First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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T HEOREM .–Amatrix R isarotationmatrixifandonlyif:
RT R = I and det R =1
P ROOF .–Thescalarproductispreservedby R if,forany u and v in Rn ,wehave:
(Ru)T · (Rv )= u T RT Rv = u T v .
Therefore RT R = I.Thesymmetriesrelativetoaplane,aswellasalltheother improperisometries(isometriesthatchangetheorientationofspace,suchasamirror), alsoverifytheproperty RT ·R = I.Thecondition det R =1 allowsustobelimited totheisometrieswhicharedirect.
1.1.2. Liegroup
Thesetofrotationmatricesof Rn formsagroupwithrespecttothemultiplication. Itisreferredtoasa specialorthogonalgroup (special because det R =1, orthogonal because RT R = I)anddenotedby SO (n).Itistrivialtocheckthat (SO (n), ) isa groupwhere I istheneutralelement.Moreover,themultiplicationandtheinversion arebothsmooth.Thismakes SO (n) a Liegroup whichisamanifoldofthesetof matrices Rn×n .
Theset Rn×n of n × n-matricesisofdimension n2 .Sincethematrix RT · R is alwayssymmetric,thematrixequation RT · R = I canbedecomposedinto n(n+1) 2 independentscalarequations.Forinstance,for n =2, wehave 2(2+1) 2 =3 scalar equations:
Asaconsequence,theset SO (n) formsamanifoldofdimension d = n2 n(n+1) 2 . – for n =1, weget d =0.Theset SO (1) isasingletonwhichcontainsasingle rotationmatrix: R =1;
– for n =2, weget d =1.Weneedauniqueparameter(orangle)torepresentthe rotationsof SO (2);
– for n =3, weget d =3.Weneedthreeparameters(orangles)torepresent SO (3).
1.1.3. Liealgebra
An algebra isanalgebraicstructure (A, +, ×, ·) overabody K,if(i) (A, +, ·) is avectorspaceover K;(ii)themultiplicationrule × of A × A → A isleft-andrightdistributivewithrespectto +;and(iii)forall α, β ∈ K,andforall x,y ∈ A, α · x × β · y and (αβ ) (x × y ).Noticethatingeneral,analgebraisnon-commutative(x × y = y × x)andnon-associative((x × y ) × z = x × (y × z )).A Liealgebra (G , +, [] , ) isanon-commutativeandnon-associativealgebrainwhichmultiplication,denoted byaso-called Liebracket,verifiesthat(i) [ , ] isbilinear,i.e.linearwithrespectto eachvariable;(ii) [x,y ]= [y,x] (antisymmetry)and(iii) [x, [y,z ]]+[y, [z,x]]+ [z, [x,y ]]=0 (Jacobirelation).
Considertherotationmatrix I of SO (n) correspondingtotheidentity.Ifwemove I byaddingasmallmatrix,say A dt of Rn×n ,wegenerallydonotobtainarotation matrix.Weareinterestedinmatrices A suchthat I + A dt ∈ SO (n).Wehave (I + A dt)T (I + A dt)= I, i.e. A · dt + AT · dt = 0 + o (dt).Therefore, A shouldbeskew-symmetric.Thismeans thatweareabletomovein SO (n) around I byaddinginfinitesimalskew-symmetric matrices A · dt thatarenotelementsof SO (n). Thiscorrespondstoanewoperation in SO (n) whichisnotthemultiplicationthatwealreadyhad.Formally,wedefinethe Liealgebra associatedwith SO (n) asfollows:
Lie (SO (n))= A ∈ Rn×n | I + A dt ∈ SO (n) anditcorrespondstotheskew-symmetricmatricesof Rn×n .
Ifwenowwanttomovearoundanymatrix R of SO (n),wegeneratearotation matrixaround I andwetransportitto R.Weget R (I + A · dt) where A is skew-symmetric.Thismeansthatweaddto R thematrix R · A · dt.
Inrobotics,Liegroupsareoftenusedtodescribetransformations(suchas translationsorrotations).TheLiealgebracorrespondstovelocitiesorequivalentlyto infinitesimaltransformations.Liegrouptheoryisuseful,butrequiressome non-trivialmathematicalbackgroundsthatarebeyondthescopeofthisbook.Wewill trytofocuson SO (3) orusemoreclassicaltoolssuchasEuleranglesandrotation vectors,whichareprobablylessgeneralbutaresufficientforcontrolpurposes.
1.1.4. Rotationvector
If R isarotationmatrixdependingontime t,bydifferentiatingtherelation RRT = I,weget ˙ R · RT + R · ˙ RT = 0.
Thus,thematrix ˙ R RT isaskew-symmetricmatrix(i.e.itsatisfies AT = A andthereforeitsdiagonalcontainsonlyzeroes,andforeachelementof A,wehave aij = aji ).Wemaythereforewrite,inthecasewhere R isofdimension 3 × 3:
Thevector ω =(ωx , ωy , ωz ) iscalledthe rotationvector associatedwiththepair R, ˙ R .Itmustbenotedthat ˙ R isnotamatrixwithgoodproperties(suchasfor instancethefactofbeingskew-symmetric).Ontheotherhand,thematrix ˙ R · RT has thestructureofequation[1.1]sinceitallowspositioningwithinthecoordinatesystem inwhichtherotationisperformed,andthisisduetothechangeofbasisperformedby RT .Wewilldefinethe crossproduct betweentwovectors ω and x ∈ R3 asfollows:
whichcanbeinterpretedasthematrixassociatedwithacrossproductbythevector ω . Thematrix Ad (ω ) isalsowritten ω ∧.
