An apollogie confirmation explanation and addition to the Vranicall astrolabe by the author.

¶ An Apollogie confirmation explanation and Addition to the Vranicall Astrolabe by the Author .

FIrst for Apollogie and Confirmation of the thinge I doe not see but I may well ( without flattering or ouermuch affecting my owne doinges auouch it to be the best & rarest deuise for an Astrolabe that euer came forth in any age , & to most vse & pleasure , and do wish any man that thinketh hee can contradict it to doe it in writing that I may eyther yeild or aunswere . For , to backbyte or detract by wordes any man that hath openly published himselfe by writing is neyther honest nor credible , rather it bewrayeth himselfe to bee eyther partiall , malicious , ignorante , or vaineglorious . But to the matter I say for an Astrolabe not comparing it ( though the deuise be more new & rare ) with the Catholicon or Iewell , which contayneth in it the seuerall vses , not onely of all Astrolabes but also of the Sphere and Globes themselues , and of all other Instruments both Astronomicall and Geometricall , being fitted accordingly as hereafter if God permit I will yet more manifest . Therefore very vnaptly by Gemma Frisius named Astrolabum Catholicon had he not added thereunto in his title . In quo quicquid vspiam rerum mathematicarum tradi possit continetur &c. And for that cause I hauing amplyfied the deuise both in fashion & vse , named it the Mathematicall Iewell But to fetsorth an Astrolabe : and no more but an Astrolabe : then I say , that this and no other is Astrolabium Catholicon or Generale . Which must needes be confessed , if with Messahala we do but define the word Astrolabe who ( as Io. Stophlerus reporteth ) sayth , that Astrolabium est nomen grecum cuius Interpretatio est acceptio stellarum . In latin it is called ( as Stophler saith ) Astrolat sus : quasi astrorum casus vel lapsus , vel quia metitur lapsus vel transitus astrorumet signorum . Wherefore seeing it appeareth by the diffinition of the name : that the very Intendement of the deuise of the Astrolabe , was but to comprehende the lapses , courses , risings , and settings of the signes , starrs , and planets , vnto the sundry Horizons in seuerall countries and latitudes : and no further . All which this my Vranicall Astrolabe so effectually , so plausibly , so pleasingly , so easily , and so suffitiently performeth . That ( the round globe it selfe excepted ) neyther he Iewell , neither any other deuise of Astrolabe either generall or particular , of which many haue been , and with great ioy and applausure heretofore receiued , commeth any way neere it . Therefore I can not see howe or with what face any man can iustly detract from it . Let me here examine the matter publicquely , and to make short & come speedily to the point omitting such horoscopes as Apian & others published with their cylinders shipps , rings & many other learned Ludicra & deuises in dyalls framed for the rising and setting of the signes & starrs , and leauing out such Astrolabes as old Sconer and diuers other did set foorth which weare neuer much accompted of , and straight iumpe in with that high commended and so euer wel accepted Astrolabe , euen the very foundation of al that haue been since which Io. Stophlerus last established in his booke de Fabrica et vsu Astrolabij . euen that Astrolabe without which on the backside , Gemma Frisius woulde not allowe his Catholicon currant : although in my Iewell I supply it on the face , and that more generall . Well then , if you throughly consider of , or looke into that so excellent deuise : whether to a particular Horizon as Io. Stophlerus framed him : or generall to all Horizons as G. Frisius appointed him : or more general to al Horizons & both Poles , as my selfe in my Iewell deuised him : yet by all those deuises , in regard that of a 1000. starts & vpwards wel known & recorded by Images in the heauens conceiued , there could be no vse of aboue 40. or 50. of them . & that vse halfe purblind for want of the Images . And that in this Vranicall Astrolabe there is equal vse of the whole trope & of euery Image & constellation in his full proportion , as plaine as the day to be feene : a few towards the South pole which with vs are neuer seen excepted : and that many more Astrolabicall propositiones are to be performed thereon , generall to al Horizons , then before could be done by that of Stophlers to one Horizon . Therefore I can not but most confidently defend , confirme , and preferr the deuise thereof before them all in his kinde . And adding another face thereunto , for those few constellations wanting & considering the small charge , the easie portabillity & the readines of 〈◊〉 ( for which 3. causes the Astrolabe was first deuised & euer accompted of ) it might seeme a deuise now sprung vp to put downe the globe it selfe . Add to these my newe deuise of the Horarlum Planetarium on the top theareof , many waies before attempted , yet neuer made generall before : also the deuise in the two side semicircles ( wheare once I had thought to haue placed the said south constellations wanting ) for equating & rectifiyng of the fixed starres , for all ages , past and to come : neuer ioyned to any other Astrolabe . These I hope may satisfie for an Appollogie & confirmation of the deuise : yf not : Sure I am , they maye make you looke a Tearme or two longer for my Organum Vranicum than I promised , because I meane once againe to peruse & write ouer the coppy perceauing by this that the Incorporating of th examples with the precepts , which I vsed here and thought hereafter more to vse for breuity of wryting , wil make the more prolixitie of understanding for yong practisers . Vnto whom , my harsh loose & deliuer of that I conceaue , ( nature affoording me no better ) I perceiue is of it selfe hard inough : which as I am sorie for , so haue I been alwaies willing , as I haue still offered & that gratis , viua voce to remedie it , as diuers by experience can testifie .

