, their authority would be considerable. But I do not find that any are now standing within thirty miles of that city, nor that any of these have been measured, or even any in Italy; and provincial measures are not of equal authority. There is still another method, whereby we may discover the measure of the Roman foot; which is, from the remains of the antient buildings now standing at Rome: and though many have made use of some single parts of them for this purpose, yet no one hath hitherto compared the measures of the principal parts of any one of them with each other, which is the only way to discover the measure whereby a building was constructed. With this view, therefore, I shall carefully examine the measures of the buildings contained in that inestimable treasury of antient Roman architecture, intituled, Les Edifices antiques de Rome, and published at Paris, by Monf. Desgodetz, in the year 1682. In order to this, it will first be necessary to ascertain the proportion of the Paris foot (the measure used by this author) to some known English standard. The Paris foot is one sixth part of the toise in the Chatelet; which was renewed in the year 1668 (4), and the new standard has continued in use ever since. (4) See De la Hire's Dissertation on the Roman Foot, in the Memoirs of the Academy of Sciences for the year 1714; where he gives an account of this renewal of the standard in the Chatelet. In the year 1742 (5), the Royal Academy of Sciences at Paris, at the request of the Royal Society of London, sent over a measure of half the toise of the Chatelet; from which Mr. Graham determined the proportion of the Paris foot to that of London, to be as $1065.41\frac{2}{3}$ to 1000. Mons. Le Monnier, of the Royal Academy of Sciences, from the same originals, found their proportion as 864 to 811, or as $1065.351$ to 1000. The difference is inconsiderable, and we may, without injustice to Mr. Graham's known skill and accuracy in these matters, suppose their true proportion to be as $1065.4$ to 1000. Mr. Graham's measure of the London yard, together with that of half the toise of the Chatelet, are deposited in the archives of the Royal Society at London, and of the Royal Academy of Sciences at Paris; and whenever I shall mention the London foot, without specifying any particular standard of it, I would be understood to mean this measure. In this enquiry, we are to seek a common measure to the several parts of each building, that shall not differ very widely from some assumed magnitude of the Roman foot: and though we might take this assumption from any of the antient foot-rules now remaining, yet the nearer it is taken to the truth, the better guide it will be to us, and the more it will facilitate our enquiry. Now, as a mean measure, derived from these rules, will, probably, be nearer the truth, than either the greatest or the least of them, so one that shall include such other remains of antiquity, as have hitherto been made of to discover (5) See Philos. Trans. No 465. the measure of the Roman foot, will be still more unexceptionable, as the writers on this subject are not agreed, which of the different authorities is to be preferred. The representations of this foot in sculpture, are four in number; one on the sepulchral monument of Cossutius, formerly in the Colotian gardens at Rome; another on that of Statilius, in the Belvedere; a third on that of M. Ebutius, in the Villa Mattei (6); and the fourth on a marble, without inscription, dug up of late years in the Via Aurelia, which, being in the possession of the Marquis Capponi, is called, by P. Revillas, the Capponian foot. Most of the early writers on this subject have expressed their measure of the Roman foot by a diagram; and Snellius (7) observing, that the paper contracted in drying, after the impression was taken off, endeavoured to make a proper allowance for it. But Greaves, finding the measures of these figures to differ in different copies of the same impression, took another method; and seems to have been the first that compared the original figures on the monuments of Cossutius and Statilius with a modern standard. This he did with such care and diligence, that his measures deserve a particular examination. The London foot, which he used upon this occasion, was taken from the iron standard of 3 feet in (6) Picard and Auzout (in a paper called De Mensuris, hereafter quoted) have given the measure of the foot on this monument of Ebutius for the Colotian foot. (7) See his Dissertation on the Roman Foot, in the 3d vol. of Saggi di Difertazioni Academiche di Cortona. the Guild-hall, London (8); which having been long since lost, or destroyed, we have nothing left whereby to discover its true magnitude, but the measures others have taken of it, and those which have since been taken of such magnitudes as Greaves had compared with his copy of it. Snellius, from a measure sent him of this iron standard (9), determined the proportion of the Rhynland to the London foot, as 1000 to 968. The Rhynland foot, according to Picard (1), contains 696 such parts as the Paris foot contains 720: whence the proportion of the latter to this measure from the iron standard, is as 1065.4 to 997 nearly. Eifenschmid found the Rhynland foot to contain 1391.3 such parts as the Paris foot contains 1440 (2); which gives 1065.4 to less than 996½, for the proportion of the Paris foot to that of the iron standard. Huyghens makes the Paris to the Rhynland foot as 144 to 139 (3); whence the proportion of the former to Snellius's London foot, will be nearly as 1065.4 to 995½. But there is reason to believe, that Huyghens's measure of the Rhynland foot was too small (4). By (8) Greaves, p. 223. (9) Eratothenes Batavus, p. 125. 141. (1) See a paper intituled De Mensuris, in Divers Ouvrages de Mathematique et de Physique, par Messrs. de l'Academie Royale des Sciences, Paris 1693, in folio, and afterwards printed in the 4th vol. of Ouvrages adoptez. (2) Eifenschmid de Pond. et Menf. vett. p. 94. (3) Horolog. Oscill. part iv. prop. 25. (4) Picard, in his Voyage d'Uranibourg, p. 64. edit. Amst. says, "Passant par Holland je pris l'occasion de verifier la proportion By these comparisons it appears, that Snellius's measure of the London foot, from this iron standard, was at least 3 parts in 1000 shorter than Graham's London foot. Our countryman Norwood, in the year 1635, measured the distance between London and York, in order to determine the length of a degree on the meridian; which he found to contain 367196 London feet of this iron standard (5). The French found the measure of a degree, in the latitude $66^\circ 20'$, to be 57438 toises, and at the equator 56783. Hence the measure of a degree in $52^\circ 44'$ (the middle latitude between London and York) will be found to be 57276 toises, or 343656 Paris feet. These numbers give the proportion of the Paris foot to that of the iron standard, as 1065.4 to 997.1 —, wanting somewhat less than 3 parts in 1000 of Graham's London foot. Picard's paper De Mensuris, quoted above, and another on the same subject by Auzout, printed with it, contain some measures, which Greaves had before compared with his London foot. Both these papers were written after the renewal of the standard of the Chatelet, in 1668. The former is so full of inaccuracies and mistakes, that little use can be made of it; but Auzout's measures appear to be accurate; and as he seems to have taken his Paris foot from the portion du pied de Paris à celui du Rhin, dont l'original est à Leyde; laquelle proportion me parut être exactement comme 720 à 696, au lieu de 720 à 695, que j'avois supposée dans la mesure de la terre." This latter is Huyghens's proportion. (5) See Norwood's Seaman's Practice. toise in the Chatelet for this purpose, it was probably a correct measure of that standard. Such of his measures as answer to Greaves's, are here reduced to thousandth parts of the London foot, reckoning his Paris foot to contain 1065.4 such parts. | Measure | Greaves | Auzout | Diff. | Diff. per foot | |--------------------------|---------|----------|-------|----------------| | The Statilian foot | 972. | 969.96 | -2.04 | 2.1 | | The braccio of Florence | 1913. | 1908.84 | -4.16 | 2.18 | | The braccio of Siena | 1974. | 1973.21 | -0.79 | 0.4 | | Pætus's palm | 732. | 731.35 | -0.65 | 0.89 | All these differences fall the same way, and shew, that Greaves's London foot bore a less proportion to Auzout's Paris foot, than that of 1000 to 1065.4. The Statilian foot, and the braccio of Florence, give the greatest differences; one is carved in relievo, and the other is a rod; so that both are subject to diminution, by time and use. The palm is engraved; and Auzout, from the deviation of Pætus's Greek and Roman foot, from the common received proportion of 25 to 24, suspected, that the latter was lengthened by wear; if so, the like must have happened to the palm. How their measures of the braccio of Siena come to differ so little, I cannot account for. But we can draw