P ROPOSITION .–If R(t) isarotationmatrixthatdependsontime,itsrotationvector isgivenby:
P ROOF .–Thisrelationisadirectconsequenceofequation[1.1].
P ROPOSITION .–If R isarotationmatrixin R3 andif a isavectorof R3 ,wehave:
whichcanalsobewrittenas
R a) ∧ = R (a∧) RT
P ROOF .–Let x beavectorof R3 .Wehave:
P ROPOSITION .–(Duality)Wehave:
Thisrelationexpressesthefactthatthematrix RT ˙ R isassociatedwiththerotation vector ω ,associatedwith R (t) butexpressedinthecoordinatesystemassociatedwith R,whereas ˙ R RT isassociatedwiththesamevector,butthistimeexpressedinthe coordinatesystemofthestandardbasis.
P ROOF .–Wehave:
1.1.6. Rodriguesrotationformulas
Matrixexponential.Givenasquarematrix M ofdimension n,itsexponentialcan bedefinedas:
e M = In + M + 1 2! M2 + 1 3! M3 + ··· = ∞ i=0 1 i! Mi
where In istheidentitymatrixofdimension n.Itisclearthat eM isofthesame dimensionas M.Herearesomeoftheimportantpropertiesconcerningthe exponentialsofmatrices.If 0n isthezeromatrixof n × n andif M and N aretwo matrices n × n,then:
e0n = In
eM .eN = eM+N (ifthematricescommute)
d dt eMt = MeMt
Matrixlogarithm.Givenamatrix M,thematrix L issaidtobeamatrixlogarithm of M if eL = M.Asforcomplexnumbers,theexponentialfunctionisnotaone-toonefunctionandmatricesmayhavemorethanonelogarithm.Usingpowerseries,we definethelogarithmofasquarematrixas
log M = ∞ i=1 ( 1)i+1
Thesumisconvergentif M isclosetoidentity.
Rodriguesformulas.Arotationmatrix R hasanaxisrepresentedbyaunitvector n andanangle α withrespecttothisaxis.From n and α wecanalsogeneratethematrix R.Thelink R ↔ (n, α) ismadebythefollowing Rodriguesformulas (i) R =
Thefirstequation(i)willbeshowninexercise1.4and(ii)isthereciprocalof(i).In theseformulas, αn∧ isanotationtorepresentthematrix Ad (αn).Aswillbeshown inexercise1.4, R = eαn∧ isarotationmatrixand n isaneigenvectorassociatedwith theeigenvalue1of R.
Axisandangleofarotationmatrix.Givenarotationmatrix R,theaxiscanbe representedbynormalizedeigenvector n associatedwiththeeigenvalue λ =1 (two ofthemcanbefound,butweneedone).Tocompute α and n,wemayusetherelation
αn =log R,whichworkswellif R isclosetoidentity.Inthegeneralcase,itismore suitabletousethefollowingproposition.
P ROPOSITION .–Givenarotationmatrix R,wehave R = eαn∧ where
P ROOF .–Takearotationmatrix R witheigenvalues 1, λ1 , λ2 andeigenvectors n, v1 , v2 .Considerthegeneralizedpolynomial f (x)= x x 1 ,where x isthe indeterminate.Fromthecorrespondencetheoremofeigenvalues/vectors,the eigenvaluesof f (R)= R R 1 = R RT are f (1)=0,f (λ2 ),f (λ3 ) andthe eigenvectorsarestill n, v1 , v2 .Since f (R) isskewsymmetricand f (R) · n = 0, wehave f (R) ∝ Adn.Thus,thevectorAd 1 R RT isaneigenvectorof R associatedwiththeeigenvalue 1.Itprovidesustheaxisof R.Tofindtheangle α,we usethepropertythatthetraceofamatrixis similarity-invariant,whichmeansthat foranyinvertiblematrix P,wehavetr(R)= tr(P 1 R P) Take P astherotation matrixwithtransforms R intoarotationalongthefirstaxis.Weget
Here atan2 isthetwo-argumentarctangentfunctiondefinedby
1.2.2. RotationvectorofamovingEulermatrix
Letusconsiderasolidbodymovinginacoordinatesystem R0 andacoordinate system R1 attachedtothisbody(refertoFigure1.5).Theconventionschosenhere arethoseoftheSNAME(SocietyofNavalandMarineEngineers).Thetwo coordinatesystemsareassumedtobeorthonormal.Let R(t)= R
Deadreckoning.Fordeadreckoning(i.e.withoutexternalsensors)thereare generallyextremelypreciselasergyrometers(around 0.001 deg/s.).Thesemakeuse oftheSagnaceffect(inacircularopticalfiberturningarounditself,thetimetakenby lighttotravelanentireround-tripdependsonthepathdirection).Usingthreefibers, thesegyrometersgeneratethevector ω r =(ωx , ωy , ωz ).Therearealso accelerometerscapableofmeasuringtheacceleration ar withaveryhighdegreeof precision.Inpureinertialmode,wedetermineourpositionbydifferentiating equations[1.12]onlyusingtheacceleration ar andtherotationspeed ω r ,both expressedinthecoordinatesystemofthebox.Inthecasewherewearemeasuring thequantity vr (alsoexpressedintheframeoftheinertialunitattachedtotherobot) witha Dopplervelocitylog,weonlyneedtointegratethefirstandthelastofthese threeequations.Finally,whentherobotisacorrectlyballastedsubmarineora terrestrialrobotmovingonarelativelyplaneground,weknow apriori thaton averagethebankandtheelevationareequaltozero.Wemaythereforeincorporate thisinformationthroughaKalmanfilterinordertolimitthedriftinpositioning.An efficientinertialunitintegratesanamalgamationofalltheavailableinformation.
