# 1911 Encyclopædia Britannica/Homoeopathy

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​HOMOEOPATHY (from the Greek ὅμοιος, like, and πάθος, feeling). The distinctive system of therapeutics which bears the name of homoeopathy is based upon the law similia similibus curentur, the originator of which was S. C. F. Hahnemann, a native of Meissen in Germany, who discovered his new principle while he was experimenting with cinchona bark in 1790, and announced it in 1796. The essential tenets of homoeopathy—with which is contrasted the “allopathy” (ἄλλος, other) of the “orthodox” therapeutics—are that the cure of disease is effected by drugs that are capable of producing in a healthy individual symptoms similar to those of the disease to be treated, and that to ascertain the curative virtues of any drug it must be “proved” upon healthy persons—that is, taken by individuals of both sexes in a state of health in gradually increasing doses. The manifestations of drug action thus produced are carefully recorded, and this record of “drug-diseases,” after being verified by repetition on many “provers,” constitutes the distinguishing feature of the homoeopathic materia medica, which, while it embraces the sources, preparation and uses of drugs as known to the orthodox pharmacopoeia, contains, in addition, the various “provings” obtained in the manner above described.
Besides the promulgation of the doctrine of similars, Hahnemann also enunciated a theory to account for the origin of all chronic diseases, which he asserted were derived either directly or remotely from psora (the itch), syphilis (venereal disease) or sycosis (fig-wart disease). This doctrine, although at first adopted by some of the enthusiastic followers of Hahnemann, was almost immediately discarded by very many who had a firm belief in his law of cure. In the light of advancing science such theories are entirely untenable, and it was unfortunate for the system of medicine which he founded that Hahnemann should have promulgated such an hypothesis. It served as a target for the shafts of ridicule showered upon the system by those who were its opponents, and even at the present time there still exists in the minds of many misinformed persons the conviction that homoeopathy is a system of medicine that bases the origin of all chronic disease on the itch or on syphilis or fig-warts.
Another peculiar feature of homoeopathy is its posology or theory of dose. It may be asserted that homoeopathic posology has nothing more to do with the original law of cure than the psora (itch) theory has, and that it was one of the later creations of Hahnemann’s mind. Most homoeopathists believe more or less in the action of minute doses of medicine, but it must not be considered as an integral part of the system. The dose is the corollary, not the principle. Yet in the minds of many, infinitesimal doses of medicine stand for homoeopathy itself, the real law of cure being completely put into the background. The question of dose has also divided the members of the homoeopathic school into bitter factions, and is therefore a matter for careful consideration. Many employ low potencies, i.e. mother ​ tinctures, first, second, sixth dilutions, &c., while others use hundred-thousandths and millionths.
Some homoeopathists of the present day still believe with Hahnemann that, even after the material medicinal particles of a drug have been subdivided to the fullest extent, the continuation of the dynamization or trituration or succussion develops a spiritual acurative agency, and that the higher the potency, the more subtle and more powerful is the curative action. Hahnemann says (Organon, 3rd American edition, p. 101), “It is only by means of the spiritual influence of a morbific agent that our spiritual vital power can be diseased, and in like manner only by the spiritual operation of medicine can health be restored.” This is absolutely denied by others. Thus there exist two schools among the adherents of homoeopathy. On the one hand there are the Hahnemannians, the “Purists” or “High Potency” men, who still profess to regard the Organon as their Bible, who believe in all the teachings of Hahnemann, who adhere in their prescriptions to the single dose, the single medicine, and the highest possible potency, and regard the doctrine of the spiritual dynamization acquired by trituration and succussion as indubitable. On the other side there are the “Rational” or “Low Potency” men, who believe in the universality of the law of cure, but think that it cannot always be applied, on account of an imperfect materia medica and a lack of knowledge on the part of the physician. They believe that in many cases of severe and acute pain palliatives are required, and that they are free to use all the adjuvants at present known to science for the relief of suffering humanity—massage, balneology, electricity, hygiene, &c. The American Institute of Homoeopathy, the national body of the United States, has adopted the following resolution and ordered it to be published conspicuously in each number of the Transactions of the society: “A homoeopathic physician is one who adds to his knowledge of medicine a special knowledge of homoeopathic therapeutics. All that pertains to the great field of medical learning is his by tradition, by inheritance, by right.”