no certain conclusion from this comparison, till we know whether any, and what, allowance is to be made for the diminution or increase of these magnitudes, in the interval between Greaves's and Auzout's measuring them. For this purpose, I shall compare Auzout's measures, with those taken by P. Revillas, with a cor- rect London foot (6), between fifty and sixty years after. These he has converted into Paris measure, by the proportion of 864 to 811; and I have reduced them back again, according to the same proportion. | | Auzout. | Revillas. | Diff. | Diff. per foot | |----------------|---------|-----------|-------|---------------| | The Statilian foot | 969.96 | 969.79 | -0.17 | 0.18 | | The Ebutian foot | 972.91 | 972.22 | -0.69 | 0.71 | | Pætus's palm | 731.35 | 732.65 | +1.3 | 1.78 | | Pætus's foot | 966.26 | 968.74 | +2.48 | 2.57 | | or 967 | | | | | Hence the carvings appear to have suffered very little diminution in this interval; but Pætus's measures, which are engraved, and in constant use as public standards, have been considerably lengthened. Therefore, allowing for both these circumstances, we may conclude, that Greaves's London foot wanted 2 parts in 1000, to be to Auzout's Paris foot, in the proportion of 1000 to 1065.4. Greaves's measure of the door-case of the Pantheon, 9 inches within the jambs, is 19.602 of his London feet and decimal parts (7). Defgodetz found the measure of the same on the inside 18 feet 4\(\frac{1}{8}\) inches Paris measure, and on the outside 18 feet 4\(\frac{1}{3}\) inches, their difference is \(\frac{1}{24}\) of a Paris inch; the depth of the jambs is 4 feet 10\(\frac{3}{4}\) inches Paris measure; and (6) Esattissimo piede Inglese. See Saggi di Dissertazioni Academiche di Cortona, vol. iii. p. 113. I have been credibly informed, that this London foot was given him by the late Martin Folkes, Esq; and that it was made by Sisson, according to Graham's measure. (7) Greaves, p. 348. 9 London inches are equal to 8,447 Paris inches and decimals. Hence the measure of this door-case 9 London inches within the jambs, should, according to Desgodetz, be 18,3676 Paris feet and decimals. But 19,602 is to this number, as 1065,4 to 998,3; and Greaves's London foot wanted $\frac{1}{10}$ to be to Desgodetz's Paris foot, in the proportion of 1000 to 1065,4. Greaves says, most of the white marble stones on the pavement of the Pantheon contained exactly 3 of those Roman feet on Cossutius's monument (8). Their Paris measure, according to Desgodetz, is 2 feet 8$\frac{1}{2}$ inches, which is equal to 2885,46 London parts. This number divided by 3, gives but 961,82 such parts; whereas Greaves's measure of the Cossutian foot was 967 parts of his London foot; which must therefore have wanted above 5 parts in 1000 of the proportion before-mentioned. But it is probable they measured different ranges of these stones. The measure of the Paris foot, which Greaves received from Mons. Hardy, was taken from the old standard in the Chatelet (9), and contained 1068 such parts as his London foot contained 1000. These numbers are in the proportion of 1065,4 to 997,57; therefore, if the new standard did not differ from this old one (and no such difference appears to have been intended), Greaves's London foot must have been 2,43 parts in 1000 shorter than Graham's. (8) Greaves, p. 211. (9) See Mons. Hardy's Letter, in Greaves's Works, p. 447. The measure he sent was only half the Paris foot; but he tells Greaves, it was taken from the standard, under his own inspection, with great care. All these comparisons shew Greaves's measure of the London foot to have been shorter than Graham's. But they are not all of equal authority. The measures of Snellius depend upon the correctness of three several feet; that is to say, his copy of the iron standard, the Rhynland, and the Paris foot; whereas the rest (except Norwood's) depend only on the Paris foot, applied to the same magnitudes that Greaves himself had measured. Norwood's measure is rather to be wondered at, that, notwithstanding all the difficulties he had to encounter, it should come so near to the true magnitude of a degree, than to be depended on for the exact measure of the iron standard; and is an instance of what the diligence and sagacity of a private and obscure man, unassisted by the public purse, or the contributions of friends, could accomplish; whose labours may, perhaps, be forgotten, when the measures of the French, taken under a royal commission, shall be known to the latest posterity. The three more immediate comparisons of Greaves's measure with the Paris foot, are by the measures of Auzout, Desgodetz, and Hardy, which afford as clear a proof as can well be expected in this matter, that his measure of the iron standard was about 2 parts in 1000 deficient of Graham's London foot. Mr. Greaves found the foot on the monument of Statilius to contain 972 such parts as his London foot contained 1000; and that on the monument of Cof-sutius 967 (1). These measures reduced to Graham's foot, are 970, and 965. (1) Greaves, p. 208, 209. Auzout's measure of the foot on the monument of M. Ebutius, is nearly equal to 973 London parts. Revillas makes this $972\frac{1}{4}$, and the Capponian foot $968\frac{3}{4}$. The mean of these measures of the Roman foot on the marbles, is nearly 969 thousandth parts of the London foot. Philander, in his notes on Vitruvius (2), says, the foot from a porphyry column, inscribed ΠΟΔ. Θ. exceeded the Cossutian foot by the ninth part of an inch; therefore it contained nearly 974 London parts. This column seems to have been a more authentic measure of the Roman foot, than the figures on the monuments; since the inscription shews it was intended for such, whereas they were only representations of it. And notwithstanding the inscription was Greek, the foot must have been Roman; for Philander's measure of it exceeds the mean measure of those on the marbles, by less than the sixteenth part of a London inch, and is less than some of the antient foot-rules, hereafter mentioned; whereas it wants above the third part of an inch to be equal to the Greek foot. It might, perhaps, be used by some Greek mason at Rome, to adjust his workmens rules by; for when the Romans began to be expensive in their buildings, they had most of their matons from Greece. And though this column is not now to be found, and the accuracy of Philander's measure of it may be doubted, it is still a proof, that, at one time or other, the measure of the Roman foot was sensibly greater than that on the monument of Cossutius. (2) P. 117. of the Lyons edit. 1552. Lucas Pætus compared five antient foot-rules with the marbles; he seems not to have been a person of great accuracy; for though in one place he allows there was some small difference between the Statilian foot and that on the monument of Cossutius (3), yet he every-where else speaks of them as equal, and has given but one figure for the measure of both. His comparisons seem to have been made only with the latter, the measure of which he had before set up in the Capitol, as the true Roman foot (4). Three of these rules, which, he says, were exactly equal to each other, were shorter than the Cossutian foot by one seventh part of a Cossutian inch; they must then have been nearly $953\frac{1}{2}$ London parts. Of the other two, one appears to have been $958$, and the other $974$, equal to that from Philander's column. He decides in favour of the three shortest, to which perhaps he was induced, partly by their agreement with each other, and partly by his opinion, that the Roman foot had been continually increasing; whence it must follow, that the shortest measure of it was the most antient. (3) Cum quo sane convenit modico discrimine hic Statilianus. Pætus de Mens. et Pond. lib. i. (4) Greaves, p. 209. says, the Roman foot engraved by Pætus in the Capitol, agreed exactly with that on the monument of Cossutius. Pætus therefore seems at that time to have acquiesced in the opinion of Portius, which, he says, was generally received. (Leonardus Portius Vicentinus, vir sane doctus, primus omnium nostræ ætatis qui de hac re scripserit, mensuram Colotiani pedis pro vera tradidit.