Nathanial Bowditch, Quoted in F Cajori, The Teaching and History of Mathematics in the United States
(LaPlace's classic, "Mecanique Celeste", is a very difficult book, and Biot, who helped him prepare it for printing said that Laplace himself would frequently get lost in following his own line of reasoning and insert, "il est aise a voir".
Bowditch translated the work and Legendre wrote of the translation, "Your work is not merely a translation with a commentary; I regard it as a new edition, augmented and improved, and such a one as might have come from the hands of the author himself, ... if he had been solicitously studious of being clear." )
The 189th day of the year;
There are 14 prime years in the 21st Century (2017 will be the third), but the 189th century would be the first to contain as few as five prime years (18803, 18839, 18859, 18869 and 18899). When is the next ?
Narayana, an Indian mathematician in the 14th century, (I believe this refers to Narayana Pandita (about 1325 – 1400)) came up with an interesting Fibonacci-like series: A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. How many cows and calves are there altogether after n years? For the 15th year, the total is 189. (How many mature and immature?)
A "magic diagram" or yantra (a geometric diagram to be used for meditation)by Narayana,
2357 is a prime number. 23357 is also prime. 233357 is also prime but 2333357 is not, and then 23333357 is; and yes, this is somehow related to the number 189. I came across a sequence on OEIS which gave "Numbers k such that (7*10^k + 71)/3 is prime." Like you may have, I wondered, "Why would someone search for primes of so unusual a sequence?" Well, if you take those exponents, and subtract 2 from them, you get the number of threes that when placed between the digit 2 and the digits 57, will produce a prime. So I can inform you today that not only is (7*10189 +71 )/3 a prime number, but that prime number is a 2 followed by 187 threes followed by 57. And you thought 189 was just some hum-drum number!!!!!
189 is the sum of consecutive cubes, 4^3 + 5^3 = 189. and also \(189 = 95^2 - 94^2 \) and it is 6^3 - 3^3; and \(189 = 15^2 - 6^2= 17^2-10^2\)
See More Math Facts for every year date here
1672 Newton's first publication is in a letter to the Philosophical Transactions: “A Serie’s of Quere’s Propounded by Mr. Isaac Newton, to be Determin’d by Experiments, Positively and Directly Concluding His New Theory of Light and Colours; and Here Recommended to the Industry of the Lovers of Experimental Philosophy, as they Were Generously Imparted to the Publisher in a Letter of the Said Mr. Newtons of July 8.1672” (Thanks to Thony Christie) *Philosophical Transactions
1680 Hooke demonstrates sound vibration at the Royal Society. This was done by putting flour on a glass plate, and bowing on the edge of glass. Hooke had observed that the motion of the glass was vibrate perpendicular to the surface of the glass, and that the circular figure of the flour changed into an oval one way, and the reciprocation of it changed it into an oval the other way. This phenomenon was rediscovered by Chladni in the eighteenth century, and given his name "Chladni figures". *Daniel P McVeigh, "An Early History of the Telephone 1664-1865"
Hooke also famously theorized that sound is a wave, and he made one of the first attempts to visualize sound vibrations. According to historical accounts, he used rotating wheels with teeth to create tones at different frequencies, helping to demonstrate that pitch is related to vibration frequency.
Hans Jenny (16 August 1904, Basel – 23 June 1972) was a natural scientist and physician who coined the term cymatics to explain the acoustic impacts of sound wave phenomena.
Chladni Plate
1706 de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for π he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result. He did so and Hermann quickly discovered a proof that Machin's series converges to π. He produced techniques that show other similar series also converge rapidly to π and he wrote on 21 August 1706 to Leibniz giving details. Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π.
In 1706 William Jones published a work Synopsis palmariorum matheseos or, A New Introduction to the Mathematics, Containing the Principles of Arithmetic and Geometry Demonstrated in a Short and Easie Method ... Designed for ... Beginners. (This is the book in which Jones first uses Pi in the mathematical sens it is now used) This contains on page 243 the following passage:-
There are various other ways of finding the lengths or areas of particular curve lines, or planes, which may very much facilitate the practice; as for instance, in the circle, the diameter is to the circumference as 1 to (16/5- 4/239) - 1/3(16/53- 4/2393) &c. = 3.14159 &c. = π. This series (among others for the same purpose, and drawn from the same principle) I received from the excellent analyst, and my much esteemed friend Mr John Machin; and by means thereof, van Ceulen's number, or that in Art. 64.38 may be examined with all desirable ease and dispatch.