Inertialunit .Apureinertialunit(withouthybridizationandwithouttakinginto accountEarth’sgravity)representstherobotbythekinematicmodelofFigure1.7, whichitselfusesthestateequationsgivenin[1.12].Thissystemiswritteninthe form ˙x = f (x, u) where u =(ar , ω r ) isthevectorofthemeasuredinertialinputs (accelerationsandrotationspeedsviewedbyanobserverontheground,butexpressed intheframeoftherobot)and x =(p, vr , ϕ, θ , ψ ) isthestatevector.Wemayusea numericalintegrationmethodsuchastheEulermethod.Thisleadstoreplacingthe differentialequation ˙x = f (x, u) withtherecurrence: x (t + dt)= x (t)+ dt · f (x (t) , u (t)) .
Figure1.7. Navigationmechanizationequations
Withgravity.Inthecasewherethereexistsagravity g (p) whichdependson p, theactualacceleration ar correspondstothegravityplusthemeasuredacceleration ames r .Inthiscase,theinertialunitcanbedescribedbythefollowingstateequations:
[1.13]
Inthiscase,theinputsofthesystemare ames r and ω r whichareobtainedby theaccelerometersandthegyroscope.Tointegratethisstateequationforlocalization purposes,weneedtoknowtheinitialstateanda gravitymap g (p)
UsingrotationmatrixinsteadofEulerangles.Fromequation[1.4],wehave ˙ R = R · Ad RT ω .Thus,thestateequations[1.13]canbewrittenwithoutany Euleranglesas
[1.14]
Themainadvantageofthisrepresentationisthatwedonothavethesingularity thatexistsin[1.13]for cos θ =0.Butinstead,wehaveredundancies,since R of dimension 9 replacesthethreeEulerangles.Fornumericalreasons,whenwe integrateforalongtime,thematrix R(t) mayloosetheproperty R · RT = I.To avoidthis,ateachiteration,arenormalizationisneeded,i.e.thecurrentmatrix R shouldbeprojectedonto SO (3).OnepossibilityistouseaQRfactorization.A QR-decompositionofasquarematrix M providestwomatrices Q,R suchthat M = Q R where Q isarotationmatrixand R istriangular.When M isalmosta rotationmatrix, R isalmostdiagonalandalsoalmostarotation,i.e.onthediagonal of R theentriesareapproximately ±1 andoutsidetheentriesarealmostzeros.The followingP YTHON codeperformstheprojectionof M on SO (3) andgeneratesthe rotationmatrix M2
analyticfunctionsoftheforcesthatareattheoriginoftherobot’smovement.Forthe dynamicmodelingofasubmarinethereferenceworkisFossen’sbook[FOS02],but therelatednotionscanalsobeusedforothertypesofsolidrobotssuchasplanes, boatsorquadrotors.Inordertoobtainadynamicmodel,itissufficienttotakethe kinematicequationsandtoconsidertheangularandtangentialaccelerationscaused byforcesanddynamicperformance.Thesequantitiesbecomethenewinputsofour system.ThelinkbetweentheaccelerationsandtheforcesisdonebyNewton’s secondlaw(orthe fundamentalprincipleofdynamics).Thus,forinstanceif f isthe netforceresultingfromtheexternalforcesexpressedintheinertialframeand m is themassoftherobot,wehave
mar = fr , where fr istheappliedforcesvectorexpressedintherobotframe.Thesametypeof relation,knownasthe Euler’srotationequation,existsforrotations.Itisgivenby
where τ r istheappliedtorqueand ω r istherotationvector,bothexpressedinthe robotframe.Theinertiamatrix I isattachedtotherobot(i.e.computedintherobot frame)andwegenerallychoosetherobotframetomake I diagonal.Thisrelationis aconsequenceofEuler’ssecondlawwhichstatesthatinaninertialframe,thetime derivativeoftheangularmomentumisequaltotheappliedtorque.Intheinertial frame,thiscanbeexpressedas
where p isthepositionand (ϕ, θ , ψ ),aretheEuleranglesoftherobot.Thefirstthree equations(i),(ii),(iii)correspondtothekinematicequationsandhavealreadybeen derived(seeequation[1.12]).Notethatinequation(iii),theacceleration ar (expressed intherobotframe)comesfromNewton’ssecondlawwheretheforcesarethetotal force τ0 andtheweight mg :
1)Showthattheeigenvaluesof Ad (ω ) are {0, ω · i, − ω · i}.Givean eigenvectorassociatedwith 0.Discuss.
2)Showthatthevector Ad (ω ) x = ω ∧ x isavectorperpendicularto ω and x, suchthatthetrihedron (ω , x, ω ∧ x) isdirect.
3)Showthatthenormof ω ∧ x isthesurfaceoftheparallelogram A mediatedby ω and x.
E XERCISE 1.2.– Jacobiidentity
TheJacobiidentityiswrittenas: a ∧ (b ∧ c)+ c ∧ (a ∧ b)+ b ∧ (c ∧ a)= 0.