—Quem juniores quoque fecuti sunt.) But he changed his opinion when he wrote his book, which was after he had set up the measures in the Capitol. Greaves says, the Cossutian foot exactly agreed with some very antient and perfect Roman feet in brass; and from his comparison of two that differed from it (which seem to be the greatest and least he had met with), one of them appears to contain nearly $972\frac{3}{4}$ London parts, and the other $962\frac{3}{5}$ (5). P. Revillas has given the measures of three foot-rules of an uncommon length; one of which contains $972$ London parts, another $976$, and the third $983$ (6). There are many more of these rules in being, that we have no account of; for most of the writers on this subject, preferring the authority of the marbles, have been negligent in their accounts of the foot-rules; therefore I think a mean measure from these above-mentioned of no great authority, especially as some of them have been produced only on account of their difference from the more common measure of the rest. If a cubic Roman foot contained a quadrantal of wine, the cube of half a foot must have contained a congius, which was the eighth part of the quadrantal. Pætus, Villalpandus, and Auzout measured the standard congius of Vespasian with water; and have (5) Greaves, p. 222. But, p. 227, he says, there is not one of ten measures of the Roman foot (besides those on the monuments), that arrive to the proportion of that deduced by Villalpandus from the congius, by $27$ parts in $2000$; whereas the greater of the two here mentioned contained $974\frac{3}{4}$ parts of his London foot, and he makes the foot from the congius of Villalpandus, but $986$ such parts; their difference is but $23$ two thousandth parts of the latter. (6) Saggi di Diss. Academ. di Cortona, vol. iii. given the weight of its contents. The two first differ from each other by almost 5½ modern Roman ounces, which is nearly the one-and-twentieth part of the whole weight, though they both measured up to the same part of the neck. The weight of water given by Pætus, is deficient of the mean weight of Auzout's measures by 3 Troy ounces; and that given by Vil-lalpandus exceeds the same by 2; therefore they must both have used bad ballances. Auzout measured this vessel twice, with the water of Trevi. His greater weight of the water is 63024 Paris grains, his lesser 62760 (7). Picard found a cubic Paris foot of spring water to weigh 641326 Paris grains (8); and that foot containing 2089.69 cubic London inches, Auzout's greater weight will give 205.36 such inches for the solid content of the congius, his lesser 204½. According to Picard's experiment, a cubic London inch of spring water should weigh but 251½ Troy grains; whereas, according to Snellius, it should weigh above 254 (9). Our countryman Wybard made many experiments to discover the weight of water (1), from which the weight of a cubic inch of rain water should be 253 Troy grains. His experiments seem to have been made with great attention, and his inch must have been that of the iron standard, which I have already (7) See the paper De Mensuris, above quoted, p. 366. 371. of the folio edit. where Picard gives these weights as Auzout's, without any farther account of them. (8) Ibidem. (9) Snellius, in Eratosthene Batavo, p. 155. (1) See Wybard's Tactometria, from p. 269 to p. 287. shown to be less than Graham's; therefore, I think a cubic inch of Graham's measure of the water of Trevi, cannot weigh less than 253 Troy grains. This will reduce Auzout's two measures of the congius to 204.35 and 203.49 cubic London inches; which give 981.7 and 980.3 for the measure of the Roman foot. Greaves measured the congius with millet seed (2), and comparing it with our measures of capacity, found it to contain 7\(\frac{1}{2}\) pints wine measure, and 6\(\frac{1}{6}\) pints corn measure. When he wrote, the wine gallon was universally allowed to contain 231 cubic inches (3); and the dimensions of the bushel were published yearly, by the Lord Mayor of London, to be 19 inches in diameter, and 7\(\frac{1}{2}\) in depth (4). But Greaves's inches were according to the iron standard in the Guildhall, and were 2 parts in 1000 shorter than Graham's. Therefore reducing these measures in that proportion, the wine gallon will be found to contain 229.62 cubic London inches, and the corn gallon 264.22. Hence by the wine measure, the congius contained 204.5 cubic London inches, and by the corn measure 203.7; which give 981.9 and 980.6 for the measure of the Roman foot. These measures agree so well with those of Auzout, that I think a mean between them must be very near the truth, and that we may reckon the (2) Greaves, note f. p. 303. (3) See Wybard's Tactometria, p. 264. Oughtred's Circles of Proportion, p. 57. who quotes Gunter and Briggs for the same opinion. (4) Wybard's Tactometria, p. 282. foot from the congius to contain about 981 London parts. Greaves took likewise another measure of this congius, by the cube of his Cossutian foot, which he found to contain 7 congi, and about an half, of millet (5). Now, the quadrantal should contain 8; hence the side of the quadrantal to this congius will contain 988 parts of the iron standard: for 988 is to 967 (Greaves's measure of the Cossutian foot) in the subtriplicate proportion of 8 to 7½. But this is a greater measure than even that of Villalpandus, and was probably owing to an erroneous cube of the Cossutian foot; for the joiners at Rome might be as bad workmen as the scalemakers; and Greaves says, he could not get a ballance there, fit to examine the congius by weight (6). Mons. Afruc, in his Natural History of Languedoc, says, he had caused the distance between the ninth and tenth mile-stones on the road from Nismes to Beaucaire to be measured, and that it was found to be 754 Paris toises. They were both set up by Tiberius, and the road lies in a strait line between them. This measure gives a Roman foot of 964 London parts. He does not say he took this measure himself (7), nor has he given any account how it was taken. Perhaps by a provincial measure, and (5) Greaves, p. 225. (6) Ibid. note f. (7) Memoires pour l'Histoire Naturelle de Languedoc, p. 225. His expression is, "J'ai fait mesurer la distance de l'un à l'autre, et elle s'est trouvée de 754 toises de Paris." Where he speaks of the Marquis Maffei, he says, "Il a mesuré;" but, in the same paragraph, speaking of himself, he again says, "J'ai fait mesurer." afterwards reduced to Paris toises; or by a wheel, which is liable to great errors: therefore we cannot allow much authority to it, unless we were better assured of its accuracy. He quotes the Marquis Maffei for a measure of 756 toises between two other mile-stones on the same road; but observes, that they were set up by different emperors, Augustus and Tiberius, and that the road formerly made an elbow between them, whose position is now not certainly known. Therefore this measure, which gives a foot of 966\(\frac{1}{2}\) London parts, is likewise of small authority. All that can be determined from such uncertain and discordant data, as I have here collected, is a measure that shall probably be neither the greatest nor the least magnitude of the Roman foot. And for this, I shall take a mean from all the measures above recited, which is nearly 968 thousandth parts of the London foot. Before I enter upon the examination of the antient buildings, it may be proper to say something concerning the nature of the evidence to be expected from them. All buildings are planned and executed by some measure of the country where they are built. At Rome this measure was the foot, which was divided by the workmen into 4 palms, and each palm into 4 digits (8). (8) Vitruvius, lib. iii. c. i. Frontinus de Agrorum Qualit. Both these authors are technical writers, and give this as the division used by workmen; and the antient foot-rules are so divided. They If the Roman buildings were correctly executed, and we had the true dimensions of their several parts in any known measure, some divisors consisting of Roman feet, and parts of those feet, applied to these measures, must, in the same building, give the same quotient to all; and this quotient will be the measure of the foot, by which that building was constructed, in parts of the known measure. Therefore, where a range of simple divisors, applied to the principal parts of any building, give as nearly the same quotient as can be expected from the common inaccuracies of workmanship, we may reasonably conclude, that these divisors were the architects numbers; and the foot derived from them, that by which the building was constructed. As an architect cannot be supposed to be limited to a few digits in the extent of the front, or of the depth of large buildings, it is probable such measures consisted of whole feet. These, and the diameters of circular buildings, I call prime measures. In all large prime measures, the preference is to be given to a round number for the divisor; as it is more probable a building should be designed for 100 feet in front, than for 99 or 101: and because the passus was 5 feet, I reckon any multiple of 5 a round number. The diameters of columns are of less authority than any other horizontal measures; not only on account of the difficulty of measuring them correctly, They both mention the duodecimal division, which seems to have been used by the vulgar; for the Romans divided every integer into 12 unciae. but because errors of workmanship, to which they are more liable than square members, more sensibly affect the magnitude of the foot in small measures, than in large ones. Uprights, of any considerable height, are of less authority than horizontal measures, from the difficulty of taking them correctly; and being designed by modules, few of them answer well to the