Jones also reports that this formula allows π be calculated:-
... to above 100 places; as computed by the accurate and ready pen of the truly ingenious Mr John Machin. No indication is given in Jones's work, however, as to how Machin discovered his series expansion for π.
1798 After making a parachute drop from beneath a Balloon, (the first person to make a successful high-altitude parachute descent without a rigid frame, making him a pioneering figure in the history of parachuting.) André-Jacques Garnerin went on to stage regular tests and demonstrations at Parc Monceau, Paris. His demonstrations became a cause célèbre when he announced in 1798 that his next flight would include a woman as a passenger. Although the public and press were in favour, he was forced to appear in front of officials of the Central Bureau of Police to justify his project.
After a ban of several months was lifted Garnerin was ready to proceed. He advertised the ascent:
The young Citoyenne who will accompany me is delighted to see the day approach for the journey. I shall ascend with her from the Parc Monceau, some time during the next ten days.
On 8 July 1798 a large number of spectators gathered in the Parc Monceau to witness the ascent. By all accounts Citoyenne Henri was young and beautiful, and she and Garnerin took several turns around the park to the applause of the crowd before she was assisted into the basket of the balloon by the astronomer Jérôme Lalande. The balloon trip passed without incident and the journey ended at Goussainville about 30 kilometres (19 mi) to the north of Paris *Wik
Citoyenne Henri is often credited as the first woman "who ever had the courage to trust herself in the regions of air", although several women had already made ascents in a balloon: on 20 May 1784, the Marchioness and Countess of Montalembert, the Countess of Podenas and a Miss de Lagarde had taken a trip on a tethered balloon in Paris, and Élisabeth Thible had made an ascent in an untethered balloon on 4 June 1784.
Shortly after this event , on 10 November 1798, Garnerin's future wife Jeanne Geneviève Labrosse was also taken up in his balloon. Subsequently, both his wife and his niece Élisa Garnerin became well known for their parachuting feats and Madame Blanchard was celebrated for her solo ballooning, relegating the less impressive exploits of Citoyenne Henri to a footnote in history.
1831 Quetelet officially uses the term, "l'homme moyen" (the average man) in an article about the different ages at which men commit crimes. *Statistics on the Table: The History of Statistical Concepts and Methods By Stephen M. Stigler
1835 Liberty bell cracked. *VFR Or maybe not In the 1830s, the bell was adopted as a symbol by abolitionist societies, who dubbed it the "Liberty Bell". It acquired its distinctive large crack sometime in the early 19th century—a widespread story claims it cracked while ringing after the death of Chief Justice John Marshall in 1835.*Wik
1842 Francis Baily, an English astronomer, is most famous for his observations of 'Baily's beads' during an eclipse of the Sun.
His observations of "Baily's Beads", during an annular eclipse of the sun on 15 May 1836, at Inch Bonney in Roxburghshire, started the modern series of eclipse expeditions. The phenomenon, which depends upon the irregular shape of the moon's limb, was so vividly described by him as to attract an unprecedented amount of attention to the total eclipse of 8 July 1842, observed by Baily himself at Pavia. *Wik
1842 Dominique Francois Jean Arago (1786-1853), French astronomer. Studied solar eclipse of July 08, 1842 and concluded the sun exist of gas.*NSEC
1881, a patron came into Edward Berner's drug store in Two Rivers, Wisconsin, and sat down at the soda-fountain counter. Since it was the Sabbath, the customer couldn't have the desirable, but scandalous, flavored soda water. Berner compromised by putting ice cream in a dish and poured over it the chocolate syrup that was previously only served as flavoring in ice-cream sodas. That was an ice cream Sunday! The name became "sundae", after the day on which Berner served it. TIS There is limited documentation for this event. There is another claim for the creation of the ice cream sundae with stronger evidence of the story.