It is claimed that the effect produced upon both the laity and the general profession of medicine by the introduction of homoeopathy was salutary in many ways. It diminished the quantity of medicine that was formerly considered necessary for the eradication of disease, and thus revealed the fact that the vis medicatrix naturae is often sufficient, with occasional and gentle assistance, to cure many diseases, especially those fevers that run a definite and regular course. Corroboration of the law similia similibus curentur is seen, according to homoeopathists, in the adoption of the serum therapy, which consists in the treatment of the most malignant diseases (diphtheria, lock-jaw, typhoid fever, tuberculosis, bubonic plague) by introducing into the system a modified form (similar) of those poisons that produce them in the healthy individual. Hahnemann undoubtedly deserves the credit of being the first to break decidedly with the old school of medical practice, in which, forgetful of the teachings of Hippocrates, nature was either overlooked or rudely opposed by wrong and ungentle methods. We can scarcely now estimate the force of character and of courage which was implied in his abandoning the common lines of medicine. More than this, he and his followers showed results in the treatment of disease which compared very favourably with the results of contemporary orthodox practice.
Homoeopathy has given prominence to the therapeutical side of medicine, and has done much to stimulate the study of the physiological action of drugs. It has done service in directing more special attention to various powerful drugs, such as aconite, nux vomica, belladonna, and to the advantage of giving them in simpler forms than were common before the days of Hahnemann. But in the medical profession homoeopathy nevertheless remains under the stigma of being a dissenting sect. It has been publicly announced that if the homoeopathists would abolish the name “homoeopathy,” and remove it from their periodicals, colleges, hospitals, dispensaries and asylums, they would be received within the fold of the regular profession. These conditions have been accepted by a few homoeopathists who have become members of the most prominent medical association in the United States.
Homoeopathy as it exists to-day can, in the opinion of its adherents, stand by itself, and its progress for a century in face of prolonged and determined opposition appears to its upholders to be evidence of its truth. There are still, indeed, in both schools of medical thought, men who stand fast by their old principles. There are homoeopathists who can see nothing but evil in the practice of their brothers of the orthodox school, as there are allopathists who still regard homoeopathy as a humbug and a sham. There are, however, liberal-minded men in both schools, who look upon the adoption of any safe and efficient method of curing disease as the birthright of the true physician, and who allow every man to prescribe for his patients as his conscience may dictate, and, provided he be educated in all the collateral branches of medical science, are ready to exchange views for the good of suffering humanity.
Great Britain.—Homoeopathy is not rapidly extending in Great Britain, and its recognition has been slow. The first notice taken of the new system of therapeutics was by the Medical Society of London in 1826. In 1827 the physician of Prince Leopold of Saxe-Coburg, Dr F. H. F. Quin (1799–1878), who had previously studied homoeopathy in Germany and practised it in Italy, came to England, and it was through his efforts that the system was introduced. Three other physicians, Dr Belluomini, Dr Romani and Dr Tagliani, claimed priority, but careful research established Dr Quin’s title. Quin was a successful man professionally and socially, and brought upon himself in a short time the anathema of the Royal College of Physicians. In 1844 Dr William Henderson, professor of pathology in the university of Edinburgh, embraced the Hahnemannian system. A storm of opposition arose, and Professor J. Y. Simpson (the discoverer of chloroform anaesthesia) published a volume, with the alliterative title, Homoeopathy, its Tenets and Tendencies, Theoretical, Theological, and Therapeutical. This brochure was answered by Professor Henderson, the title of his book being Homoeopathy Fairly Represented. From 1827 to 1837 there were but a dozen practitioners of homoeopathy in London, but during 1837 to 1847 the number increased to between seventy and eighty. In 1857 there were upwards of two hundred practitioners in the kingdom, with thirty-three institutions in which the law of similars was used as a basis of practice. In 1867 the increase was not so rapid, the number being 261. A society was formed about this period for “the protection of homoeopathic practitioners and students,” which proved of great value in binding the sect together. In 1870 congresses were established, and annual meetings held, which have continued to the present time. In 1901 there were over three hundred homoeopathic physicians in the British Isles, of whom between seventy and eighty were in London alone. There were seventy-nine chemists, of whom seventeen were located in London, and