foot measure. But here we must except such shafts of columns as are of one block of marble; which seem to be as good authority as any part of a building: for the necessity of making them all exactly of the same length, must produce accuracy; and the doing this was no difficult piece of workmanship. Being likewise commonly (if not always) wrought at the quarry, to save expense in the carriage, they were probably bespoke to some simple measure; and we shall find all such shafts answer to some number of whole palms. In the following enquiry, I suppose that Desgodetz's Paris foot contained 1065.4 such parts, as our London foot contains 1000: since it appears, from the comparison of his measure of the door-case of the Pantheon with Greaves's, that if Greaves's London foot was not above 2 parts in 1000 shorter than Graham's, Desgodetz's Paris foot was but $\frac{3}{1000}$ of the London foot shorter than that in the archives of the Royal Society. This difference (if such there was) does not amount to the 277th part of an inch; which is much nearer than we can expect to find the measure of the Roman foot, from all the remains of antiquity now in being. The temple of Fortuna virilis. When this temple was built, is not known; but it is commonly thought, from the poorness of the materials, and the style of the architecture, to be one of the most antient buildings now remaining in Rome. The great simplicity of its parts, and their near agreement with each other, afford so clear an evidence of the measure by which it was constructed, that I have chosen to begin with it, as an introduction to the rest. It has four Ionic columns in front, and seven in depth. The depth, taken between the centers of the angular columns, is 54 feet 8\(\frac{1}{4}\) inches Paris measure, equal to 58264.06 thousandth parts of the London foot; which contain just 60 Roman feet of 971 + such parts each. The distance between the centers of the columns in this range, is 9 feet 1\(\frac{3}{8}\) inches, equal to 9710.6 London parts; containing 10 Roman feet of the same measure with the former. The extent of the front, taken between the centers of the angular columns, is 28 feet 8\(\frac{1}{2}\) inches, = 30556.25; which contain 31\(\frac{1}{2}\) Roman feet of 970 + London parts each. The distance between the centers of the angular columns, and those on each side the entrance, is exactly equal to the distance between the centers of the columns on the sides; being 10 Roman feet of 971 + parts each. There remain therefore 11\(\frac{1}{2}\) Roman feet, for the distance between the centers of the columns on each side side the entrance. This measures 10 feet $5\frac{1}{2}$ inches (9), equal to 11134.9; which, divided by $11\frac{1}{2}$, gives 968$\frac{1}{4}$ for the Roman foot. This disagreement between the measures in the front, cannot be reconciled by any probable divisors; therefore it must be owing to an error in workmanship. And if the distance between the centers of the middle columns were but a little more than $\frac{3}{8}$ of a London inch greater, both that, and the whole extent of the front, would answer to the same measure of the foot with the sides. Now, as the middle intercolumniation was intended to be greater than the rest, the workman might be less exact in laying out the front, than the sides, where they were all to be equal; for the front intercolumniations next the angles being kept equal to those on the sides, the symmetry of the building would be preserved, and the whole error fall on the middle intercolumniation, where it could not be discovered. Or if, on the other hand, we suppose the measure of the extent in front to have been correct, and the error to have lain in the sides, the workman must still have made the intercolumniations on the sides equal to each other, and those in the front next the angles, equal to them: and, in this case, 970 would be the true measure of the foot, instead of 971. Therefore, I think, it cannot be doubted, that one or the other of them was the measure by which this building was constructed. (9) Desgodetz has figured this 10 feet $5\frac{1}{2}$ inches; which will be found to be a mistake, by comparing the correspondent numbers, and by his measure of this interval in modules. The temple of Vesta, at Rome. This building is remarkable for the style of its architecture, and the elegance of its workmanship. The pointed abacus shews the architecture to be Greek; which is a strong presumption in favour of its antiquity, as it is the only example of the kind now remaining at Rome (1). None of the diameters will answer to the same measure of the foot with the other parts, by probable divisors. This was likely enough to be the case, if the circles of the plan were described by a cord, whose stretching might increase their diameters beyond the intended measures. But the rest of the measures agree pretty nearly with each other, by as simple divisors as can be expected in so small a building. At the door-case, the arch of the wall is discontinued, and the groundfil is strait. The width between the jambs is 9 feet $1\frac{2}{3}$ inches, equal to $9736.6$; which, divided by $10$, gives a foot of $973.7$ London parts. From the wall to the centers of the columns, is 8 feet $8\frac{1}{8}$ inches, = $9244.6$; which, divided by $9\frac{1}{2}$, gives a foot of $973.1$ parts. From the wall to the extremity of the basement, is 10 feet 6 inches, = $11186.7$; which, divided by $11\frac{1}{2}$, gives $972.7$. (1) That is to say, it is probably older than the Pantheon; for, according to Pliny, in his thirty-sixth book, there seems to have been no marble temple in Rome above sixty years older than that building. And his history of marble is confirmed by the saying of Augustus, "That he found Rome brick, and left it marble." Sueton, in Aug. c. 28. From the top of the surbase within, to the pavement of the cell, is 7 feet 9\(\frac{1}{6}\) inches, \(= 8271.65\); which, divided by 8\(\frac{1}{2}\), gives 973.1. From the top of the surbase without, to the outer pavement, is 9 feet 1\(\frac{5}{6}\) inches, \(= 9751.4\); which, divided by 10, gives 975.1. The shafts of the columns are 27 feet 5 inches, \(= 29209.8\); which, divided by 30, gives 973.7. The columns, with their bases and capitals, are 32 feet 0 inch, \(= 34092.8\); which, divided by 35, gives a foot of 974 London parts. The mean measure of the foot from this building, is 973.6; which agrees nearly with the foot of Philander's porphyry column above-mentioned, and with one derived from Greaves's measure of the monument of Cestius, whose side within the city (as he says in p. 151 of his works) is completely 78 feet English; which, reduced to Graham's measure, is 77.84, and contains 80 Roman feet of 973 London parts each. The temple of Vesta, at Tivoli. The measures of the parts of this building disagree so much with each other, by any probable divisors, that I shall only mention the width of the door-way between the jambs, which is 7 feet 3\(\frac{3}{4}\) inches, \(= 7790.7\), and answers to 8 Roman feet of 973.8 parts each. The Pantheon. This is a circular building, with a portico before the entrance; which, having been added after the body of the work was finished (2), has induced many (2) See Desgodetz, p. 3. & p. 14. to think the temple was more antient, and that Agrippa only adorned it with marble, and added the portico. But the inscription on the freze, "M. Agrippa L. F. Cos. tertium fecit," seems to refer to the whole. And Pliny, enumerating the most magnificent buildings in Rome, says, "Pantheon Jovi ulti ab Agrippa factum; cum Theatrum ante texerit Romae Valerius Ostiensis architectus, ludis Libonis (3)." If the latter part of this passage alludes to the dome of the Pantheon (as it seems to do), it will imply, that Agrippa built it. In another place (4), Pliny calls this building Agrippa's Pantheum. Now, temples of this form were usually peripteres; that is to say, encompassed with columns (5), as the two temples of Vesta; but such a colonnade to this building (supposing the entablature to reach up to the top of the wall) could not consist of fewer than twenty-six columns of 9 London feet in diameter; which, with their entablature, must have cost much more than the whole building, as it now stands. Therefore, it seems as if no columns were at first intended on the outside, and that the portico was an after-thought. This temple is said to have been burnt in that great conflagration, which happened in the reign of Titus (6); and again by lightning, in the thirteenth year of Trajan (7), and to have been restored by (3) Pliny, lib. xxxvi. c. 15. (4) Ibid. c. 5. (5) Vitruvius, lib. iv. c. 7. (6) Xiphilin. in Tito. Suetonius likewise mentions this fire; but does not say what buildings were burnt by it. Titus, c. 8. (7) Eusebius in Chronico. Adrian Adrian (8). But it does not seem to have been totally destroyed either time; for the inscription of Severus says, it was "vetustate corruptum;" which could hardly be true, if it was entirely