THen for Addition to the parts of this Astrolabe , let the Mater or Celestiall thereof remaine as at the first , onely I thinke good in steede of the letters I. F.M.A &c , for more readines to write the words Ianuarie , Februarie , March &c. vnto the same places of the Ecliptick circle whereas the Sonne entereth the 12. Monethes of the yeare .

Let the Zenitfer also remaine as before sauing that , I would not haue the hower-circle thereon discribed , neyther any numbers wtitten in capitall letters , because they are combersom , but in steede of them smaller figures set vpright the standing way as in this figure of the Almcantifer you see done from 50. to 70. and those are needefull , but in the Noone lyne for the double numbring from the centre out-wards , but in the North line there neede not any such .

Let the Cursor now carrie one quadrant of 90 deg. feruing chieflye for the Azimuthes , and therefore called the Azimuthall quadrant , numbred double vz. vpwards and downewards , and another of 6. howers double numbred also : but vpwards , al one way , called the hower quadrant : all which by his figure marked with A B C D. you may here see whose line A D is his fiducyall line which must alwayes run verye square crosse the Zenitfer as in the 1. Chap. is shewed , his line A B keeping in that motion euen with the fiducyall line of the Zenitfer , at whose point A must be a centre loope hole for the Almicantifer to play vpon . And at B let be another centre loope hole with a loope of thread , or a hooke alwaies fixed therein to hitch a thread and perle vnto , when neede requireth .

Also I would nowe haue euen for ease as much as for vse three narrowe Cursors O P & Q which may bee called lydgers rather then Cursors , to be planted for all questions of one latit . euen fixed on the Zenitfer : the one at the verticall point in the Noonelyne , the other at the Horozontall poynte in the North lyne , the other at the Azimuthall poynt in the North line also , which poynte is alwayes twise so many radiall degrees from the centre as the latitude of your place amounteth vnto and was not mentioned any where in the Booke : and the Verticall and Azimuthall lydgers to haue each a centre loone hole to hitch a thread and pearle vnto , when cause requireth .

NOw to the Explanation which methinks should not nede any vntil past the first 12 chapters because those containe nothing but what is common in euery Astrolabe in a manner : most easie to be vnderstood of any that hath read but the first principles of the globe or speare . For al the Excentrick circles of the Mater , vz. such as are not described on the centre of the Celestiall or Mater of the Astolabe , are no other ( as in the 2 cap. is sayd ) but th ecliptick circle of the Zodiack marked with BC , with euery fift of his circles of longitude , distinguishing the whole face into the 12 signes & their parts , yssuing from his poles : and euery fift of his parallel circles of latitude , passing on ech side towards ech of his poles , or so many & so much of them as the proiectment wold allow , which admitteth but one of the Zodiacks poles within it marked with the letter D. And theron is only the Aequinost . circle described concentrick cutting th ecliptick at the beginnings of Aries & Libra at B & C , whose numbers & deuifiōs because they should not disturbe the Mater , are set in the outer limbe thereof : where note that B standeth at the tip of the taile of Cetus , C. in the right wing of Virgo , D vnder the foote of Draco though some hath said there be no such letters . And now , who can not easily by the squares made with the crossings of those circles of long . & latit . easily descrye the longitudes & latitudes of any star , carect , or constellation thereon set or drawen as in the 2 cap. is taught . And seing the centre of the Astrolabe marked with A , representeth the north pole of the world about which the Zenitfer moueth . The radiall degrees of whose nooneline are numbered from th equinoctials intersection , where the ✚ standerh either way towards ech pole . Who can be so ignorant but to know that by the leading about thereof by the degrees of the Maters limbe , that his fiducyall or noone-line maye shewe the 360 Meridians of the globe , though to auoyde confusion of lines they be not drawen on the Astrolabe , & that his radial degrees do lymit out the parallelles of latitude or declination from th equinoctiall by that leading about .

