Two Letters, Written by Dr. John Wallis to the Publisher; One, Concerning the Variety of the Annual High-Tides In Respect to Several Places: The Other, Concerning Some Mistakes of a Book Entitaled SPECIMINA MATHEMATICA Francisci Dulaurens, Especially Touching a Certain Probleme, Affirm'd to Have Been Proposed by Dr. Wallis to the Mathematicians of All Europe, for a Solution

A LETTER. Written by Dr. John Wallis to the Publisher, concerning the Variety of the Annual High-Tides, as to several places; with respect to his own Hypothesis, deliver'd No. 16, touching the Flux and Reflux of the Sea. Sir, In my Hypothesis for Tydes, you may remember, that I cast the Annual High-Tides not on the Two Equinoxes, about the 21st of March and September; nor yet on the Apogeeum and Perigeeum of the Sun, about the middle of June and December; but (as proceeding from a Complication of those two Causes) on a Middle time between the Perigeeum and the two Equinoxes, (like as is the greatest Inequality of the Natural days, proceeding from a Complication of the same Causes.) And particularly, for the Coast of Kent (and consequently the Rivers of Thames and Medway) about the beginning of November and February: which agrees with Observations on those Coasts, and particularly with that of yours of Febr. 5th this year. The last year, when I was present in the R. Society, I remember, an account was brought us of the Annual High-Tides on the Severn, and at Chepstow-bridge, to be about the beginning of March, and the end of September. Which though they agree not with the particular times on the coast of Kent, yet in the general they agree thus far, That the one is about as much before the one Equinox, as the other is after the other Equinox. You now acquaint me with High-Tides about February 22nd, about the coast of Plymouth, which is later than that of the coast of Kent, but sooner than that on the Severn. And I doubt not but in other parts of the world will be found other Varieties. The reasons of these Varieties are (as I have formerly signified) to be attributed to the particular Position of those parts, rather than to the general Hypothesis. Of which this, in brief, may serve for some account at present. The General Hypothesis of the Earth's diurnal Motion from West to East, would cast that of the Waters, not following so fast, from East to West; which causeth the constant Current within the Tropicks, where the Circles are greatest, west-ward from the Coast of Africa. Africa to that of America, (which is also the Cause of the constant Eastern Brize blowing in those parts.) But the Sea thus beating on the Coast of America, is cast back as with an Eddy on either hand, and consequently returns from the American shore East-ward towards the Coast of Europe; where, the Parallel Circles to the Equator being less, and consequently the Diurnal Motion slower, doth not cast the waters so strongly West-wards, as between the Tropicks, and so not strong enough to overcome the Eddy, which it meets with from the other Motion, which gives the Sea a North-Eastly Motion (on these Coasts) as to its usuall course. The Current therefore of our Seas being North-Eastly, we are next to consider, at what times it runs more to the North, and at what more to the East. When it runs most Northerly, it runs up the Irish Sea, and so up the Severn: When most Easterly, it runs straight up the Channel, and so to the Coast of Kent: When between these, it beats against Devonshire and Cornwall, and those parts. We are therefore to consider (as to the Annuall periods) that the Annuall Motion of the Earth in the Zodiack, and the Diurnal in the Equator, are not precisely in the same direction, but make an Angle of $23\frac{1}{2}$ deg. at the Equinoxes; but run, as it were, parallel at the Solstices: And as they be nearer, or farther from these points, so is the Inclination varied. Which several directions of Motion, do cause the Compound Motion of both to vary from the East and West more or less, according as the Sun's Position is farther or nearer the Solstices. And therefore, nearer to the Equinoxes, this Inclination doth cast the Constant current of our Seas more to the North and South; and further from it, more to the East and West. Which is the reason why the Current up the Irish Sea is nearer to the Equinoxes (at the beginning of March and end of September) and up the Channel or Narrow Seas, farther from it (at the beginning of February and of November;) and against the Coasts of Devonshire and thereabout, at some intermediate time. And thus much I thought fit to signifie upon this occasion. Dat. Oxford the 7. of March An. 1667. Aaaa Another Another Letter Written by the same Hand, concerning some Mistakes, to be found in a Book lately publish'd under the Title of SPECIMINA MATHEMATICA Francisci Du Laurens, especially touching a cer- tain Probleme, affirm'd to have been proposed by Dr. Wallis, to the Mathematicians of all Europe, to solve it. Accipi (V.C.) ante quatriduum, quem mihi misisti Francisci Du Lau- rens Tractatum, cui titulus, SPECIMINA MATHEMATI- CA, &c., eumq; mox evolvi, quo Tibi possum (quod petis) quid de eo sentiam, paucis ostendere. Videtur autem plus fronde polliceri, quam opere absolutum. Prioris libri pars magna, ex Oughtredi meisque scriptis (utut necntrius