rebuilt by Adrian, who began his reign but seventy-six years before Severus; but is very applicable to a building that had stood above 200 years, and been twice damaged by fire. Neither does it seem possible, that a building of this size and solidity, without any timber in it, except the roofing of the portico, should be totally consumed by the fire of the neighbouring buildings. Therefore, we may suppose the measures of the principal parts of it to be according to the foot of the Augustan age. If the execution of this noble building had been answerable to the beauty of the design; the multiplicity of its parts, and the accuracy with which Defgodetz has measured them, would have afforded ample materials for the discovery of the foot by which it was constructed: but the dimensions of such parts of it as should be equal to each other, differ so widely, that scarce any but the prime measures are to be depended on. These were, probably, the diameter of the circle passing through the centers of the columns within; for on this, the diameter of the cupola depended; and the diameter of the whole building, from out to out. Defgodetz has given twelve measures of the diameter within the shafts of the columns. The mean from them all, is 133 feet 2\(\frac{2}{3}\) inches Paris measure: (8) Spartanus in Adriano. to this, add the mean diameter of the columns, 3 feet 4\(\frac{1}{2}\) inches, their sum, 136 feet 7\(\frac{1}{2}\) inches, equal to 145533.6 London parts, is the diameter of the circle passing through the centers of the columns; which, divided by 150 (as round a number as can be desired), gives 970.2 such parts for the measure of the Roman foot. The mean depth of the four long chapels, between the shafts of the columns and the wall, is 9 feet 1\(\frac{4}{8}\) inch, = 9764.3; which, divided by 10, gives a foot of 976.4 London parts; but by 10\(\frac{1}{4}\), one of 952.6; which is certainly too small. And if this was an arbitrary measure, we cannot admit the fraction of a palm in it. Therefore, we may suppose these chapels were intended to be 10 Roman feet deep in the clear between the columns and the wall. The mean diameter of the columns, 3 feet 4\(\frac{1}{2}\) inches, = 3622.36 London parts, divided by 3\(\frac{3}{4}\), gives a foot of 966 parts. The shafts of these columns measure 27 feet 1\(\frac{1}{3}\) inches, whose eighth part (reckoning the shaft at 16 modules) is 3 feet 4\(\frac{2}{3}\) inches; which is somewhat less than the mean of their diameters. Yet even this measure (which is equal to 3610.5 London parts) does not want half a Roman digit of 3\(\frac{3}{4}\) feet of 970.2 parts each; which the architect might safely neglect, and allow 3\(\frac{3}{4}\) feet for these diameters, in his general plan. Now, if to twice the depth of these chapels, or 20 Roman feet, we add twice the semidiameter of the columns, or 3\(\frac{3}{4}\) such feet; and to their sum, the diameter of the circle passing through the centers of the columns, 150 feet; we shall have 173\(\frac{3}{4}\) Roman feet. feet for the diameter from wall to wall of the opposite chapels. Desgodetz has given four measures of this diameter, whose mean, 158 feet \(2\frac{2}{3}\) inches, is equal to 168570 London parts, which, divided by 173\(\frac{3}{4}\), give 970.2 —, for the measure of the Roman foot; agreeing with that found from the diameter of the circle passing through the centers of the columns. Or if, instead of taking a mean of the four diameters from wall to wall, we add twice the mean depth of these chapels, and twice the mean semi-diameter of the columns, to the mean diameter of the circle passing through their centers, we shall have 158 feet 4 inches for the diameter from wall to wall, equal to 168688 London parts; which, divided by 173\(\frac{3}{4}\), give 970.9, for the measure of the foot. But if this diameter from wall to wall was a prime measure, as well as that of the circle passing through the centers of the columns, the depth of these chapels must have been a residuary measure, and might be more or less than 10 Roman feet. And if we divide the diameter last mentioned by 174 (which being composed of 150, and twice 12, may here be reckoned a round number), it will give a foot of 969\(\frac{1}{2}\) London parts. But if, instead of twice the mean depth of these chapels, we take twice the greatest depth, 18 feet 6 inches, for part of this diameter, the same divisor will give a foot of 970\(\frac{1}{2}\) such parts. The measure of the diameter from out to out cannot be so well determined as either of the former; for Desgodetz has given but one measure of the outer wall, which is 5 feet 4\(\frac{1}{2}\) inches; to this we must add 1 foot 10\(\frac{1}{3}\) inches, for the mean depth of the niches in in the long chapels; and twice their sum, 14 feet 5\(\frac{2}{3}\) inches, added to 158 feet 2\(\frac{2}{3}\) inches, the inside diameter between the walls of these chapels, will give 172 feet 8\(\frac{1}{3}\) inches for the diameter from out to out. This is equal to 183988.6 London parts; which answer to 190 Roman feet, of 968.4 such parts each. But this diameter being got by the ad- dition of many small parts, one of which is a single measure, is of small authority. Though if the outer wall is 5 feet 4\(\frac{1}{3}\) inches thick in brick-work, and the backs of the niches in the long chapels are in- crusted with marble (as I suppose they are), a proper allowance for the incrustation will make this diameter answer to a foot of 970 London parts. The principal parts of the circular range within, are too unequal to answer to any regular division into compartments; and the two largest measures in it sufficiently evince, how little use is to be made of the rest: for the width of the grand chapel opposite the entrance, and the width of the entrance between the pilasters in the range, will not answer to the same measure of the foot, by any probable divisors. But the inequality of the faces of the pilasters, that terminate the solids in this range, affords the clearest proof, that no certain measure of the foot is to be ob- tained from the lesser parts of this building: for two of them, at the extremities of the same solid, differ in breadth by a Paris inch and three quarters. One of the two on each side the grand chapel exceeds the other in breadth, by a Paris inch and a quarter; and it exceeds the least in the circular range, by no less than 4\(\frac{1}{3}\) such inches, though the diameter of the col- umn before it is less than that of some in the range. Vol. LI. The diameter of the circular light at top is 27 feet 5 inches, equal to $29209.7$; which, divided by 30, gives $973\frac{1}{3}$ for the Roman foot. Whether Desgodetz took more measures than one of this diameter, does not appear, nor how he took it. It is probable, that 30 Roman feet was the measure here intended, being a round number, and one fifth of the diameter between the centers of the columns: but the making this light a true circle, precisely answering to a given measure, was a difficult piece of workmanship. Therefore, this measure is of small authority. The square compartments on the pavement measure 9 feet $0\frac{2}{3}$ inch, or $9647.79$ London parts; which, divided by 10, give a foot of $964.8$ parts. But Desgodetz observes (9), that this is not the original pavement: and this measure of the foot answers to the age of S. Severus, as I shall shew hereafter. This pavement being considerably raised above the old one, no measures of uprights taken from it are of any authority. The height of the attic, above the cornice it stands upon, is 27 feet $2\frac{1}{3}$ inches, equal to $29010$; which, divided by 30, gives a foot of $967$ parts. But the walls of this building being brick, this measure may have been diminished by their settling, as the whole error occasioned by it might be made to fall on this part, rather than on the order below. The length of the shafts of the columns in the circular range, is 27 feet $1\frac{1}{3}$ inches, = $28884.18$; which answers to $29\frac{3}{4}$ feet, of $970.9$ parts each. (9) Desgodetz, p. 41. The width of the door-case, between the jambs, has been supposed, by Picard, De la Hire, Eiffenschmid, and others, to have been a measure of whole feet; for this reason, because they thought it an arbitrary one, and therefore not likely to contain a fraction (1). But in this, I think, they were mistaken; for it appears to have been determined by the width of the passage within, of which it takes up just three fourths. The proportion is too simple, and too exactly executed, to be accidental; and it is highly improbable, that the width of this passage, which is a principal part of the plan of the inside, should be determined by that of the door-case. The width of the passage, is 24 feet $5\frac{3}{4}$ inches, = 26098.6; that of the door-way, next the passage, is 18 feet $4\frac{1}{3}$ inches, = 19610; next the portico, 18 feet $4\frac{1}{3}$ inches, = 19561.9; none of which will give a measure of the foot, answering to the