But to come vnto the residue of the booke wherein our newe deuised tackling is applyed to so excellent vses generall to all Horizons : which as it seemeth hath more want of Explanation . The first thing you haue now to do when any question is proposed , is to set the Index of the hower circle vnto the place of the ☉ , for the day or time of your question by helpe of the 5 cap. the & noneline , that is to say bring the nooneline of the Zenitfer vnto the degree of th ecliptick aunswering your daye proposed , and then apply the Index of the hower-circle euen therewith , so is the hower circle planted for that day , and in like sort you shall from day to daye by helpe of the nooneline remoue the Index further & further for euery day a degree rebating the odd minuts in the 5 cap. mentioned . So shall the nooneline now shew you still theron the hower in euery question vnasked as they say : without labelling to the son , as in most chapters of the booke was vsed , although he that desireth variety may now vse both : For the Amicantifer labelled from the centre of the Astrolabe vnto the Sonnes place in th ecliptick as in the booke is vsed , either it selfe or one of his three crosse finnes shall shew yow the hower in the hower quadrant of the Cursor , so you take those hower numbers which fit vnto the direction of the fin written thereon : that is to say , if the Matutine or Pomeridian fin come thereon then the numbers from 1 to 6 must serue : if the Antemeridian or Vespertine then those from 6 to 12.

The second thing most materiall to be done is as in the 10 & 17 cap. I indeuoured to shew to chuse out on the Zenitfer the pointes & lines appertaining to your latitude , Horizon , or place of your abode , & to prepare him with his furniture to the Horizon of that place : Which hauing nowe foure Cursors you shall doe & vnderstand I hope with ease . And for more facilitie as well in working as numbering , you shall reiect the odd minutes of your latitude , if any happen , and worke vpon the neerest entire degree as in all auntient Astrolabes the vse euer was vntill practise hath made you more perfect for the minutes . For example , to prepare the Zenitfer to the latitude & Horizon of Reading , where the latit . or poles eleuation is by the 10 cap. to be found about 51 2/3 deg. you shall in steed of 51 2/3 take 52 deg. and accompt that for our latit . neglecting the od 20 min. which for common vse wil hurt little . Wherefore you shall seeke out this latit . 52 amonge the radiall degrees of the nooneline , where I haue in this figure set E , & there to plant the fiducial line of your Verticall lidger in the former figure marked with O. Then likewise seke the same 52 latit . among the radial deg. of the North line where I haue set R , & there to plant the Horizontall lidger Q. And between those 2 lidgers thus placed you shal euer find 90. radial deg. of the Zenitfer as if you tel them with a stick as they say you may easily try , now the 20 min : be reiected , & those 90 ra . deg. are the 90 intersections of the 90 Almicantares with the North quadrant of the Meridian in euery latit . & therfore I cal them the North deg. Then seke out twise 52 deg. vz. 104 degrees in the Northline reckoned from the centre outwards which will fal out at 14 deg. without the ✚ , where I haue set X and there plante the Azimuthall lydger P. And these 3 thus planted shall neuer be stirred so long as you keepe in the 52 latit . Then to proceede further if now you follow the fiducyall line of the Verticall lydger thus planted crosse the Zenitfer from E , you shal perceiue it of it selfe , to finde out & touch the Zenith point of your latitude , that is to say , the innermost end of that Zenith line which appertaineth to the 52 latit . marked with F , whose outer end is marked with B , this line B F is the Zenith line of the 52 latit . & all the rest of his fellowes are of no vse in that country , yet is he ioyntly with them deuided by the crosse slope lines which you see into 90 parts , though vnequall which are as in the 1 cap. is sayd , the 90 Almicantar roots that is to say out of these 90 roots seuerally doth spring the 90 seuerall Almicantares or circles of altitude of the 52 latit . which roots are easily reckoned by the vpright numbers set in the left margent vnto euery 10 of them . And because these roots begin from the outer-most ends , & thence are numbered inwards , therefore the point B being the outer end of the 52 Zenith line , is the roote of the first or 0 Almicantare of this 52 latit . which in dede is alwaies the Horizon it selfe . For the Almicantares begin from the Horizon & thence grow lesse & lesse til they come to nothing at the Zenith , as the parallells grow from th equinoctiall to the pole , where they end at 90 , so that euer the Horizon circle is the first or 0 Almicantare . This wel vnderstood them shal you set the fiducial line A D of the Cursor euen with the point B on the Zenitfer & there plant him fast with a screw pin , if your work be of mettal & so shal he cut the noneline at A , thence apply the fiducial line G F of the Almicantifer in the former figure euen with the noone line , & theron slide downe the finger of the pointer til it touch the Horizontall lydger at the point X there screw fast the pointer also . Lastly you shall from the point X where the Azimuthall lidger resteth extend his thread euen with the noone line , & there slide vp the perle til it touch the Verticall lidger at E , & there knit the pearle fast with a knot neuer to be stirred in the 52 latit . And so at the last ( your three lidgers thus placed at E R & X & the Cursor to B A & the pointer distant from A according to the quantitie A R & the perle from X according to X A ) your Zenitfer is prepared to the Horizon of this 52 latitude , which being done ( once for all as they say ) performeth al the Chapters of the booke without any altering or remouing , some 2 or 3 Chapters excepted , for which the Cursor onely is to be remoued but none of the rest , and therefore in regarde thereof the labour is short though it seeme long .