In 1892 in Ithaca, New York, Reverend John M. Scott and local pharmacy owner, Chester Platt, created the first sundae by topping vanilla ice cream with cherry syrup and a candied cherry at Platt's drugstore.This is supported by a dated newspaper ad from 1892 in Ithaca promoting "Cherry Sundaes."
1933 Jansky announced detection of radio radiation from galactic center.*VFR Before Jansky observed the Milky Way in the 1930s, physicists speculated that radio waves could be observed from astronomical sources. In the 1860s, James Clerk Maxwell's equations had shown that electromagnetic radiation is associated with electricity and magnetism, and could exist at any wavelength. Several attempts were made to detect radio emission from the Sun by experimenters such as Nikola Tesla and Oliver Lodge, but those attempts were unable to detect any emission due to technical limitations of their instruments.
Karl Jansky made the discovery of the first astronomical radio source serendipitously in the early 1930s. As an engineer with Bell Telephone Laboratories, he was investigating static that interfered with short wave transatlantic voice transmissions. Using a large directional antenna, Jansky noticed that his analog pen-and-paper recording system kept recording a repeating signal of unknown origin. Since the signal peaked about every 24 hours. *Wik
Jansky and his rotating directional radio antenna (early 1930s), the world's first radio telescope.*Wik
2002 Well into his mid-nineties, Donald Coxeter gives the keynote address at the Janos Bolyai Conference on Hyperbolic Geometry with a new paper on the Descartes circle theorem as extended by Phillip Beecroft. His Opening remarks:
The Absolute property of four mutally tangent circles that I am describing seems to have been discovered by Mr. Phillip Beecroft, of Hyde Academy, Cheshire, England, and published in The Lady and Gentleman's Dairy... In Beecroft's own words, "If any four circles be described to touch each other mutually, another set of four circles of mutual contact may be described whose points of contach shall coincide with those of the first four." He then proceeded to give a new and elegant proof of the 1842 theorem, which extended the four circle theorem of Descartes to be, in essence, an eight circle theorem.
*Siobhan Roberts, King of Infinite Space
1760 Christian Kramp (July 8, 1760 – May 13, 1826) was a French mathematician, who worked primarily with factorials.
As Bessel, Legendre and Gauss did, Kramp worked on the generalized factorial function which applied to non-integers. His work on factorials is independent of that of James Stirling and Vandermonde. He was the first to use the notation n! (Elements d'arithmétique universelle, 1808). In fact, the more general concept of factorial was found at the same time by Arbogast.*TIS
The notation n! was introduced by Christian Kramp (1760-1826) in 1808. In his élémens d'arithmétique universelle (1808).
An early factorial symbol, was suggested by Rev. Thomas Jarrett (1805-1882) in 1827. It occurs in a paper "On Algebraic Notation" that was printed in 1830 in the Transactions of the Cambridge Philosophical Society and it appears in 1831 in An Essay on Algebraic Development containing the Principal Expansions in Common Algebra, in the Differential and Integral Calculus and in the Calculus of Finite Differences (Cajori vol. 2, pages 69, 75). This symbol was still in use after 1960 *PBNotes
For more on the symbols and history of the factorial see here.
1777 Daniel Friedrich Hecht (8 July 1777 in Sosa – 13 March 1833 in Saxony) was a German mathematician. He was a mine manager, then a teacher and finally a professor of mathematics. He is most notable for writing high school textbooks on maths and geometry.
During the 18th century, the Kingdom of Prussia was among the first countries in the world to introduce free and generally compulsory primary education, consisting of an eight-year course of basic education, Volksschule. It provided not only the skills needed in an early industrialized world (reading, writing, and arithmetic), but also a strict education in ethics, duty, discipline and obedience. Children of affluent parents often went on to attend preparatory private schools for an additional four years, but the general population had virtually no access to secondary education.
In 1810, after the Napoleonic wars, Prussia introduced state certification requirements for teachers, which significantly raised the standard of teaching. The final examination, Abitur, was introduced in 1788, implemented in all Prussian secondary schools by 1812 and extended to all of Germany in 1871. The state also established teacher training colleges for prospective teachers in the common or elementary grades.