rest of the building, by whole divisors. The width of the door-way next the portico (which has been supposed to be just 20 Roman feet), divided by 20, gives 978. But the diameters of the temple (taking either the greatest or the least, instead of a mean), will not answer to a foot of this measure, by probable divisors. The principal measures in the portico are, its extent in front and depth, and the shafts of the columns. (1) Mr. Greaves says, this door-case is one entire marble stone. See p. 348. and p. 494. And Ficoroni (a late writer), in his Vestigia di Roma antica, p. 132, speaking of this door-case, says, "Essendo amendue i lati, col di sopra, e il di sotto di un solo pezzo di marmo." But Desgodetz, p. 18, says, that each side of the architrave consists of two stones, and the top, of one; and corrects Serlio, for saying the whole was of one piece. The extent in front, taken between the centers of the angular columns, is 98 feet $4\frac{3}{8}$ inches, equal to 104797.6; which contain 108 Roman feet of 970.3 parts each. Its depth on the right-hand side of the entrance (2), from the center of the angular column to the center of the pilaster, is 41 feet $5\frac{1}{3}$ inches, = 44136.4, and contains 45$\frac{1}{2}$ feet of 970 + parts. The shafts of the columns, which are each of one single stone, measure 36 feet $7\frac{7}{8}$ inches (3), equal to 39053.6; which answer to 40$\frac{1}{4}$ Roman feet of 970.3 London parts. I can find no measures in this building of equal authority with the diameters, the shafts of the columns in the circular range, and these measures in the portico. All of which give the measure of the Roman foot greater than 970 such parts as the London foot contains 1000. In the rest of the buildings, where the measures do not require a particular explanation, I shall first give Desgodetz's measures, with their reductions to thousandth parts of the London foot, and then set down the divisors and quotients with two hooks between them; the divisor on the left-hand, expressing (2) By some expressions of Desgodetz, his plans seem to be inverted, as gravings commonly are. I speak according to the plan, supposing the reader to view it in front. (3) Desgodetz, p. 19. has figured the shafts of the pilasters in the portico 36 feet $7\frac{1}{3}$ inches, and in modules 16 mod. $8\frac{3}{4}$ parts; which is somewhat too great for his foot measure, his module being 27 Paris inches. But p. 16. he has figured the shafts of these columns 16 mod. $8\frac{3}{4}$ parts, which is 36 feet $7\frac{7}{8}$ inches. And that this is no mistake, appears from his measures of the capitals of each. a number of Roman feet; and the quotient on the right, their magnitude in parts of the London foot. The temple of Peace. The length on one side within the walls, including the pronaus or porch, is 269 feet 8\(\frac{1}{2}\) inches, \(= \frac{287347}{26} = 296\) ... (970,8 The width of the nave 77 feet 5\(\frac{1}{2}\) inches, \(= \frac{82524}{1} = 85\) ... (970,9 The width of the side-building 48 feet 0 inch, \(= \frac{51139}{2} = 52\frac{3}{4}\) ... (969,5 The length of the first division of the side 71 feet 5 inches, \(= \frac{76087}{3} = 78\frac{1}{2}\) ... (969,3 The length of the second division 71 feet 8 inches, \(= \frac{76353}{87} = 78\frac{3}{4}\) ... (969,6 The length of the third 70 feet 3\(\frac{1}{2}\) inches, \(= \frac{74888}{75} = 77\frac{1}{4}\) ... (969,4 The width of two of the passages in the cross walls, is 22 feet 4 inches, \(= \frac{23793}{9} = 24\frac{1}{2}\) ... (971,2 The width of the exedra in the middle division of the side 53 feet 11 inches, \(= \frac{57442}{82} = 59\frac{1}{4}\) ... (969,5 Its depth is equal to the width of the passages in the cross walls. The two middle entrances through the porch, 15 feet 3 inches, \(= \frac{16247}{35} = 16\frac{3}{4}\) ... (970 The side entrances 14 feet 4\(\frac{1}{2}\) inches, \(= \frac{15315}{13} = 15\frac{3}{4}\) ... (972,4 The entrance into the porch on the right-hand 14 feet 4 inches, \(= \frac{15270}{74}\) \(= \frac{15\frac{3}{4}}{}(969.5)\) The shaft of the column (of one stone) 49 feet 3 inches, \(= \frac{52470}{9}\) \(= \frac{54}{54}(971.7)\) The diameter of the column 5 feet \(8\frac{1}{4}\) inches, \(= \frac{6058}{47}\) \(= \frac{6\frac{4}{4}}{}(969.4)\) The height of the order 71 feet \(0\frac{1}{2}\) inch, \(= \frac{75687}{79}\) \(= \frac{78}{78}(970.4)\) A mean from these twenty-one measures gives a Roman foot of 970.3 parts. But the measures in this building seem to be very incorrectly executed. The temples of Bacchus, and of Faunus. I shall not particularly recite the measures of these buildings, because they seem to be the works of a much later age than any of the rest. By the different workmanship of the shafts of the columns in the temple of Bacchus (4), it seems to have been compiled out of the ruins of other buildings. In the temple of Faunus, most of the columns are of granite, but some of them of white marble; the cross wall was evidently designed to support a flat roof, the outer wall being too weak to bear a cupola (5); and the windows (which I take to be as old as the rest of the building) are in the Gothic style: all which are marks of a late age. Therefore both these temples seem to be more modern than they are commonly thought to be. The temple of Bacchus (4) See Desgodetz, p. 68. (5) Ibid. p. 79. answers to a foot of 968 parts, that of Faunus to one less than 967. The amphitheatre of Vespasian. The plan of this building is elliptical; and the area in the middle being allotted to the exhibition of the shews, seems to be the part on which the magnitude of the whole depended. Therefore one of its axes must be the prime measure. The greater axis measures 263 feet 11 inches, = 281176.8 London parts; the lesser 165 feet 1 inch, = 175879.7. These numbers are so nearly in the proportion of 8 to 5, that there is little reason to doubt their being intended for it. The only two whole numbers in this proportion that will answer our purpose are 288 and 180, by which the greater axis will give 976.3 London parts for the measure of the foot; the lesser, 977.1. Either the distance from the inner to the outer wall, or that from the inner wall to the extremity of the upper step, was probably a prime measure too. The former is 155 feet 5\(\frac{1}{2}\) inches, = 165610.5; which, divided by 170, gives 974.2. The latter is 158 feet 0\(\frac{1}{2}\) inch, = 168340.6; which, divided by 172, gives 978.7. But the axes will admit of other divisors, which are in my opinion as probable at least, as those above-mentioned. For if we suppose the greater axis to have been the prime measure from which the lesser was determined in the proportion of 5 to 8, and assume 290 Roman feet for it (which is a more probable measure than 288), the lesser axis must then be 181\(\frac{1}{4}\) such feet. And the measure of the foot by the the greater axis will be 969.6 parts; by the lesser 970.3. The distance from the inner to the outer wall will not answer to this measure of the foot by any whole divisor. And that from the inner wall to the extremity of the upper step, divided by 174, gives but 967\(\frac{1}{2}\). But this disagreement will not appear extraordinary, when we consider that these distances are not single measures, but the sums of nine or ten smaller ones (6), and may therefore be liable to an error of some inches. The lesser parts are so incorrectly executed, that no use can be made of them. And the heights of the four orders, and of their parts, are so irregular as not easily to be accounted for. The amphitheatre at Verona. The principal measures in this building answer to the Greek foot; which is not the subject of the present inquiry. The arch of Titus. This edifice is so ruined, that we have not the measure of the extent of its front. | Measure | Value | |--------------------------|----------------| | The depth of the solid | 14 feet | | 7 inches | = 15537.1 | | The width of the entrance| 16 feet | | 4 inches | = 17401.5 | | The shafts of the columns| 16 feet | | 2 inches | = 17223.97 | (6) See the section, Pl. 3. The arch of Constantine. The measures of this elegant design are so incorrectly executed, that I can make no use of them. The fronts of the two outermost solids differ by a Paris inch and one third; the side entrances by $1\frac{1}{2}$; the distances of the columns from the pilasters (a measure of little more than 18 inches) differ an inch and $\frac{1}{2}$; and the depth of the solid at one end, exceeds that at the other end by no less than $7\frac{1}{4}$ Paris inches. The temple of Antonine and Faustina. Desgodetz has given the measure of no part of this building in feet and inches, but the diameters of the columns; which he says do not differ from each other, as in many of the rest. The diameter of the columns is 4 feet $6\frac{1}{2}$ inches, $= 4846.09$ The extent of the portico taken between the centers of the angular columns, 25 mod. $16\frac{1}{2}$ parts, $= 61935.7$ The shafts of the columns, 15 mod. 28 