The Zenitfer thus prepared to the Horizon of the 52 latitude & the Howercircles Index set to the ☉ as before , for the day and time proposed , you may then presently euen with one bare inspection see almost halfe the conclusions of the booke performed without any labour at all more then to apply the nooneline vnto that planet carrect or starre which you meane to worke vpon .

Example : the 5 day of Marche 1598. proceeding as in the 5. cap. is taught , you shall finde the ☉ his place in th ecliptick 25 degr. 3 min. in ♓ thereto dyrect the Index of the hower-circle by helpe of the Noone-lyne . Where let him rest fixed for all that day , so haue we no more troble with that part . And now will we here worke vppon the great starre of the first magnitude which you see in the face of Taurus commonly called Palilicium Aldebaram or Oculus tauri to knowe his state in the heauens . His longitude is easily descryed as in the 2 cap. is shewed by the circles described on the Celestiall , to be about 4 deg. & an halfe , and his latitude in like sort to bee almost 5 degrees Southwards from th ecliptick , for all the Starres which haue North latitudes , euen the one halfe of the globe are in this Astrolabe comprehended within the Zodiacke .

Then do but apply the Noone line of the Zenitfer as before prepared , vnto the middle pricke or centre of this starre Oculus tauri , it shall there shew you in the Maters limbe 62. deg. and better his right ascention and in th eclipticke circle foure degrees and better in ♊ his deg. of culminacon as in the 8 cap. and in the hower circle almost halfe an hower past 4 of the clock after-noone the time when as hee shall that day culminate or come vnto the meridian in the South part , as in the 16 cap. And thus much the bare line sheweth you . Now do but marke what point or radiall degree of the nooneline cutteth vppon the middle pricke of the Starre , & that point shall play as good a part , for if you reckon that point or degree according to the numbers set on the noone line , you shall find it 16 deg. and better , and so much is the declination of Oculus Tauri as in the 8 cap. and that Northwards from th equinoctiall circle because hee is within it . Then reckon by the radiall de . on the nooneline counting them from the vertical lidger ( set to the latitude as before ) til you com to that 16 degree ouer the cnetre of the star , you shall finde them amount to 36 , & so much wil his distance from our Zenith be , when he commeth to the meridian , which 36 deg. taken out of 90 leaueth 54 degrees his Meridian altitude as in the 16 cap. And if you turne about the Zenitfer till the North line of Media nox come on this starr , so shall the nooneline also ( whose office alone is now to shew the howers ) shew you halfe hower past 4 in the morning and better , the iust time when as Oculus Tauri shall come vnto the Meridian in the North vnder the earth . And now if you list to remoue the hower circles , Index from daye to day the noone line ( resting still on the starre ) may shew you what hower that Oculus ♉ commeth to the Meridian euery day of the yeare .