When the German Empire was formed in 1871, the school system became more centralized. In 1872, Prussia recognized the first separate secondary schools for females. As learned professions demanded well-educated young people, more secondary schools were established, and the state claimed the sole right to set standards and to supervise the newly established schools. *Wik
1838 Count Ferdinand von Zeppelin Germany aviation pioneer who built the first rigid dirigible airships, named Zeppelins. He patented his idea on 31 Aug 1895 and formed a company to build airships in 1898. Many thought his invention incredible, and called him "Foolish Count". His first airship took off in 2 Jul 1900 at Lake Constance, where it had been assembled in a floating assembly shed. He continued to improve the design and built a fleet of airships for commercial passenger service. During WW I, Zeppelins were used to bomb Britain beginning 19 Jan 1915 with attacks on Great Yarmouth and King's Lynn. After the war, passenger service included transatlantic flights. Zeppelin use ended after the 6 May 1937 Hindenburg fire disaster at Lakehurst, N.J., U.S.A.*TIS
First flight of the LZ 1
1949 Torsten Carleman (8 July 1892, Visseltofta, Osby Municipality – 11 January 1949, Stockholm), born Tage Gills Torsten Carleman, was a Swedish mathematician, known for his results in classical analysis and its applications. As the director of the Mittag-Leffler Institute for more than two decades, Carleman was the most influential mathematician in Sweden. *Wik
1895 Igor Yevgenyevich Tamm (8 July 1895 – 12 April 1971) Soviet physicist who shared the 1958 Nobel Prize for Physics with Pavel A. Cherenkov and Ilya M. Frank for his efforts in explaining Cherenkov radiation. Tamm was an outstanding theoretical physicist, after early researches in crystallo-optics, he evolved a method for interpreting the interaction of nuclear particles. Together with I. M. Frank, he developed the theoretical interpretation of the radiation of electrons moving through matter faster than the speed of light (the Cerenkov effect), and the theory of showers in cosmic rays. He has also contributed towards methods for the control of thermonuclear reactions. *TIS
one of my favorite math stories is from George Gamow's autobiography and is about Tamm.
"Here is a story told to me by one of my friends who was at that time
a young professor of physics in Odessa. His name was Igor Tamm (Nobel
Prize laureate in Physics, 1958). Once when he arrived in a neighboring
village, at that period when Odessa was occupied by the Reds, and was
negotiating with a villager as to how many chickens he could get for
half a dozen silver spoons, the village was captured by one of the
Makhno bands, who were roaming the country, harassing the Reds. Seeing
his city clothes (or what was left of them), the capturers [sic]
brought him to the Ataman, a bearded fellow in a tall black fur
hat with machine-gun cartridge ribbons crossed on his broad chest and
a couple of hand grenades hanging on the belt.
'You son-of-a-bitch, you Communist agitator, undermining our Mother
Ukraine! The punishment is death.'
'But no,' answered Tamm, 'I am a professor at the University of Odessa
and have come here only to get some food.'
'Rubbish!' retorted the leader. 'What kind of professor are you ?'
'I teach mathematics.'
'Mathematics?' said the Ataman. 'All right! Then give me an estimate of
the error one makes by cutting off Maclaurin's series at the nth term.
Do this, and you will go free. Fail, and you will be shot!'
Tamm could not believe his ears, since this problem belongs to a rather
special branch of higher mathematics. With a shaking hand, and under
the muzzle of the gun, he managed to work out the solution and handed
it to the Ataman.
'Correct!' said the Ataman. 'Now I see that you really are a professor.
Go home!'
Who was this man? No one will ever know. If he was not killed later, he
may well be lecturing now on higher mathematics in some Ukrainian
university."
I tell this story every other year or so to my physics students when
they cannot be bothered to remember the form of the remainder in Taylor
expansions....
1897 Sir Austin Bradford Hill CBE (8 July 1897 – 18 April 1991) was an English epidemiologist who pioneered the modern randomised clinical trial and, together with Richard Doll, demonstrated the connection between cigarette smoking and lung cancer. Hill is widely known for pioneering the "Bradford Hill" criteria for determining a causal association.