eighty-two towns and cities in the country contained from one to ten homoeopathic practitioners each, together with many established chemists for dispensing homoeopathic medicines. The British Homoeopathic Society was founded by Quin in 1844, and has numerous members and fellows, besides corresponding members in all portions of the world, including Australia, India and Tasmania. The London Homoeopathic Hospital was founded in 1850, also largely through the efforts of Quin, and a few years afterwards moved to Great Ormond Street. During the cholera epidemic of 1854 the statistics of this hospital showed a mortality of 16.4%, against 51.8% of other metropolitan charities. The London Homoeopathic Hospital has a convalescent home under its management at Eastbourne. There are also dispensaries in Ealing and West Middlesex, Kensington, Notting Hill and Bayswater. Similar institutions are located in Bath, Birkenhead, Birmingham, Bootle, Bournemouth, Brighton, Bristol, Bromley, Cheltenham, Cheshire, Croydon, Dublin, Eastbourne, Edinburgh, Folkestone, Hastings and St Leonards, Ipswich, Leeds, Leicester, Liverpool, Newcastle, Northampton, Norwich, Oxford, Plymouth, Torquay, Tunbridge Wells, Weston-super-Mare. The homoeopathic journals include the Homoeopathic World, the London Homoeopathic Hospital ​ Reports, the Journal of the British Homoeopathic Society, and the British Homoeopathic Review, the last being issued by the British Homoeopathic Association, which was founded in 1902 for the purpose of developing and extending homoeopathy in Great Britain. The British Journal of Homoeopathy was first published in 1843, and was edited by Drs Drysdale, Russell and Black. For many years it was the foremost homoeopathic journal in the world. Its motto was In certis unitas, in dubiis libertas, in omnibus charitas. One reason why homoeopathy has not advanced as rapidly in the British Isles as in America is said to be the discrimination exercised against it by the General Medical Council, and another is want of cohesion amongst the homoeopaths themselves.
United States.—Homoeopathy was introduced into the United States by Dr Hans Birch Gram, who was born in Boston. His father being Danish, Gram in his eighteenth year went to Copenhagen, where he graduated in 1814. In 1823 he became acquainted with homoeopathy, and brought a knowledge of it to America in 1825 when he settled in New York. The first homoeopathic association was formed in 1833 in Philadelphia, the second in New York, 1834, and homoeopathy became known in the different states somewhat in the following order: New York, 1825; Pennsylvania, 1828; Louisiana, 1836; Connecticut, 1837; Massachusetts, 1837–1838; Maryland, 1837; Delaware, 1837; Kentucky, 1837; Vermont, 1838; Rhode Island, 1839; Ohio, 1839; New Jersey, 1840; Maine, 1840; New Hampshire, 1840; Michigan, 1841; Georgia, 1842; Wisconsin, 1842; Alabama, 1843; Illinois, 1843; Tennessee, 1844; Missouri, 1844; Texas, 1848; Minnesota, 1852; Nebraska, 1862; Colorado, 1863; Iowa, 1871. After 1871 the spread of the system was rapid throughout every state in the Union, and it is in the United States that homoeopathy principally flourishes. There are thousands of homoeopathic physicians, and their clients number several millions. It may be noted that departments of homoeopathy are connected with the universities of Boston, Michigan, Iowa, Minnesota and Kansas City.
Canada.—The early history of homoeopathy can be traced back nearly to 1850 in the province of Quebec. In the Dominion of Canada the various provinces control the licensing of physicians, excepting in Quebec, which is the only province having a separate homoeopathic board of examiners. This is under the control of the Montreal homoeopathic Association, and is known as the College of Homoeopathic Physicians and Surgeons of Montreal. Three examiners are annually appointed by the association. Successful candidates receive the diploma of the college, and are entitled to add to their degree the letters M.C.H.P.S. A certificate of successful examination is forwarded to the lieutenant-governor at Quebec, who, “if satisfied of the loyalty, integrity and good morals of the applicant, may grant him a license to practise surgery, physic and midwifery, or either of them, in the province of Quebec.” The word “loyalty” has been decided by the provincial secretary to mean a British subject. This is the only government medical license now issued in the British empire, the others being by provincial boards or colleges of physicians and surgeons. In 1894 there was no homoeopathic institution in the province; at present the Montreal Homoeopathic Hospital is in active operation. Two homoeopathic papers are published monthly—the Homoeopathic Record in Montreal, and the Homoeopathic Messenger in Toronto. In 1870, in the province of Ontario, the three schools, allopathic, homoeopathic and eclectic, united for examining purposes into one board called the medical council, seventeen members representing the old school and five the other two systems. Finally the eclectics were merged in the old school, the board appointing five of Hahnemann’s followers for examining purposes. Grace Hospital at Toronto (erected 1892) was begun as a dispensary in 1887.