And as all those Meridian conclusions , ( as I may terme them ) are soone had by bare inspection without mouing . So you may as redily attaine al the Horizontall conclusions with a verie easie kinde of motion : but euen with leading about the Zenitfer by the Almicantifer , as one shoulde leade a horse by the halter rounde about an horse mill , and all the skill is but to knowe which way to leade him , for if your question proposed bee , to bee performed on thest halfe of the Horizon , which I call the East hemisphere ( for the Meridian or nooneline alwaies deuideth euery Horizon in two equall parts ) then must your Almicantifer play on the East or left side of the nooneline where Oriens is written , and leade him about leftwards till the pointer touch vpon the planet or star proposed , but if on the West hemisphere then right-wards : as for example , to know what hower the ☉ shal rise the said 5 of March 1598 play your Almicantifer in the East hemisphere and thereby leade about the Zenitfer left-wards till the pointer may be brough to touch the said 25 deg , 3 mi. of ♓ there shal the noone-line shew you in the hower circle almost a quarter after 6 of the clock , the time of son rising that day , and the Almicantifer labelled to the East line of the Zenitfer , vz , to L M sheweth as in the 17 Chap. in the Celestialls limbe 355 1/2 deg. his oblique ascention for that daye . Then if you list to leade him on leftwards till the pointer come to touch the middle prick of Oculus Tauri , there you shall see the nooneline to shew in the howercircle a little past 9 in the forenoone the time of his rising that daye , and the Almicantifer labelled or applyed from L on M , shall shew in the Maters limbe 41 deg. his oblique ascention for that day as in the 18 cap. And if now you would know what time they will set , then play the Almicantifer on the West hemispheare , and by him leade about the prepared Zenitfer right-wards till the pointer do touch eyther of them , and you shall so finde the nooneline to shew for the the son almost 6 of clock , and for Oculus Tauri the times of their setting , almost 12 at midnight . And the Vespertine fin of the Almicantifer G I applyed to L M so shall the fiduciall line G P F of the Almicantifer which representeth now the West lyne shewe in the Maters limbe the 354 1/● deg. and for the starre 86 deg. their oblique descentions for that day . Then if you reckon the howers betwene the Son rising and setting the same is the length of the day , and those taken out of 24 leaueth the length of the night also , the howers betweene the rising & setting of Oculus Tauri are called his diurnall arch , those taken out of 24 leaueth his Nocturnall arch . Likewise the right ascention of Oculus ♉ being as before 62 deg. and nowe his oblique ascention found 41 degree take the bigger out of the lesse there resteth 21 deg. his difference of ascention in this 52 latitude from that of the right Horizon .

To performe the 15 Chap. applye the Noone-lyne to the ☽ and from thence reckon in the hower-circle set , so many howers eyther rightwardes or left-wardes according as you finde the hower giuen by his shade , distante from the Index and at the end of that reckoning is your desire .

Now to the example of the 23 cap. yf you plant the noneline on the great Starre Oculus Tauri , the howercircles Index planted at 26 deg. 3 min. of ♓ seeke thereon , 10 of the clocke at night the hower giuen and thence reckon thereon the howers vnto the noone lyne , you shall finde then 5 2/3 howers those shall be the howers distaunce of Oculus Tauri from the South there sought , and that Westwards because the reckonning goeth Westwards from 10 of the clocke .

Next to the 24 cap. the howercircles Index planted ( for still that must first be done ) at 20 de. 10 min. ♎ . In steed of labelling to thee ☉ you shall applye the nooneline to 10 of clocke in the morning , in the hower-circle , and there holding still the Zenitfer , prepare it as before is taught to some altitude at auenture , admit to 20 degrees : then playing about the Almicantifer if the poynter so set will come to touch the Sun his place or starre proposed , then bee sure that altitude prepared is your desire , but if the poynter ouer reach as at that proffer it will . Then holding still the Zenitfer at the hower giuen , prepare it to a greater altitude , admit to 25 deg. at auenture , and then playing about the Almicantifer you shall find the poynter to cut th eclipticke in the 20 deg. 10 min. of ♎ at K being the Sonnes true place for that instant . Therefore you may conclude that the 10 of October 1596 at 10 of clocke in the morning the Son was 25 deg. high in our 52 latit . yf 25 had not hit you must haue assaied againe if neede had been .