In 1922, Hill went to work for the Industry Fatigue Research Board. He was associated with the medical statistician Major Greenwood and, to improve his statistical knowledge, Hill attended lectures by Karl Pearson. When Greenwood accepted a chair at the newly formed London School of Hygiene and Tropical Medicine, Hill moved with him, becoming Reader in Epidemiology and Vital Statistics in 1933 and Professor of Medical Statistics in 1947.
Hill had a distinguished career in research and teaching and as author of a very successful textbook, Principles of Medical Statistics, but he is famous for two landmark studies. He was the statistician on the Medical Research Council Streptomycin in Tuberculosis Trials Committee and their study evaluating the use of streptomycin in treating tuberculosis, is generally accepted as the first modern randomised clinical trial. The use of randomisation in agricultural experiments had been pioneered by Ronald Aylmer Fisher. The second study was rather a series of studies with Richard Doll on smoking and lung cancer. The first paper, published in 1950, was a case-control study comparing lung cancer patients with matched controls. Doll and Hill also started a long-term prospective study of smoking and health. This was an investigation of the smoking habits and health of 40,701 British doctors for several years (British doctors study).
On Hill's death in 1991, Peter Armitage wrote, "to anyone involved in medical statistics, epidemiology or public health, Bradford Hill was quite simply the world's leading medical statistician."
1904 Henri (-Paul) Cartan, (July 8, 1904 – August 13, 2008) mathematician born in Nancy, France. His father, Elie Cartan, was also a mathematician. Henri made fundamental advances in the theory of analytic functions, worked on the theory of sheaves, homological theory, algebraic topology and potential theory. Along with others, such as Weil and Dieudonné, Henri Cartan wrote under the name Bourbaki. Bourbaki's Eléments de mathématique contains more than 30 volumes and aims to present mathematics so as to illustrate the axiomatic structure of modern mathematics. *TIS
1915 Kenneth O. May (July 8, 1915, Portland, Or. – December 1,1977) was an American mathematician and historian of mathematics, who developed May's theorem [In a two-candidate election with an odd number of voters, majority rule is the only voting system that is anonymous, neutral, and monotone, and that avoids the possibilites of ties.] The Kenneth O. May Prize is awarded for outstanding contributions to the history of mathematics. Ken May established Historia Mathematica, and preserved it by separating it from its creator, "The distinguished predecessors of HM were associated with their founders and died with them. If HM is to avoid this fate, we must prepare and carry through a prompt transfer of editorial responsibility to younger hands." His list of publications numbers above 300. *Henry S. Tropp, E'loge, Isis 70, Sept 1979, Pgs 419-422
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1390 Albert of Saxony (Latin: Albertus de Saxonia; c. 1320 – 8 July 1390) was a German philosopher and mathematician known for his contributions to logic and physics. He was bishop of Halberstadt from 1366 until his death. He was bishop of Halberstadt from 1366 until his death.. He wrote an excellent logic text and published two works on squaring the circle. *VFR ert was born at Rickensdorf near Helmstedt, the son of a farmer in a small village; but because of his talent, he was sent to study at the University of Prague and the University of Paris.
At Paris, he became a master of arts (a professor), and held this post from 1351 until 1362. In 1353, he was rector of the University of Paris. After 1362, Albert went to the court of Pope Urban V in Avignon as an envoy of Rudolf IV, Duke of Austria, in order to negotiate the founding of the University of Vienna. The negotiations were successful, and Albert became the first rector of this University in 1365.
In 1366, Albert was elected bishop of Halberstadt (counted as Albert III), Halberstadt being the diocese in which he was born. As Bishop of Halberstadt, he allied himself with Magnus with the Necklace, Duke of Brunswick-Lüneburg, against Gebhard of Berg, Bishop of Hildesheim, and was taken prisoner by Gebhard in the battle of Dinckler in 1367.