parts, $= 38607.2$ These shafts were probably intended for 16 modules, and their measure is more to be relied on than that of the diameters, both on account of its size, and because the correct execution of it was not only more easy, but more necessary. Therefore, though these measures are too few, and too unequal to determine the exact magnitude of the foot used in the con- struction of this building, yet it seems to have been less than any we have hitherto met with. The arch of Septimius Severus. This building deserves our particular attention, both on account of the correctness of the workmanship, and because it exhibits a different measure of the Roman foot from any of the preceding. It is all of white marble. The stones are laid without mortar, and fastened together with cramps. And the shafts of the columns are each of one stone. The whole extent in front, is 71 feet 5\(\frac{1}{6}\) inches, \(= 76161.3\) ———— 79) ... (964,1 The depth at one end, is 21 feet 8\(\frac{3}{4}\) inches, \(= 23150.3\) ———— 24) ... (964,6 The depth at the other end, 21 feet 8\(\frac{5}{6}\) inches, \(= 23157.6\) ———— 24) ... (964,9 The width of the middle entrance, 20 feet 10 inches, \(= 22195.8\) ———— 23) ... (965 The width of the side entrances, 9 feet 2 inches, \(= 9766.17\) ———— 10\(\frac{1}{3}\)) ... (964,5 From the middle entrance to the extremity of the outer solid, on one side, is 25 feet 3\(\frac{3}{4}\) inches, \(= 26967.9\) ———— 28) ... (963,1 The like on the other side, 25 feet 4\(\frac{1}{2}\) inches, \(= 26997.5\) ———— 28) ... (964,2 The distance between the shafts of the outermost columns, is 60 feet 7\(\frac{5}{6}\) inches, \(= 64619.5\) ———— (7) 67) ... (964,5 (7) This and the two following measures are inserted only to shew the probability, that the diameters of the columns were designed The distance between their centers, 63 feet $4\frac{1}{3}$ inches, = 67504.9 The distance including their shafts, 66 feet $\frac{5}{6}$ inch, = 70390.4 The diameter of the columns, 2 feet $8\frac{1}{2}$ inches, = 2885.46 The uprights will not answer to any one measure of the foot by whole divisors; nor will they agree with each other by broken ones, so nearly as the hor- izontal measures do, which may partly be owing to the difficulty of taking them correctly. The shafts of the columns measure 21 feet $8\frac{1}{4}$ inches, = 23105.9 The pedestals, 12 feet $3\frac{1}{2}$ inches, = 13088.14 The whole order, 33 feet $4\frac{1}{4}$ inches, = 35585.53 The attic, 15 feet $5\frac{1}{3}$ inches, = 16454.51 The blocking, 1 foot $9\frac{2}{3}$ inches, = 1923.64 The whole height, 62 feet $10\frac{2}{3}$ inches, = 67001.8 I have produced these uprights, rather as an in- stance how little such measures are to be depended on, even in buildings correctly executed, than as of any authority for the measure of the foot. For they cannot be reconciled to a common measure without *Signed to be three Roman feet, and not as of any authority in the determination of the measure of the foot.* fractions of the digit in the divisors, by which means any measure of the foot might be derived from any building. A mean from the seven first horizontal measures gives a Roman foot of 964\(\frac{3}{4}\) parts. A mean from the uprights one of 964\(\frac{1}{2}\). But the depth of the solid, and the widths of the entrances are most to be depended on, and these give a foot of 964\(\frac{3}{4}\) London parts; which will be found to be the greatest measure this building will answer to by probable divisors. For though the extent of the front, or the width of the middle entrance, must probably have been the prime measure in a building of this kind; yet no part seems more likely to consist of some number of whole feet (or at least whole palms) than the depth of the solid; since there appears no reason why the architect should use odd digits in the measure of it. Now if we suppose the extent of the front to have been 78\(\frac{1}{2}\) Roman feet, they will contain 970 parts each; but to make the depth agree with this measure, it must have been 23\(\frac{7}{8}\) such feet, which does not seem probable, for the reason above-mentioned. The middle entrance must then have been 22\(\frac{7}{8}\), instead of 23 whole feet, and the extent on each side 27\(\frac{1}{8}\), instead of 28. And there will not be one whole measure in the plan of this building; which is improbable. Neither do these broken measures serve for the subdivision of the sides better, or even so well as the whole ones. The shafts of the columns are so nearly 16 modules, that they seem to have been designed for that proportion; and \(\frac{1}{2}\) of a Paris inch added to the diameters meters would make them so. But this is not sufficient to make those diameters answer to 3 feet of 964\(\frac{3}{4}\) parts, of which they want almost \(\frac{1}{16}\) of a Paris inch. This may be partly an error in the workmanship, and partly in Desgodetz's measure; but if this quantity were added to them, they would not only answer to the same foot with the rest of the building, but all the parts in which their measure is included, would agree better with the depth of the solid, and the widths of the entrances. The height of the blocking was probably intended for 2 Roman feet, though it answers to no greater measure of that foot than the diameters of the columns do. But measures of 2, 3, and 4 feet, seldom agree with the larger ones in any of the buildings; for a small error, either in the workmanship, or in the measuring, makes a very sensible difference in the length of the foot derived from such intervals. And this blocking does not want the \(\frac{1}{13}\) part of a London inch, of 2 feet of 965 parts each. The bankers arch, and the portico of Severus, are contemporary buildings with this, and seem to agree with it in the measure of the foot. The former is a very small building, and its parts will not answer to any one common measure by divisors of whole digits. However, the depth of the solid (which, as I have before observed, is most likely to be a whole measure) taken at three different parts, is 6 feet 4 inches, which answer to 7 Roman feet of 963,93 London parts. There are but few measures in the portico of Severus. Those most to be depended on are, the whole extent in front, and the shafts of the columns, which are are each of one block. The former measures 56 feet \(11\frac{7}{8}\) inches, = 60708.3 —, which contain 63 Roman feet of 964.6 parts each. The latter measures 26 feet \(11\frac{1}{4}\) inches, = 28699.22, which contain 29\(\frac{3}{4}\) such feet of 964.7 parts. The whole column, with the base and capital, measures 32 feet 6\(\frac{1}{2}\) inches, = 34669.9 London parts, which contain 36 Roman feet of 963 parts each. It appears by the inscription on the architrave of the portico of the Pantheon (8), that Severus repaired that temple; and Desgodetz observes, that the pavement is not so old as the building. That it was laid by Severus is probable, from the correctness of its measures, which is a remarkable characteristic of the workmanship of that age. The square compartments on this pavement answer to 10 Roman feet, of 964.78 parts, agreeing with a mean from the depth of the solid of this arch, and the widths of the entrances, which are the most unexceptionable measures in it. The breadth of the strait bands between these compartments, is equal to the measure Desgodetz has given of the diameters of the columns of this arch, and answers to 3 feet of 961.8 parts. But these bands might be purposely made somewhat short of the full measure, to leave room for one of the same breadth with the rest between the outermost compartments and the circular range; for notwithstanding this, the workmen found themselves obliged to diminish both the outermost compartment, and the band beyond it. (8) Imp. Cæs. Septimius Severus—et Imp. Cæs. M. Aurelius Antoninus—Pantheum vetustate corruptum cum omni cultu restituerunt. The concurring testimonies of these contemporary works, though not of sufficient authority to establish any measure of the foot by themselves, are yet very strong evidence in favour of the measure above derived from this arch. And we shall find a farther confirmation of it in Diocletian's baths. The baths of Diocletian. The whole length of this building is 439 feet 3 inches, = 467977 The length of the great hall, 180 feet 8\(\frac{1}{6}\) inches, = 192497 The breadth at one end, 74 feet 3\(\frac{5}{6}\) inches, = 79179,9 Its breadth at the other end, 74 feet 5\(\frac{1}{2}\) inches, = 79327,9 The distance between the middle solids on the side, 49 feet 8\(\frac{1}{4}\) inches, = 52937 The depth of the recess out of the great hall between these solids, is 33 feet 6\(\frac{1}{6}\) inches, = 35705,7 The distance between one of these solids and the nearest angular solid, 32 feet 11\(\frac{5}{8}\) inches, = 35124,9 The distance between the other of them and the other angular solid, 33 feet 1\(\frac{1}{3}\) inch, = 35276,57 Entrance at one end of the hall, 36 feet 2\(\frac{1}{6}\) inches, = 