Germany.—In 1810 Hahnemann published his Organon, which was the starting-point of homoeopathy in Germany. In 1811 an endeavour was made to found an institution in Leipzig in which practitioners might learn the new method of treatment theoretically and practically, but it was not a success, as the entire tide of professional opinion was against the system. In 1829, at the celebration of the fiftieth anniversary of Hahnemann’s doctorate, the German Central Society was organized, holding its first meeting in 1830. In the university hospital of Munich some experiments were made to test the efficacy of homoeopathic medicines, but these were not successful. In 1831 the government prohibited homoeopathists from dispensing their own medicines; this was a severe blow to the system. In 1834 there was a division among the homoeopathists themselves, which much retarded the progress of the school. A homoeopathic hospital was established about this time (January 1833) in Leipzig, but there was such constant wrangling among the physicians connected with it that its sphere of usefulness was curtailed, and it was finally converted into a dispensary. The Baden Homoeopathic Society was established in 1834. The homoeopathic hospital in Munich was established in 1836, but suffered a similar fate to that of Leipzig, and was converted into a dispensary. The rather equivocal success of these hospitals in Saxony and Bavaria was in direct contrast to the fate of two newly established hospitals in Austria, one in Vienna and the other in Linz, which were very successful, and aroused great interest both among physicians and laymen. During the political confusion of 1846 and 1849 there was complete stagnation of everything medical in Germany. But during all these years, though the public institutions were few, the literature on homoeopathic subjects became very extensive, and exercised a significant influence upon the system in all parts of the world. Hahnemann died in 1843, and on the 10th of August 1851 a bronze monument to him was unveiled at Leipzig. The Leipzig dispensary lived thirty-three years. From 1842 to 1874 there were treated in this institution 65,106 patients. In 1901 there were about 250 homoeopathic physicians in Germany; they appeared to be strongest at Berlin, in the province of Brandenburg, in Pomerania and Westphalia, Saxony, Hessen and in Württemberg.
Austria-Hungary.—Homoeopathy was introduced into Austria about 1817, and in 1819 its practice was forbidden by law. Shortly afterwards the physician attending the archduke John became a homoeopath. In 1825 the doctrine was introduced into Vienna. To test the efficacy of the system Francis I. ordered that experiments be made with homoeopathic medicines, and for this purpose a ward furnished with twelve beds was allotted. The results were satisfactory to the new system, and it made gigantic strides in Vienna. During the cholera epidemic of 1836 an increased impetus was given to the new school by the reported brilliant successes of the treatment. Societies were founded and journals published. In 1846 a second hospital was founded. In 1850 a third hospital was opened, and clinical lectures upon the system were delivered. In 1873 the Society of Homoeopathic Physicians was formed. Between the years 1873 and 1893 homoeopathy declined. In 1901, in thirty-seven cities and towns there were to be found about fifty physicians and two hospitals, and it was estimated that about seventy-five more were scattered in Moravia, Bohemia, Tirol, Salzburg and the coast provinces. There is a professorship of homoeopathy at the University of Budapest, and homoeopathic clinics are held at the new Rochus Hospital in Üllöi Street, and also in the homoeopathic department of the Hospital Bethesda of the Reformed Community. The Elizabeth Hospital, exclusively homoeopathic, has existed for many years.