ibi meminerit) videtur desumpta, idque tam manifeste, ut non modo peculiares loquendi formulas, sed & ipsa symbola Notaeque paetim retineat. Posterioris, non parum ex Vieta, Schotenio, aliisque ab eo editis (quorum & subinde meminit) desumptum. Occurrunt inibi aliqua parum sana, & minime ac- curata multo plura. Quamam autem sint illa Genuina Principia, Veraque Geometriae Elementa, hucusque nondum tradita, qua Titulus pollicetur, non reperio: Longeque diversimode Hic & Ego sentimus, dum pag. 141. Neminem esse, opinatur, qui haec sua non praefert insenti Euclideorum Elementorum Multitudini. In calce, manifestam mihi facit injuriam, ea de me affirmans, qua vera utique non sunt. Appendicem quippe subjungit, cui speciosum hunc fecit Titu- lum, Solutio Problematis, à D. Wallisi totius Europe Mathematicis pro- positi, sed prius ad generale revocati, A. MDCLIII. eodem tempore, quo propositum erat. Post Titulum, haec sequentur. Problema D. Wallisi, Datis Ellipses * maximis Diametris, tum puncto in transversa ejus Diame- tro assignato, reperiendi in numeris segmenta lineae * Pro, Ellipses, errore intra Ellipsim terminatae, & per datum punctum Typographi, sine dubio transeuntes, atque datum angulum cum dicta diame- tro facientis. Verum quia propositae Questionis solutio æque facilis est in numeris, ac in lineis (ut postea apparebit) melius facturum me judicavi, si prius de- monstrationem Analyticam hic afferrem, ex qua tum Numerica, tum Geo- metrica sequeretur, ad problematis solutionem pertinens, effectio. At- que ut haec solutio cum fænore detur, speciale D. Wallisi problema ad gene- rale sic revoco. (Postque hanc Præstationem Problema sequitur tanquam suis verbis expo- situm cum sua ejusdem solutione per septem continuas paginas.) Ad qua haec dicenda nunc habeo. 1. Totius Europæ Mathematicos, ob rem hujusmodi, in arenam vocare, ja- rantie genus est, cujus ego hactenus non fui, (credo) nec futurus. 2. Si libuisse (ostentandi gratia) sic fecisse, legisset certe quod vel ma- joris esset difficulatis, vel majoris momenti, Problema, quam hoc esse videtur, * apote quod mediocris Algebrista, primo intuitu, semihora spacio facile solvet. 3. Nec sane hoc Problema, nec quod huic equipollcat, unquam Ego (quod memini) ulli mortalium, neque totius Europae Mathematicis, propusui, (neceo an ulli unquam propositurus:) nec quicquam hujus, quod de me persuaderi sibi passus est, verum est. 4. Erat quidem aliquando Problema huic non prorsus absimile mihi propositum (cujus & solutionem protinus expediebam) sed à me propositum memini, quod quatenus me spectare posse, video in Epistola quadam mea, ad Nobiliss. Vice-Comitem Brounker data Maij II. 1658. (quem annum inuit D. Dulauens) eoque anno in meo Commercio Epistolico p. 171. typis vulgata, in hac verba: Sub initium Februarii jam proxime elapsi, amicum non-nemo, cui forte occurrebam sero vesperi, quaestionem sequentem mihi porrexit in scriptis, quam jam nuperrime intelligo typis vulgatis esse cum hac Epigraphie; Speciatissimos viros, Matheos Professores, & alios praecarios in Anglia Mathematicos, ut Problema solvere dignentur, Jean de Montfort maxim me desiderat. "Extremis Ellipseos Diametris, distantiæ centri ab aliquo puncto in Axii transverso, ubi linea eundem secet sub angulo dato, in numeris datis; segmenta ejusdem lineæ (si opus est) productæ, & intra transversum Axem & Ellipsin terminata, in numeris invenire. Hanc Ego quaestionem, suam ratus (neque enim vel innuebat Ille, vel Ego tum sciscitabam cujus erat,) paulo adhuc universalius expositam, sub ea fere, qua subest, forma (neque enim ipsissima verba memini) postero mane solvebam: Nec eram de illa ultra solicitus (quippe qua nec magna videbatur difficultatis, nec momenti,) quam etiam, ut nunc audio, variis variis modis solvebant, ut ut eorum solutiones nondum viderim. (Ae deinde sequitur mea istius Problematis, universalius adhuc propusiti, solutio, cum annexa demonstratione, brevis & perspicua; saltem si excipias praelibata, qua tamen qui hæc intelligit, facile restituet.) Atque hoc omne illud est, quod Ego de hæc Problemate fecerim, quo idquatenus ad me spectare videatur. Num autem hoc sit (quod vult D. Dulauens) Problema illud totius Europæ Mathematicis, a Me propositum esse; Ego nihilbet judicandum permitto, qui Latina intelligit, utcumque scribit Mathematicos ignarus. Qui illud mihi monstrabat Problema (scriptum primò, postea typis impressum) est Dr. Richardus Rawlison. Quis autem fuerit ille Jean de Montfort qui praefuit ignorantibus autem impressa Chartula Londini tum temporibus sat p. o. labus, pluribusque Mathematicos peritis propusit, cujus Problema exhibens ex Galia delatum, quod & ex nostris antiquis Londinis solvebant: Quorum unus (Dr. Christoph. Wren, tunc quidem in Collegio Greshamensi Londini, nunc, Oxonii Astronomiae Professor) solutionem eam typis editam publici juris fecit: simulque (in eadem charta) reposuit Problema aiunt, quod ipse praestantissimis in Gallia Mathematicis (uti illud indic nobis in Anglia) solvendum proposuit; quod illorum nemo (quod seiam) habentem solutionem dedit. Sua vero ista solutio ex est in meo de Cycloide Tractatu. p. 71-73. Cum itaque stat hæc omnia (quod dictatur) publica & historia, non pejsum non mirari, quo animo D. Dulauens palam & in publicum ederet rem tan ab omnibus innotescen.