1)Showthatthisidentityisequivalentto: Ad (a ∧ b)= Ad (a) Ad (b) Ad (b) Ad (a) where Ad (ω ) istheadjointmatrixofthevector ω ∈ R3 .
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“Where are you going?”
“Up to see this man—this mate of the Silver Swan.”
“Oh yes. Well, you tell him I’m coming up to see him myself, today. It’s a mystery to me why he should go to that place. I don’t understand it. How was he looking when you saw him—for I take it you have seen him?”
“How do you mean—sick or well?”
“Yes.”
“Oh, he appeared in pretty fair health, I should say,” replied Brandon, beginning to think that there was something queer about it all.
“Well, I’ll see him myself,” declared the merchant, rising and giving the boy his hand. “I tell you what we’ll do, Brandon. If you don’t get back here by noon, I’ll step up and get you, and we’ll go to lunch together; then afterward we’ll take a look at the whaleback, if you like.”
Brandon thanked him and opened the door into the outer office, almost falling over Mr. Alfred Weeks, who had his head suspiciously near the keyhole.
“Lo—looking for my ruler that I dropped,” declared the red haired clerk, as his employer’s eyes rested sternly upon him.
But as he passed out, Brandon noticed that the ruler was on the high desk holding open the leaves of a much tattered paper novel.
“Funny sort of fellow for a respectable ship owner to employ,” Brandon decided, as he made his way along the crowded thoroughfare. “In fact, I guess I’ll withhold my opinion of all three of these people till I know ’em better—Wetherbee, Pepper, and his clerk.”
By closely scanning the signs on the buildings as he passed, the captain’s son finally discovered the place he sought. He came within an ace of not doing so, however, for the words “New England Hotel” were simply painted on a small strip of tin on one side of the
doorway, the rest of the sign space being devoted to the words: John Brady, Wines, Liquors, and Cigars.
Brandon hesitated a moment before entering the place. It was plainly a saloon of the worst type, the “hotel” part evidently being but a “blind” by means of which the bar could be kept open all night.
Two or three disreputable men—sailors or longshoremen by appearance—were hanging about the door, but Brandon Tarr had a good deal of confidence in his ability to take care of himself, and finally ascended the steps.
A sickening odor of stale tobacco smoke and bad liquor assailed his nostrils as he stepped within the room, and he was almost tempted to back out and give up his intention of seeing Wetherbee. But the man behind the bar—a villainous looking fellow with a closely cropped head and red face—had seen him and came briskly forward.
“Well, young felley, what kin I do fur ye?” he asked, in what was intended as a pleasant tone.
Deciding that he was in for it, the captain’s son walked forward to the bar and replied:
“Nothing to drink, thank you. I’m looking for a man who’s stopping here—Caleb Wetherbee.”
The bartender eyed him curiously and repeated:
“Caleb Wetherbee, eh? Well, I’ll see ’f he’s here.”
He stepped back to a door leading into an inner room and, opening it a crack, called to somebody inside. There was a whispered conversation between the men, and the bull necked individual came back to the bar.
“All right, m’ duck; he’s in dere,” he said, with a grin, and a motion of his thumb toward the inner door. “Yer don’t have ter send in no kyard.”
Taking this as a permission to enter, Brandon walked across the long saloon, littered with tables and chairs, and its door covered with
sawdust, and opened the door
The apartment beyond was as badly furnished as the outer room, there being only a square deal table and several wooden bottomed chairs. In one of these chairs before the table, with his head bowed upon his arms, was the sailor whom Brandon had seen two days before in the woods on his uncle’s farm back in Chopmist, the only occupant of the place.
CHAPTER XIII
THE OLD SAILOR WITH THE WOODEN LEG
I was only in the country—in the woods and sheltered fence corners —that the patches of snow still remained on this sixth day of April. In New York the sun shone warmly upon the sidewalks, washed clean by the shower of the night before, and the tiny patches of grass in the parks and squares were quite green again.
About the middle of the forenoon a man stumped along a street leading to what remains of the Battery park—a man dressed in a half uniform of navy blue, and with a face (where the beard did not hide the cuticle) as brown as a berry.
At first glance one would have pronounced this person to be a sailor, and have been correct in the surmise, too.
The man’s frame was of huge mold, with massive development of chest and limbs, and a head like a lion’s. But his bronzed cheeks were somewhat hollow, and his step halting, this latter not altogether owing to the fact that his right leg had been amputated at the knee and the deficiency supplied by an old fashioned wooden leg.
Still, despite his evident infirmity, the old seaman looked cheerfully out upon the world on this bright April morning, and pegged along the sidewalk and into the park with smiling good nature.
Not a beggar had accosted him during his walk down town without having a nickel tossed to him, and it was with vast contentment that the wooden legged sailor at length seated himself upon a bench, from which vantage point he could overlook the bay and its multitudinous shipping.
“Ah!” he exclaimed, sniffing the air which blew in from the sea, like a hungry dog. “This is life, this is! Thank heaven I’ve got away from them swabs of doctors at last. Another week at that ere hospital
would ha’ been the death o’ me. Still, I reckon they meant well ’nough.”
He sat there for some time in cheerful silence, and drank in the exhilarating air, his pea cloth jacket thrown open to the breeze, baring the broad expanse of flannel shirt beneath.
“A few days o’ this’ll put me right on my feet,” he said, with delight, “better’n all the tonics the old sawbones ever invented. Lord! if I’d had this breeze a-blowin’ inter my winder up there to the hospital, I’d been out a fortnight ago.