Russia.—The homoeopathic system was introduced into Russia in 1823. In 1825 great impetus was given to the new doctrine by the conversion of Dr Bigel, physician to the grand duke Constantine. In 1829 the grand duke ordered a series of experiments to be conducted to prove the truth or fallacy of homoeopathy, and they demonstrated the success of the new school. In 1841 a hospital was established in Moscow, and in 1849 similar institutions were founded in Nizhniy-Novgorod. Since then homoeopathy has been steadily practised, and has penetrated to the remotest parts of Russia. In 1881 the civil engineers proposed to commemorate the virtues of the emperor Alexander II. by the erection of a hospital; a committee for collecting funds was created, and 58,064 roubles were handed to the Charity Society of the followers of homoeopathy at St Petersburg for the erection and founding of a homoeopathic hospital. The foundation stone of the edifice was laid on 19th June 1893, the emperor Alexander III. giving 5000 roubles. The inauguration of a new dispensary and a pharmacy took place on the 19th of April 1898, and the hospital itself, intended originally for fifty beds, was opened on the 1st of November 1898. There are sixteen free beds, three of them being in the name of the emperor Nicholas, the empress Maria Feodorovna, and the emperor Alexander III. On the 28th of January 1899 an imperial edict was issued granting the rights of public service to the doctors of the hospital and dispensaries of the Charity Society, thus placing them on an equality with the doctors of the prevailing medical school.
France.—Homoeopathy was first introduced into France in 1830 by Count de Guidi, doctor of medicine, doctor of science, and inspector of the university, who practised in Lyons. About the same year Dr Antoine Petroz, widely known by his Grand dictionnaire des sciences médicales, began practising homoeopathy in Paris, and his establishment became the headquarters of the new system there. In 1835 Hahnemann himself came to the capital. In 1832 the homoeopathic method of treating disease was introduced into the Hospice de Choisy, and in 1842 into the hospital of Carentan. Tessier practised the new doctrine in his wards in the Hospital St Marguérite, and in the Children’s Hospital up to the year 1862, when he retired. The first homoeopathic society was established in 1832 (the Société Gallicain), Hahnemann becoming president in 1835; in 1845 the Société de Médecine Homéopathique was organized; and in 1860 the two were united for the better interests of the school. In 1901 there were at Paris three hospitals—the Hospital St Jacques with fifty-five beds, the Hahnemann Hospital with thirty-five beds, and the new Protestant Hospital for Children with twenty-five-beds. At Lyons there is the Hospital St Luc. The medical journals include L’Art médical, La Revue homéopathique belge, Journal belge d’homéopathie, La Thérapeutique Intégrale, La Revue homéopathique française. In the year 1900 the medical officers of the republic having supervision over the medical department of the International Exhibition officially recognized the members of the homoeopathic school, and arranged for the proper accommodation and reception of the International Congress of Homoeopathic Physicians held in June. On the 30th of that month, with appropriate ceremonies, the remains of Hahnemann were removed from the cemetery of Montmartre and deposited in Père-la-Chaise, and a monument bearing a suitable inscription was erected to the memory of the founder of homoeopathy.
Italy.—The Austrians when they entered Naples in 1821 brought homoeopathy into Italy, the general in command of the army being ​a devoted friend of Hahnemann. In 1828 Dr Count Sebastian de Guidi came from Lyons and assisted in spreading the doctrine. During the period from 1830 to 1860 many physicians practised homoeopathy, and the literature on the subject became extensive. A homoeopathic clinic was established and a ward opened in Trinity Hospital at Naples, and a homoeopathic physician was appointed to the count of Syracuse. During the severe cholera epidemics of 1854, 1855, 1865 the success of homoeopathic treatment of that disease was so marked under the care of Dr Rubini that the attention of the authorities was directed to the system. In 1860 the homoeopathic practice was introduced into the Spedale della Cesarea, and since that period homoeopathy has been recognized with more or less favour in most of the cities. The Italian Homoeopathic Institute is recognized by royal warrant as an established institution, and its regulations are approved by the government. In Turin the legal seat of the Homoeopathic Institute, there is a hospital under the management of the State Association. The homoeopathic medical press consists of the Revista Omiopatica, established in 1855, and L’Omiopatico in Italia, the organ of the Italian Homoeopathic Institute, which first appeared in 1884.
Spain.—Homoeopathy was introduced into Spain in 1829 by a physician to the Royal Commission sent by the king of Naples to attend the marriage of Maria Christina with Don Ferdinand VII. Shortly after this, a merchant of Cadiz visited Hahnemann in Coethen, and was cured of a serious disorder; he returned to Spain with a supply of homoeopathic literature, and immediately sent a medical student to Leipzig to study the new system. In 1843 many cases of cholera were treated homoeopathically in Madrid. The civil war, which did not terminate until 1840, arrested all medical investigation in Spain, but in 1843 there still existed in Madrid five pharmacies and a number of homoeopathic physicians. About this time Dr Tosi Nuñez returned from an investigation of the new system with Hahnemann, and owing to his success in the treatment of disease was created one of the physicians of the bedchamber to the queen, who soon afterwards conferred upon him the title of marquis, with the grand crosses of the Charles III. and of the Civil Order of Beneficiencia. This recognition by high authority gave an impetus to homoeopathy which has continued ever since.
