Let　B-n　be　the　book　graph　which　consists　of　n　copies　of　triangles　all　sharing　a　common　edge.　Let　Cm　be　a　cycle　of　length　m.　In　1978,　Rousseau　and　Sheehan　initiated　the　study　of　the　book-cycle　Ramsey　number.　A　lot　of　effort　has　been　made　to　determine　the　value　of　r(Bn,Cm)　since　then.　In　[Ars　Combin.,　31　(1991),　pp.　239-248],　Faudree,　Rousseau,　and　Sheehan　mentioned　the　following:　＂we　know　practically　nothing　about　r(B-n,C-m)　when　m　is　even　and　greater　than　four.　Also,　the　problem　of　computing　r(Bn,C-m)　when　m　is　odd　and　m　and　n　are　nearly　equal　provides　an　unanswered　test　of　strength.＂　Answering　the　second　part　of　the　question　above,　the　second　and　fifth　authors　recently　obtained　the　value　of　r(B-n,C-m)　for　8n/9+112　＜=　m　＜=　inverted　right　perpendicular3n/2inverted　left　perpendicular+1　and　n　being　large.　However,　the　value　of　r(B-n,C-m)　is　previously　unknown　for　m　＜=　8n/9+111　and　m　being　even　as　well　as　n+13/4＜m　＜=　8n/9+111　and　m　being　odd.　In　this　paper,　for　even　m,　we　manage　to　determine　the　value　of　r(B-n,C-m)　provided　that　m　is　linear　with　n　and　m　is　large　enough.　Thus　this　makes　progress　towards　the　first　part　of　the　question　above.　In　addition,　for　odd　m,　we　are　able　to　obtain　the　value　of　r(B-n,C-m)　for　n/4　＜=　m　＜=　9/n/10　and　m　being　large.