“The old man ain’t dead yet. It was a pretty hard tug, I admit; but here I be!”
He slapped his leg with such vigor that a flock of sparrows flew up with sudden affright from the path; but this energetic gesture was taken in another sense by the group of urchins which had gathered near by to talk and fight (much after the manner of their feathered prototypes, by the way) over the morning’s sale of papers.
At the old man’s motion half a dozen of these sharp eyed little rascals broke away from the group, and ran shrieking toward him, wildly waving their few remaining wares in his face.
“’Ere you are, sir! Tribune, Sun, World!”
“Tribune,” said the old sailor, laughing heartily as though he saw something extremely ludicrous in their mistake.
“My last ’un, sir. Thankee!”
The successful Arab pocketed his money and went back to his friends, while the sailor slowly unfolded the sheet and took up the thread of his reflections again.
“Once I get my sea legs on,” he thought, fumbling in his pocket for a pair of huge, steel bowed spectacles, which he carefully wiped and placed astride his nose “once I get my sea legs on, I’ll take a trip up ter Rhode Island and see the cap’n’s boy, unless he turns up in answer to my letter
“Poor lad! he’s doubtless heart broken by Cap’n Horace’s death, and won’t feel much like goin’ into this ’ere treasure huntin’ business; but for his own good I’ll have ter rouse him up. It would be what the cap’n would wish, I know.”
He let the paper lie idly on his knee a moment, and a mist rose in his eyes.
“Never mind if the old brig has gone to pieces before we get there,” he muttered. “I’ve got a little shot in the locker yet, an’ the boy shan’t come ter want. I’ll do my duty by him as though he was my own son, that I will!”
He picked up the paper again, and turned naturally to the shipping news, which he ran over carelessly, smiling the while. Finally his eye was attracted by something near the bottom of the column.
“Eh, what’s this?” he exclaimed. “What’s this about the Silver Swan?”
With great excitement he read the following news item, following each line of the text with his stumpy forefinger:
Captain Millington, of the English steamer Manitoba, which arrived here yesterday from Brazil, reports that he passed a very dangerous wreck in latitude 22:03, longitude 70:32. It was the hull of a brig, apparently in good condition, but with her masts snapped off close to the decks, and all her rigging carried away. The name on her stern was Silver Swan, Boston.
This is the same derelict reported by the steamer Montevideo at Savannah several weeks ago. According to Captain Millington, the wreck of the brig is a great menace to all vessels plying between this and South American ports, as its course seems to be right across the great highway followed by most of the steamship lines.
It will be remembered that the Silver Swan was wrecked over two months ago on Reef Eight, southwest of Cuba, grounding, according to the report of the survivors of her crew, upright on the rock. The captain of the Montevideo sighted her not far from the reef, from which she was
doubtless loosened by the westerly gale of February 13th; but since that time she has floated some distance to the north and east, and if she follows the same tactics as many of her sister derelicts, she may zigzag across the course of the South American steamers for months.
The cruisers Kearsarge and Vesuvius are both lying in port at present, and it will be respectfully suggested to the Navy Department that one or both of those vessels be sent to destroy this and several others of the most dangerous derelicts now floating off our coast.
“Shiver my timbers, sir!”
With this forcible and exceedingly salty ejaculation, the old sailor with the wooden leg dropped the newspaper to the walk, and his spectacles along with it, and springing up, trampled upon them both.
But in his great excitement he noticed neither the torn paper nor the ruined glasses. He stumped up and down the walk for several moments before he became calm enough to think coherently.
In fact, the blue-coated policeman on the corner had begun to eye him suspiciously.
“The Silver Swan afloat—a derelict!” he muttered. “This ’ere is a sitiwation I didn’t look for. An’ then, them blasted cruisers are liable to go down there and blow her into kingdom come any minute. The Silver Swan on Reef Eight was bad enough, but the Silver Swan afloat, at the mercy of the gales as well as other vessels, is worse!
“Now, what in creation’ll I do about it? I haven’t heard from the boy yet, and there’s little enough time as it is. Why, she might sink ’most any time with all them di’monds the cap’n told about aboard her!
“I’ll take a steamer to get down there ahead of them confounded iron pots” (by this disrespectful term did he designate Uncle Sam’s cruisers), “but who under the canopy’s got a steamer to charter?
“By the great horn spoon, I have it!” he exclaimed, after a moment’s thought. “Adoniram Pepper is just the fellow.”
With this declaration he jammed his hat on his head, and stumped off as rapidly as one good leg and one wooden one could carry him, toward the shipping merchant’s office on Water Street.
CHAPTER XIV
THE OLD SAILOR’S EXCITEMENT
A the old sailor hurried along the street toward the ship owner’s office he became calmer, and, being a person who had all his life been taking greater or less chances in his business of seagoing, he began to look at the situation more composedly.
The Silver Swan was without doubt in far greater danger of destruction now than she had been while hard and fast on the reef, but no amount of worrying would better the matter, and therefore one might accept the fact coolly Then, besides, she had floated unmolested for over six weeks already, and there was a big chance for her doing so for six weeks or more to come.
“Blast these navy vessels any way, I say!” the old man muttered, stumping along now at a moderate gait. “They probably won’t be able to find her. And if nothing collides with her, I reckon she’ll keep afloat for one while, for I can swear myself that the old brig warn’t injured none below the water line—she went on that reef jest as easy!
“She’s got the same chance o’ staying above board—the Silver Swan has—as any other craft that’s become a derelict. Look at the schooner W. L. White, abandoned by her crew during the great storm of ’88. She floated about the North Atlantic for the better part of a year, before she went ashore at last on the Hebrides.