LAST of all for Addition in the vse , take one deuise more with you for the Azmuth then before I wrate of : which with no small studdie I attayned , meaning to haue reserued it with diuers other propositions that I haue in store for a second edition . But because carpers shall not say that my so commende Astrolabe wanteth any needefull part that other Astrolabes had . I will heere though I coulde performe it 2 or 3 wayes for breuity only deliuer the best . As to the example before mentioned , where according to the altit . of the ☉ taken 25 degree high by the Zenitfer prepared to that 25 altit . and the pointer applyed on the Sun his place at K , his nooneline shewed in the howercircle 10 of clock . There holde fast the Zenitfer till you haue done , if you desire also to know what Azimuth the ☉ was then in . And play about the Almicantifer ( it is no matter on which side of the nonneline ) till it will mete at G. iustly with the pearle ( prepared as before ) of the thread fastned in the loope of the Azimuthall lydger at X their hold also fast the thred X G. Then apply the fiducyall line of the Almicantifer euen with the fiducyal line of the Cursor square crossing the Zenitfer : And so slyde them square down together til the Almicantifer cut the perle at G. which wil be at H. in the nooneline , & there at H. fasten the Cursor & slice the poynter to G. & then vnhitch the thred from X & hitch it to the vertical lidger at E , & from E extend the thread on the Suns place at K , lastly playe about the Almicantifer till the poynter setled at G as before do touch the Verticall thread E K , which will be at I. H I. being equall to H G. So doth H I. the fiducyall line of the Almicantifer cut amonge the Antemeridian degrees of the Cursors Azimuthall quadrant , 57 deg. the Azimuth sought for . That is to say , the degrees cut are to be taken by the outer numbeing being it was in the morning because the Azimuthes are numbered from the East and West pointes of the Horizon either-way vnto the Meridian there ending at 90 , which is to be regarded by the letter P on the Almicantifer .

And out of this proposition ( because we will leaue no hard chap. of the booke vnexplaned ) you may easily learne how to performe the 20 chap. another way . For by another Synonomy of speech to know the amplitude of rising and setting of the Sun , or stars , is but to know vpon what Azimuth or point of the compasse they rise . Wherefore being that the Horizon circle of any place is the begining of the first circle of altititude that is to say , of the 0 Almicantare . Therfore to get this amplitude , you must first prepare your Zenitfer to that 0 altitude , that is to save , to the Horizon as before is taught by setting the Cursor to A D. and the pointer to R. from A. And being so prepared , if it be for sonne rising let the Almicantifer playe in the East Hemisphere , and his pointer there touching the Suns place , hold fast the Zenitfer till you have all done , whose noone lyne sheweth in the hower-circle his hower of rising . Then for the Azimuth work in all respects as is done last before , and you shall find the meeting of the pearle and pointer at G. will fall out in the centre line L. M. extended . Wherefore if a thread were hitched in the centre L. and his pearle extended beyond M. to that crossing at G. the Cursor neede not here bee remooued at all from A. D. when the one threade is extended out of the Verticall point E on the ☉ & this thred out of L till perle out that vercal thred as before he did at I this centre thred shall shew in the limbe of the Celestiall the amplitude desired , desired , If you reckon thence to 6 of clock that is to say ; vnto that degree which the 6 of clock , howerline of the howercircle wold shew in the Celestiall limbe , if it were extended .

Note that if any man shall affect to haue the howercircle on the Zeniifer according to the first direction : and not on a plate or flat by it selfe as now I appoint him . Yet let him quite out away the Pomeridian howers which are without the Zenitfer numbered from noone vnto media nox , to to the end that nothing may hinder the sight of the whole Hemisphere at all times and that the Cursor maye alwayes haue free passage to the centre . And then to vse in steede of the centre thread a rule or labell , like vnto the Almicantifer before described , with the Vespertine sin onely . For so shall the rule shew the Antemeridian howers from midnight to noone and the fin the Pomeridian from noone to midnight , and all by that one halfe hower circle described on the Zenitfer euen as before it was done on the quadrant with three finnes , and as readily as in the booke by the whole circle .

FINIS .