38546,8 Entrance at the other end, 36 feet 1\(\frac{1}{4}\) inch, = 38465,4 — (twice) Passage from the circular vestibule, 30 feet 9\(\frac{1}{4}\) inches, \(= 32783.2\) (twice) \(\ldots\) (964) Side entrance into the vestibule, 20 feet 9\(\frac{2}{3}\) inches, \(= 22166.2\) A measure from 4 feet 6 inches to 4 feet 6\(\frac{1}{2}\) inches, repeated 12 times, the mean from all of them is 4 feet \(6\frac{7}{8}\) inches, \(= 4818.03\) A measure from 5 feet 5 inches to 5 feet 5\(\frac{1}{4}\) inches, repeated 11 times, the mean from all is 5 feet 5\(\frac{2}{4}\) inches, \(= 5787.05\) A measure from 2 feet 8\(\frac{1}{2}\) inches to 2 feet 8\(\frac{5}{6}\) inches, repeated 4 times, the mean from all is 2 feet 8\(\frac{7}{12}\) inches, \(= 2892.86\) A multiple of 6 feet on the wall, where this measure of 3 feet is thrice repeated, is 43 feet 5\(\frac{1}{8}\) inches, \(= 46267.2\) The diameter of the circular vestibule is too prin- cipal a part to be neglected. It answers to 65 Ro- man feet of 969.1 parts, which the other measures in this building will not come up to by probable di- visors. It answers likewise to 65\(\frac{1}{4}\) feet of 965.4 parts; but an odd palm in so large a measure is not pro- bable. Therefore it seems to have been incorrectly executed, and perhaps this circle was described by a cord. The The shafts of the columns, each of one block, measure $35$ feet $11\frac{1}{2}$ inches, $= 38302.6$ inches, $= 39\frac{3}{4}$. . . (963,6 The whole column is $43$ feet $6\frac{1}{6}$ inches, $= 46359.7$ inches, $= 48)$. . . (965,8 These are the principal measures in this building, and a mean taken from all of them, except the diameter of the vestibule, gives a foot of 964,1 London parts. But the three small measures, of 5, 6, and 3 Roman feet (the first of which is the passus) are alone sufficient to discover the measure of the foot by which this building was constructed. For, being so often repeated with so little variation, they must afford a mean very near the truth, if their divisors be rightly assumed; and that these are so, can hardly be doubted. For, were the two largest to be encreased but by a single digit, they would give different measures of the foot from each other; and that from the passus so encreased, would fall short of 952 London parts; which is less than the least of the antient foot-rules. On the other hand, were they diminished by the same quantity, they would not agree together so nearly in the measure of the foot as they now do, nor give such a measure of it as the rest of the building will answer to by probable divisors. Besides, I have here reckoned only such of these measures as Desgodetz has figured on his plan; whereas they are much oftener repeated, and that not in the diameters of columns, or the members of an order, but in breaks, projections, and thicknesses of walls, where a digit or two more or less could make no visible difference, Vol. LI. 5 N difference, provided the correspondent parts were made to answer to each other: and it is highly improbable that such odd measures as the passus wanting a digit, 6 feet wanting the same, and 3 feet wanting half a digit, should be so often repeated in so large a building, and in places that do not seem to require any such precise exactness. Therefore these measures must have been whole feet, and the divisors are rightly assumed. Their mean is 964,1 London parts, which exactly agrees with that found above from all the measures taken together. The other remains of antiquity contained in this work are such fragments as can be of little use in the present enquiry. CONCLUSION. It appears from the measures of these buildings, that the Roman foot before the reign of Titus exceeded 970 parts in 1000 of the London foot, and in the reigns of Severus and Diocletian fell short of 965. Whether this difference proceeded from any alteration in the standard, or from a false measure of it being got into common use, either before the reign of Titus or after, is uncertain. We have no account of any alteration made by law in the Roman standards after the Plebiscitum Silianum, quoted by Festus (9); but as great a difference as this might arise from their having been lost or destroyed. (9) Under the words Publica pondera. They were kept in the Capitol (1), and Rigaltius, from a passage in Hyginus, observes that the standard of the foot was deposited in the temple of Juno Moneta (2). Now the Capitol was burnt no less than three times; first in the civil war of Sylla (3); then again when Sabinus was besieged in it by the troops of Vitellius (4); and the third time in that dreadful conflagration which happened in the reign of Titus (5). Whether the standards were destroyed in the first of these fires is uncertain; but they could hardly escape the fury and confusion of the second, when, according to Pliny, the temple of Juno Moneta seems to have been burnt to the ground (6). And, if we may credit Xiphilin (whose account of the third is in some measure confirmed by Spartan), not only the temple of Jupiter Capitolinus, but those adjoining to it, were burnt down in the last (7). Vespasian (1) This appears from the inscription on Vespasian's congius, and from R. Fannius. And Jul. Capitolinus in Maximinis, mentions the amphora Capitolina. (2) Hyginus de L'mit. constituend. near the end, distinguishes the Roman foot by the name of Pes Monetalis. That this temple was in the Capitol, appears from Livy, lib. vii. c. 28. Cicero de Divin. lib. i. and Pliny, lib. xxxiv. cap. 7. Suidas (v. Μονητα) tells us the mint was kept here. (3) Taciti Hist. lib. iii. This fire is likewise mentioned by Plutarch and Appian. (4) Tacitus, ibid. (5) Xiphilin in Tito. Sueton. in Domit. c. 5. (6) Pliny, lib. xxxiv. c. 7. says, "Ætas nostra vidit in Capitolio, priusquam id novissime conflagravit, à Vitellianis incensum, in cella Junonis canem ex ære, &c." And if the donatives were destroyed, it is not likely the standards should be saved in a scene of such confusion. (7) ὁ τε νέων ἐς Δίους τῆς Καπιτωλίας, μετὰ τῶν συνδέων ἀντικαταστήσει. Xiphilin in Tito. Spartan, enumerating the public buildings Vespasian rebuilt the Capitol after the second conflagration, and restored the antient records from copies of them that were got abroad (8), and probably the standards at the same time. The congius was restored by weight, according to the Plebiscitum Silianum, as the inscription on it testifies. The quadrantal was too cumbersome a vessel for common use, to which the congius (like our gallon) was well adapted; so that I do not see what other purpose it could serve, but to adjust the congius to its capacity, and the foot to its side: but here we see the congius adjusted by weight, and it is not very likely that a new quadrantal should be made for no other end but to adjust the foot by, when so many copies of the old standard were extant. Therefore it is not improbable, that the standard of the foot was at this time restored, without any regard to its relation to the quadrantal. But as to the difference between the foot derived from the congius, and that found from other authorities, I must farther observe, that the correct adjustment of weights to measures, is a very difficult matter, even in this age, and in this kingdom, where workmanship is arrived at a high pitch of accuracy. And what errors rude workmanship is liable to, sufficiently appears from the weights Pætus and Villalpandus have given of the congius. Therefore I see no reason to reject the testimonies of those authors, buildings restored by Adrian, mentions most of those that Xiphilin says were burnt in this fire. Domitian rebuilt the Capitol. Suet. in Domit. c. 5. (8) Sueton. in Vespaf. c. 8. who say that the cubic foot contained a quadrantal of wine; and as little to believe that these two standards were ever truly adjusted to each other. But had the original standard of the Roman foot been truly adjusted to the quadrantal, and continued invariable from the time of its first establishment, yet a false measure of it might at one time or other have got into common use at Rome, as well as a false measure of the French foot did at Paris; where in the year 1668, the mason's foot was found to exceed the foot of the Chatelet by $\frac{5}{72}$ of a Paris inch (9), which is above $\frac{1}{4}$ of a London inch: and the unaccountable negligence which appears in the Roman coinage, gives sufficient ground to suspect they were not more accurate in their measures. LXX. A Description of a metalline Thermometer; by Keane Fitzgerald, Esq; F. R. S. Communicated by the Right Hon. George Earl of Macclesfield, President of the Royal Society. Read May 22, 1760. It is universally allowed, that all bodies, whether solid or fluid, are expanded and contracted by heat and cold; and, as far as experiments of this kind have yet reached, it appears, that scarce any two bodies of different natures, or even of the same; are expanded, or contracted equally by the same degrees. (9) Picard, in the paper De Mensuris, quoted above.