“An’ then there was the Weyer G. Sargent, mahogany laden, floated fifty-five hundred mile, or more, ’cording to the pilot chart, a-swingin’ ’round the Atlantic from New Foundland to the Azores for two years. An’ there may be many another good ship that’s got a bigger record ’n that at this very day, down in the Sargasso sea. Oh, it might be worse.”
Nevertheless, despite this cheerful view, the old sailor’s forehead was knotted into a scowl as he opened the door of the ship owner’s dingy office and entered. The red haired clerk was alone at the desk and the door of the private office was shut.
“Well, you jail bird, are you here yet?” demanded the visitor impolitely, eying the clerk with exceeding disfavor.
“Oh, is that you, Mr. Featherbee——”
“Wetherbee, you scoundrel!” roared the sailor, in a voice like a bull.
“Oh, yes! I should say Wetherbee—er—that’s what I meant,” the clerk hastened to say.
It was remarkable to notice the difference between the greeting accorded to Caleb Wetherbee and that given young Brandon Tarr shortly before.
“So you haven’t managed to get at Pepperpod’s till and clear out, yet, eh?” demanded Caleb jocularly.
Mr. Weeks scowled and grinned at the same time, a feat that very few men can perform; but he made no verbal reply to the question.
“Where is he?” queried the sailor, nodding toward the inner office. “In his den?”
“He’s busy—engaged,” Mr. Weeks hastened to say.
“I believe you’re lying to me, Weeks,” returned the sailor, after eying the fellow a moment. “You’d rather lie than eat Where’s Pepperpod?”
“He—he really is engaged, sir,” declared Weeks, who stood in mortal fear of the brawny sailor. “That is, he told me to say so to anybody that called——”
“I don’t doubt it—that’s what’s taught you to lie,” cried Caleb, in disgust. “Well, I’m going to see him if he’s engaged fifty times. Cut along now and tell him I’m here.”
Mr. Weeks slowly descended from his stool, evidently unwilling to comply with the request.
“Get a move on you,” the sailor commanded. “If you don’t I’ll roast you over a slow fire. I’m just out of the hospital and I’ve got an appetite like an ostrich—or I’d never think of eating you.”
Mr. Weeks unwillingly went to the inner door and rapped on the panel. Then he turned the knob and went in, remaining a few moments, and on making his appearance again, held the portal open for Caleb.
The sailor entered without a word and the clerk closed the door behind him; then, as on the former occasion, he applied his ear to the keyhole with a diligence worthy of a better cause.
Mr. Pepper was sitting before his desk, which was piled high with papers and letters. The day’s mail had just been sent up from the wareroom office by Mr. Marks, the ship owner’s trusted manager, or “steward,” as Adoniram was in the habit of calling him.
Beginning business life more than fifty years before in this very office, Mr. Pepper could not bring himself, as his trade increased, to leave his old quarters, and having found his manager to be a most trustworthy man, he had shifted the burden of the more arduous duties upon his younger shoulders, and himself reposed contentedly amid the dust, the gloom, and the cobwebs of the Water Street office.
Thus it was that few people ever saw “Adoniram Pepper & Co.” to know him; but to his old friends, those of his boyhood and young manhood, Adoniram was always the same.
Naturally his acquaintance was mostly among seafaring people, and it was no uncommon sight to see old hulks of sea captains and ship owners, long past their usefulness, steering a course for the Water Street office on pleasant days, where they were sure to receive a pleasant word from the little old gentleman, if he was in, and not uncommonly a bit of silver to spend for luxuries which “sailors’ homes” do not supply.
The old gentleman sprang up at once at Caleb’s appearance, the unfortunate eye glasses jumping off the chubby little nose as though they were endowed with life. Mr Pepper gave both his hands to the
huge sailor, who indeed looked gigantic beside the little man, and begged him to sit down.
“Well, Pepperpod, how are ye?” cried the sailor, in a hearty roar that shook the light pieces of furniture in the room, just as his bulk shook the chair he had seated himself in.
“First rate, old Timbertoes!” declared the old gentleman, laughing merrily. “So you’re out of the hospital, at last?”
“I be, Adoniram, I be!” cried Caleb with satisfaction. “Never was so glad o’ anythin’ in my life. Them sawbones would have killed me if they’d kep’ me there much longer.”
“Well, well, Caleb, you was a mighty sick man—a mighty sick man.”
“I reckon I was,” responded the sailor reflectively.
“The doctor wouldn’t let me come in to see you,” said the merchant, smiling jovially; “so I had to content myself with sending up things.”
“Yes, you did,” said Caleb, turning on him sternly. “I did think, Adoniram, that you wouldn’t waste your money on such truck as that —a-sendin’ me white grapes, an’ jellies, an’ bunches o’ posies.”
He snorted in veriest scorn.
“Well, er—er—you see, Caleb, I told Frances about you and she took over the things herself,” said Adoniram hesitatingly.
“Hem!”
The old sea dog flushed up like a girl and mopped his suddenly heated face with a great bandanna, finally saying gruffly:
“You tell your sister, Miss Frances, that I am mightily obleeged for ’em, Adoniram. They—er—jest went to the right spot, you tell her; jest what I needed to tone me up!”
“You’d better come up and tell her yourself, Caleb,” said the merchant, with a sly smile.
“Well—er—mebbe I will. Thankee, Adoniram.”