Denmark.—Homoeopathy was unknown in Denmark until the year 1821, when Hans Christian Lund, a medical practitioner, adopted it. Hahnemann, however, had been both before and after that time consulted by Danes, and consequently homoeopathic therapeutics was recognized in different parts of the country. Lund translated many of Hahnemann’s works into Danish, as well as those of other eminent members of the new school. (W. T. H.)
* ↑ An interesting controversy has been carried on between the members of the homoeopathic school as to the proper construction of the Latin motto which constitutes its acknowledged basis. For many years the verb at the conclusion of the sentence was used in the indicative mood, curantur, thus making the sentence a positive one. After extended research it has been discovered that Hahnemann himself never employed the word curantur as descriptive of his law of cure, but always wrote curentur, which greatly modifies the meaning of the phrase. If the subjunctive mood be used, the motto reads, “Let similars be treated by similars,” or “similars should be treated by similars.” The reading similia similibus curentur was officially adopted as the correct reading of the sentence by the American Institute of Homoeopathy at its session held in Atlantic City, N.J., on the 20th of June 1899; and the words are so inscribed on the monument erected to the memory of Hahnemann and unveiled in Washington, D.C., on the 23rd of June 1900, and also are those carved upon the tomb of Hahnemann in Père-la-Chaise, Paris.
* ↑ Some points of Hahnemann’s system were borrowed from previous writers—as he himself, though imperfectly, admits. Not to mention others, he was anticipated by Hippocrates, and especially by Paracelsus (1495–1541). The identical words similia similibus curantur occur in the Geneva edition (1658) of the works of Paracelsus, as a marginal heading of one of the paragraphs; and in the “Fragmenta Medica,” Op. Omnia, i. 168, 169, occurs the following passage: Simile similis cura; non contrarium. “Quisquis enim cum laude agere Medicum volet, is has nugas longe valere jubeat. Nec enim ullus unquam morbus calidus per frigida sanatus fuit, nec frigidus per calida. Simile autem suum simile frequenter curavit, scilicet Mercurius sulphur, et sulphur Mercurium; et sal ilia, velut et illa sal. Interdum quidem cum proprietate junctum frigidum sanavit calidum; sed id non factum est ratione frigidi, verum ratione naturae alterius, quam a primo illo omnino diversam facimus.” It is very remarkable that in Hahnemann’s enumeration of authors who anticipated him in regard to the doctrine of Similia, he makes no mention of the views of Paracelsus, though the very words seem to be taken from the works of that physician. The other point in Hahnemann’s doctrine—that medicines should be tried first on healthy persons—he admits to have been enunciated by Haller. Roughly it has been acted on by physicians in all ages, but certainly more systematically since Hahnemann’s time. In the most characteristic feature of Hahnemann’s practice—“the potentizing,” “dynamizing,” of medicinal substances—he appears to have been original.
* ↑ Two methods of preparing medicines are recognized, one on the decimal, the other on the centesimal scale. The pure tinctures are denominated “mother tinctures,” and represented by the Greek φ. To make a first decimal dilution or first decimal trituration, 10 drops of the mother tincture, or 10 grains of a crude substance, are mixed with 90 drops of alcohol, or 90 grains of saccharum lactis (sugar of milk) respectively. The liquid is thoroughly shaken, or the powder carefully triturated, and the bottles containing them marked 1 X, meaning first decimal dilution or trituration. To make the 2 X potency, 10 drops or 10 grains of this first dilution or trituration are mixed with 90 drops of pure alcohol, or 90 grains of milk sugar, and are succussed or triturated as above described, and marked 2 X dilution or trituration. This subdivision of particles may be continued to an indefinite degree. On the Hahnemannian or centesimal scale the medicines are prepared in the same manner, the difference being that 1 drop or grain is mixed with 99 drops or grains, to make the first centesimal, which is marked 1 c or 1 simply, and so on for the second and higher dilutions.