He died at Halberstadt in 1390.*Wik
1695 Christiaan Huygens (14 April 1629 – 8 July 1695) Dutch mathematician, astronomer, and physicist, who founded the wave theory of light, discovered the true shape of the rings of Saturn, and contributed to the science of dynamics - the study of the action of forces on bodies. Using a lens he ground for himself, on 25 Mar 1655, he discovered the first moon of Saturn, later named Titan. In 1656, he patented the first pendulum clock, which he developed to enable exact time measurement while observing the heavens. Huygens studied the relation of the length of a pendulum to its period of oscillation (1673) and stated theories on centrifugal force in circular motion which influenced Sir Isaac Newton in formulating his Law of Gravity. Huygens also studied and drew the first maps of Mars. On 14 Jan 2005, a NASA space probe, named after Huygens, landed on Titan. *TIS
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| *Mars Solar Cap by Huygens |
1902 John Daniel Runkle (October 11, 1822 – July 8, 1902) was a U.S. educator and mathematician. B.S. in mathematics, 1851, Harvard College, second president of the Massachusetts Institute of Technology, was associated with the Nautical Almanac computation project from 1849 to 1884. In 1858 he founded the journal Mathematical Monthly and edited it for three years, when publication ceased. In 1860 he was a member of the committee that prepared the “Objects and Plan of an Institute of Technology” which led to the establishment of MIT. In 1862 he became MIT’s first secretary, and in 1865 he joined the new faculty as professor of mathematics, where he remained until 1902. He served as president pro-tem, 1868-1870, and was MIT’s second president, 1870-1878. He was married to Catherine Robbins Bird Runkle. *MIT History
1971 Kurt Werner Friedrich Reidemeister (October 13, 1893 – July 8, 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
He received his doctorate in 1921 with a thesis in algebraic number theory at the University of Hamburg under the supervision of Erich Hecke. In 1923 he was appointed assistant professor at the University of Vienna. While there he became familiar with the work of Hans Hahn and Wilhelm Wirtinger. In 1925 he became full professor at University of Königsberg, where he stayed until 1933, when he was forced to leave because of his opposition of the Nazis.
Reidemeister's interests were mainly in combinatorial group theory, combinatorial topology, geometric group theory, and the foundations of geometry. His books include Knoten und Gruppen (1926), Einführung in die kombinatorische Topologie (1932), and Knotentheorie (1932). He was the brother of Marie Neurath.*Wik
1979 Shin'ichiro Tomonaga (March 31, 1906 – July 8, 1979) Japanese physicist who shared the Nobel Prize for Physics in 1965 (with Richard P. Feynman and Julian S. Schwinger of the U.S.) for independently developing basic principles of quantum electrodynamics. He was one of the first to apply quantum theory to subatomic particles with very high energies. Tomonaga began with an analysis of intermediate coupling - the idea that interactions between two particles take place through the exchange of a third (virtual particle), like one ship affecting another by firing a cannonball. He used this concept to develop a quantum field theory (1941-43) that was consistent with the theory of special relativity. WW II delayed news of his work. Meanwhile, Feynman and Schwinger published their own independent solutions.
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2008 Sixto Ríos García (January 4, 1913; Pelahustán, Toledo - July 8, 2008; Madrid,) was a Spanish mathematician, known as the father of Spanish statistics.
He has held the positions of Director of the School of Statistics at the University of Madrid, Director of the Institute for Operations Research and Statistics CSIC, Director, Department of Statistics, Faculty of Mathematical Sciences at the Complutense University and President of the Spanish Society Operational Research, Statistics and Information. It was academic correspondent of the National Academy of Sciences of Buenos Aires, and organizer and founder, commissioned by Unesco, School of Statistics, University of Caracas. He was a member of the drafting committee of Statistical Abstracts and fellow of the International Statistical Institute and the Institute of Mathematical Statistics. Wik-ES
2010 David Harold Blackwell (April 24, 1919 – July 8, 2010) was Professor Emeritus of Statistics at the University of California, Berkeley, and is one of the eponyms of the Rao–Blackwell theorem. Born in Centralia, Illinois, he was the first African American inducted into the National Academy of Sciences, and the first black tenured faculty member at UC Berkeley.
Blackwell was also a pioneer in textbook writing. He wrote one of the first Bayesian statistics textbooks, his 1969 Basic Statistics. By the time he retired, he had published over 90 papers and books on dynamic programming, game theory, and mathematical statistics.
He was President of the Institute of Mathematical Statistics, in 1956 *Wik
Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell
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