He was silent a moment, and then, suddenly bethinking himself of the errand which had brought him there, he turned upon the little merchant with a slap of his knee which sounded throughout the office like a gun shot.
“But this ’ere ain’t what brought me here—not by a long chalk. Ye know the Silver Swan, Adoniram? Cap’n Horace Tarr’s brig ’t I was with when she grounded on Reef Eight, two months and more ago?”
Mr. Pepper nodded.
“Well, sir, she’s afloat.”
“Afloat!”
“That’s what I said; afloat! A-f-l-o-t-e,” responded the sailor, spelling the word very carefully, if a trifle erratically.
“How—how can that be?”
“Well, ye see she went aground jest like she was goin’ inter stocks for repairs, and if we’d stuck by her, it’s my opinion Cap’n Tarr’d ha’ been alive now.” He stopped and blew his nose hastily. “Well, what is, can’t be bettered, so we’ll say no more o’ that.
“But what I’m gettin’ at is this: she went aground all standin’, an’ the storm wot come up right arterwards, blew her off ag’in. She’s been floating, according to this morning’s paper, ever since.”
“Well, well!” exclaimed Adoniram. “It’s too bad her hull can’t be secured for the boy. If it’s still sound——”
“Sound as a dollar!”
“Where is it floating?”
“’Cordin’ to the report of a cap’n wot sighted her, she’s somewheres about latitude 22, longitude 70.”
“A pretty valuable derelict, eh, Caleb?” said the merchant, reflectively.
“Valible? Well, I should say!” The old sailor looked at his friend curiously a moment, and then leaned forward and rested his huge hand on Adoniram’s knee. “Besides a valible cargo wot we took on
at the Cape and Rio, there’s enough diamonds hid aboard that brig to make the boy a second Vanderbilt!”
“Mercy me!” exclaimed the merchant, and this time the eye glasses leaped off their insecure resting place and fell with a crash to the floor, the splintered crystal flying in all directions.
“Now you’ve done it, Adoniram!” ejaculated Caleb in disgust. “What under the canopy a man like you—with no nose to speak of—wants to try to wear such tackle as them for, is beyond me.”
“Well—er—Frances thinks they look better on me than other kinds of glasses,” remarked the merchant meekly.
“Well—hem!—I s’pose they do look some better on ye,” declared Caleb loyally, and then a slight noise from the other side of the door caused him to jump up and spring hastily to it.
When he flung the door open, however, the red haired clerk was astride his high stool with a look of perfect innocence on his face; but Caleb was not reassured. He shook his huge fist at the fellow, and then shut the door again, turning the key in the lock and hanging his hat upon the door knob for further precaution.
CALEB
CHAPTER XV
RECEIVES A STARTLING COMMUNICATION
“S of these days,” said Caleb, with decision, when he had taken these precautions, “I shall wring that scoundrel’s neck, Adoniram. I wonder at your keeping him here.”
“Well, you see, nobody else would have him,” responded the merchant, as though that fact was reason enough for his keeping the objectionable Mr. Weeks.
“Ya-as—one o’ your blasted philanthropic notions,” declared Caleb, with a snort denoting disgust. “Well, he’ll rob and murder you some day and then you’ll wish you’d heard to me. If ‘jail bird’ ain’t written on his face, then I never saw it on no man’s.”
“But, Caleb, what do you mean by the astounding remark you just made about the Silver Swan?” asked the merchant, drawing the sailor’s mind away from the subject of Mr. Alfred Weeks and his frailties.
“I’ll tell you about it,” said Caleb, in a lower tone, seating himself by the desk again. “What I said is straight, Pepper. There is hidden inside that hulk of the Silver Swan, a lot o’ di’monds—how many, I don’t know—but enough, according to Cap’n Horace’s own words to make a man fabulously rich. They belong to his boy, Brandon, and we must get ’em for him.
“I never knew a word about the stones till we was on the raft. Cap’n Horace was pretty fur gone—any one with half an eye could see that —and when we’d been out several days an’ hadn’t sighted no ship, he wrote a long letter to Brandon an’ give it to me with a package of other papers.
“I’ve got them papers right here at this identical minute; but I ain’t opened ’em, ’cause it ain’t my place to do so. They tells all about the di’monds an’ how they come into Cap’n Horace’s han’s.
“It seems that just afore we left the Cape a man come aboard the Silver Swan and brought a package of wot he thought was papers, to Cap’n Horace, from his brother Anson.”
“Why, Anson was dead long ago, I thought,” interrupted Mr. Pepper.
“So did everybody else think so; but he wasn’t. He was dead, though, when this feller seed Cap’n Horace, for he’d give the package into the man’s hands when he was dying, for him to send to Cap’n Tarr. But we put into the Cape afore the man got ’round to sendin’ ’em to the States.
“He never knew what a valible thing he was a carryin’ ’round; but when the cap’n come to open the package he found a lot o’ di’monds done up in a separate wrapper. These he hid somewhere about the brig—he tells about it in this letter to Brandon, I b’lieve.
“I wanted to know why he didn’t take ’em on the raft when we left the brig, but it seems he misdoubted himself about a rascally sailor we had with us—one Jim Leroyd.
“This ’ere Leroyd had been snoopin’ around the cabin when the cap’n was given the diamonds, and he thought the feller suspected something. So, not knowing how it might go with any of us, he left the gems on the brig, preferring to risk losin’ ’em altogether, rather than to cause strife an’ p’r’aps bloodshed on that raft.
“An’ I reckon ’twas lucky he did so, fur we had trouble enough with that swab Leroyd.”
“Why, wasn’t he the man who was saved with you?” asked the merchant.
“That’s who.”
“Tell me, Caleb,” said Mr. Pepper earnestly, “why was it he stood the experience so much better than you? Why, he was discharged from the hospital in a week, so I understand, while you show traces of the suffering you underwent even now.”
Caleb closed his lips grimly and looked at the little man in silence for several moments. Then he leaned further forward and clutched his arm with one great brown hand.
“He had food that I didn’t have,” he whispered hoarsely.
“What!” cried Adoniram, shrinking back, his eyes abulge.
Caleb nodded slowly.
“There were four of us on that raft. Paulo Montez—he went first. We divided the food and water, an’ that villain Leroyd ate his all up. Then we had ter drive him behind his chest at the other end of the raft, an’ keep him there at the point of our pistols.
“Then the cap’n went, an’—an’—I had to throw him to the sharks to keep him out o’ the clutches o’ that cannibal Leroyd!”
“Great heavens!” exclaimed the ship owner, shrinking back into his chair, his face the picture of horrified amazement.
“Yes, sir,” whispered Caleb; “he dragged poor Paulo’s body back o’ that chest—an’—well, ’taint no use talkin’! I ain’t said a word about it before to any living creature. It’s only my word ag’in his, at best. But I swear, Adoniram, I’d kill the hound with as little compunction as I would a rat.
“He’s been sneaking ’round the hospital, inquiring about me, too,” continued the sailor. “He’s got his eye on these papers, for he see Cap’n Horace give ’em to me. I reckon he don’t know what they’re about, but he suspects there’s money in it. He was ’round to the hospital only last night, so the doctor told me.
“And now, Adoniram, wot I want o’ you is to help me find this derelict before some o’ Uncle Sam’s blasted iron pots go out after her. We must get the boy down from that uncle’s place in Rhode Island——”
“Why, didn’t you see him this morning?” asked Mr. Pepper, in surprise.
“See who?”
“Why, the boy—Captain Tarr’s son, Brandon?”
“What?” roared the sailor. “Then he’s here in New York, is he?”
“Why—of—course,” responded the merchant, in bewilderment. “I thought you’d seen him again. He started out to call on you not two hours ago. He said you’d given him your address—at the New England Hotel, just below here.
“And what I want to say, Caleb is that I don’t consider it a great proof of friendship on your part, for you to go to such a place as that, even if you were low in finances. I’d only be too glad to have you come to my house and stay the rest of your natural life—and so would Frances.”
“Me!—at the New England Hotel!—why the man’s crazy!” declared Caleb.
“Ain’t you stopping there?” gasped the merchant.
“Am I? Well. I guess not! I ain’t but just got out o’ the hospital this blessed morning.”
“Why, he said he’d seen you once, and you’d told him to call at the New England Hotel.”
“Who?” roared Caleb.
“Brandon Tarr.”
“Why, man alive, I never saw the lad in all my life!”
“Then,” declared Adoniram with energy, “there’s foul play about it. When I came down this morning I found the captain’s son waiting to see me. He’d just come down from Rhode Island, I believe, and he’d got your address—said he’d already seen you once, mind you—and was going up to this place to see you again.
“I thought ’twas funny you should put up at such a house, Caleb; but I didn’t know but perhaps you were ‘on your uppers’” (Caleb snorted at this), “and had gone there for cheapness. I told Brandon I’d come up after him this noon and take him to lunch.”
But Caleb was on his feet now, and pacing the floor like a caged lion.
“I see it all—I see it all!” he declared. “It’s some o’ that swab Leroyd’s work. Why, man alive, do you know what the New England Hotel is? It’s one o’ the wickedest places in New York. I know the den well, and the feller as runs it, too. Why, the boy’s in danger every moment he stays there!”
He seized his hat and jammed it on his head again.
“Ef anything’s happened to that boy, I’ll break every bone in that scoundrel’s body!” he exclaimed, seizing the door and throwing it wide open without the formality of unlocking it.
The splintered wood and broken lock flew in all directions as he dashed through the doorway and flung himself into the street, while Mr. Pepper remained weakly in his chair, too utterly bewildered to move, and the festive Mr. Weeks dodged behind the high desk with alacrity, as the sailor went through the outer office like a whirlwind.
CHAPTER XVI
TELLING HOW BRANDON BEARDED THE LION IN HIS LAIR
A Brandon Tarr entered the apartment behind the bar room of the New England Hotel, the man at the table raised his head and surveyed him surlily. Evidently he had been drinking, and the liquor had changed his mood greatly from that of the affable sailor who had accosted the captain’s son in the Chopmist woods.
“Well, how came you here?” inquired the sailor, in no very friendly tone, gazing at Brandon, with bloodshot eyes.
“I came down on the train.”
“Ain’t you lost?”
“Guess not,” responded the boy.
The man shifted his position uneasily, keeping his eyes fixed upon his visitor.
“Can’t say as I expected to see you—just yet, any way.”
“No?” returned Brandon coolly.
“Say! wot the blazes do you want, any way?” demanded the sailor fiercely, after an instant’s silence. “It won’t pay you to be sassy here, my lad, now I can assure ye.”
“Think so? Seems to me you’re not as glad to see me as I reckoned you would be. It didn’t exactly pay you to come ’way up to Rhode Island to pump me, did it?”
The fellow hissed out an oath between his teeth and clinched his fist angrily.
